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Journal of Mathematical Sciences, Vol. xxx, No. y, 2003

DIFFERENTIAL EQUATIONS IN BANACH SPACES I I. THEORY OF COSINE OPERATOR FUNCTIONS

V. V. Vasil'ev and S. I. Piskarev

UDC 517.986.7; 517.983.6 Dedicated to Vasil'eva Aleksandra Vladimirovna and Piskareva Lidiya Ivanovna, our mothers.

INTRODUCTION More than 13 years have passed since the fundamental survey [16] was prepared, which, as the author intended, should b e the first part of a large work devoted to abstract differential equations and methods for solving them. However, the troubles b eing in the Russian science during the whole this p eriod have influenced also on the authors, and instead of two years supp osed, the preparation of the second part has occupied considerably more time. During the last 10 years, the work in the field of differential equations in abstract spaces was very active (in foreign countries), and every year several b ooks and a heavy numb er of pap ers devoted to this direction app ear in the world (of course, the most of them are not available for the Russian reader). At the same time, only two b ooks [33, 75] of such a typ e app eared b eing translated by the authors of the present survey and [20], which were edited by Yu. A. Daletskii. Therefore, the work whose second part is prop osed to the reader will b e undoubtedly useful for the Russian reader. Its style coincides with that of [16], i.e., the material is often presented without proofs, and the main attention is paid to the structure of presentation, although we present certain proofs from foreign sources that are almost inaccessible for Russian readers. From our viewp oint, this allows us to demonstrate clearly the philosophy, to describ e the results obtained, and to indicate the main directions of the development of the theory in the framework of a limited volume of the survey.
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 113, Functional Analysis, 2002. 1072­3374/03/xxxy­0059 $ 27.00 c 2003 Plenum Publishing Corporation 59


Moreover, the authors have prepared a separate edition of the bibliographical index [18], which can serve as a sufficiently complete source of information ab out the theory of differential equations in abstract spaces during the recent years. The main ob ject of the study in this part are second-order differential equations that are presented very little in Russian literature up to now. Here, we can only mention the pap er [20] written in accordance with the own interests of the author, which does not pretend to the exhausting description of all asp ects of the theory. Moreover, the material of the present survey includes the presentation of the abstract Cauchy problem for first- and second-order equations that is not considered in [17]. As was already mentioned in [16], the philosophy of the theory of C0 -cosine op erator functions is very close to the op erator semigroup theory and often is develop ed in parallel to it. Therefore, the reader easily draws analogies b etween the material presented here and that presented in [16]. At the same time, the theory of C0 -cosine op erator functions considerably differs from the op erator C0 -semigroup theory. First of all, these distinctions concern with the prop erties inherent to the corresp onding parab olic and hyp erb olic partial differential equations. We now present the main notation, a certain part of which was already introduced in [17] and which is also used here. The set of natural numb ers is denoted by N, N0 := N {0}, the set of integers by Z, the set of reals by R, and the set of complex numb ers by C. A tuple of numb ers 1, 2, ..., m, m N, is denoted by 1,m, the real semiaxis (0, ) by R+ , and [0, ) by R+ . We denote by E a Banach space over the field of complex numb ers with the norm · . For a Hilb ert space with the inner product (·, ·), we use the symb ol H . The b oundary of a set is denoted by , the interior of the set by int(), the closure in the strong top ology by , and, for example, the closure in the weak top ology by w-cl-(). As usual, the space dual to E is denoted by E , with elements x , y , ..., and the value of a functional x E at an element x E is written as x, x . The domain and range of an op erator A will b e written as D(A) and R(A), resp ectively, and the null-space (kernel) as N (A). The set of linear op erators acting from D(A) E into E is denoted by L(E ), and the set of linear continuous op erators by B (E ). Closed linear op erators with dense domain (D(A) = E ) in E are distinguished into the set C (E ) L(E ). In the case where op erators act from one space E into another F , we write L(E, F ) and B (E, F ), resp ectively. The linear variety D(A) endowed with the norm x denote it by D(A).
A

:= x + Ax in the case of a closed linear op erator b ecomes a Banach space; we

60


We use the traditional notation for the resolvent set (A) and sp ectrum (A) of an op erator A; as usual, the latter is divided into the p oint sp ectrum P (A), the continuous sp ectrum C (A), and the residual sp ectrum R (A). Sections 2.2, 2.4, 6.1, 6.3, 7.1­7.2, 9.2, 10.2, 10.3, 12.1­12.4, 12.6­12.10, 13.3­13.6 and Chapter 14 were written by S. I. Piskarev, the other part of the text was prepared by the authors in collab oration.

Chapter 1
CAUCHY PROBLEM AND RESOLVING FAMILIES Before considering the theory of C0 -cosine op erator functions, we describ e the general picture of the statement of the well-p osed Cauchy problem in a Banach space. As is easily noted, a natural generalization of the concept of solution leads to more general families: integrated semigroups and C -semigroups. These families will b e considered in a forthcoming survey in more detail. 1.1. Cauchy Problem for a Complete Differential Equation Let E b e a Banach space, and let A0 ,A1 , ..., Am-1 b e closed linear op erators on E , i.e., Ak C (E ),k 0,m - 1. In the Banach space E , let us consider the following abstract Cauchy problem of order m: m-1 (m) u (t) = Ak u(k) (t), t R+ , (1.1) k =0 (k) u (0) = u0 , k 0,m - 1, m 2. k Definition 1.1.1. A function u(·) C m (R+ ; E ) is called a classical solution of problem (1.1) if u(k) (t) D(Ak ), Ak u(k) (·) C (R+ ; E ) for t R+ , k 0,m - 1, and relations (1.1) hold. As in [17], we define the propagators Pj (t), j = 0,m - 1, which give a solution of the Cauchy problem (1.1) with initial conditions uj (0) = jk u0 (jk is the Kronecker symb ol), i.e., uj (t) = Pj (t)u0 . j j Definition 1.1.2. The Cauchy problem (1.1) is said to b e uniformly wel l-posed if Pk (·)x C k R+ ; E ,
(k -1) Pm-1 (k )



(t)x D(Ak ),t R+ , and

(k -1) Ak Pm-1

x C (R+ ; E ) for any x E and k 0,m - 1.

In the general case, in the Banach space E , problem (1.1) has b een studied incompletely. In particular cases, some concepts are introduced, which will b e considered in the next chapters in more detail. Definition 1.1.3. We say that an op erator A generates times integrated semigroup with 0 if (, ) (A) for a certain R and there exists a strongly continuous function S (·) : [0, ) B (E ) such that S (t) Met , t R+ , with a certain constant M 0 and (I - A)-1 =
0

e-t S (t) dt 61


for all > max{, 0}. The family S (·) itself is called an times integrated semigroup. Theorem 1.1.1 ([118]). Let m 2, and let Am-1 generate an r times integrated semigroup. Assume
m that D(Ai ) D(Am-1 ) for al l i 0,m - 2, and, moreover, Ai D(Am-1 ) D(Ai--1+r m m-1 j =0 +2

) for i m -

r - 1. Then problem (1.1) has a unique exponential ly bounded solution for u0 -1 D(Ar+11 ), u0 m m- k D(Ar -1 Ai ), k 0,m - 2, and for certain constants c, > 0, we have the fol lowing estimate for m

t R+ :
r +1 m-2 m-2 r m-2

u(t) + Am-1 u(t) cet
l=0

Al -1 u0 m n

-1

+
k =0 i=0 l=0

Al -1 Ai u0 + m k
k =0

( u0 + Am-1 u0 ) . k k

In [193], this theorem was slightly changed by extending the set of initial data and by the absence of the exp onential b oundedness.
m

Denote P () :=
i=0

i Ai with the domain D(P ) :=
(k )

m-1 i=0

D(Ai ).

Theorem 1.1.2 ([194]). The propagators Pk (·), k 0,m - 1, are norm-continuous for t R+ iff there exists 0 R+ such that lim (0 + i )m-1 P
-1 -1

|| ||

(0 + i ) = 0, k 0,m - 1.

(1.2) (1.3)

lim

(0 + i )k

P

-1

(0 + i )Ak = 0,

Corollary 1.1.1. Let conditions (1.2)­(1.3) hold. Then for each k 0,m - 1, the operator Pk (t) is norm-continuous for t R+ . The case of time-dep endent Ak = Ak (t), t R+ , was considered, e.g., in [227, 228]. In [295], the conditions for the existence of a unique entire solution of problem (1.1) were presented. Consider problem (1.1) with Ak C (E ) for all k 0,m - 1; let D(A0 ) D(Ak ) for k 1,m - 1. Theorem 1.1.3 ([226]). Under the assumptions described above, the fol lowing conditions are equivalent for the Cauchy problem (1.1): (i) the operator A0 generates a C0 -semigroup; (ii) for any u0 ,u1 , ..., um-1 D(A0 ), the Cauchy problem (1.1) has a unique solution u(·) C
(m-1)

(R+ , D(A0 )).

The following theorem on the uniform stability of problem (1.1) holds. 62


Theorem 1.1.4. Let an operator A0 generate a C0 -semigroup, and let u(·) be a solution of the Cauchy problem (1.1) with initial conditions uk (k 1,m - 1; l ), uk D(A) for k 1,m - 1, uk 0 in E . l l l Then ul (·) 0 uniformly on any compact set. Theorem 1.1.5 ([226]). Let A0 generate a C0 -semigroup, Ak C (E ), D(A) D(Ak ), and let be such that for Re > , there exists a generalized resolvent (pencil resolvent) R := R(; A0 , ..., Am-1 ) = (m I - m-1 Am-1 - ... - A1 - A0 )-1 (such an always exists!). Also, on D(A), let the relation Ak R = R Ak (k 1,m - 1; Re > ) hold. Then problem (1.1) is uniformly wel l-posed, and its solution has the form
m-1

u(t) =
k =0

Qm-1-k,
m-1

m-1

(t)uk ,

(1.4)

where the operator-valued functions Qm-1-k, ator semigroup

(t) are strongly continuous families composing the oper Q0,m-1 (t) . . . Qm-1,m-1 (t) I 0 0 I ... 0 , t R+ ,

Q (t) ... 0,0 . .. . G(t) = . . Qm,0 (t) ... with the generator = A0 A1 . . . . . .

... .. .. . . 0 . .. .. . . . . 0 ...

Am-1 0

0 0 . I 0

Now let us consider the Cauchy problem for the following equation of order m having the sp ecial form:
m j =1

d - Aj u(t) dt

d - Am ... dt

d - A1 u(t) = 0 dt

(1.5)

with initial conditions u(k) (0) = u0 , k and op erators Aj C (E ),j 1,m. Definition 1.1.4. The Cauchy problem (1.5) is said to b e uniformly wel l-posed if the following conditions hold: 63 k 0,m - 1, m 2,


(i) there exists a solution of the Cauchy problem (1.5) for u0 , ..., um-1 taken from a certain dense set D in E ; (ii) for u0 , ..., um-1 D, the solution of the Cauchy problem (1.5) has the prop erty
k j =1

d - Aj u(t) C dt

m-k

(R+ ,E )

(1.6)

for k 1,m - 1; (iii) the uniform stability of the solution of (1.5) is complemented by the following condition on any compact set: the convergence
k j =1

d - Aj up (0) 0 dt

implies the convergence
k j =1

d - Aj up (t) 0 dt

uniformly on each compact set in R+ (here, k 1,m - 1; p ). Theorem 1.1.6 ([20]). In the Cauchy problem (1.5), let Aj C (E ) (j 1,m), let the intersection of the resolvent sets (Aj ) of the operators Aj be nonempty, and let the set ~ D {D(Ai1 ...Aim ) : ik 1,m} (1.7)

be dense in E . Then problem (1.5) is uniformly wel l-posed iff Aj generates a C0 -semigroup for each j 1,m. Theorem 1.1.7 ([20]). Under the conditions of Theorem 1.1.6, let the operators Aj generate C0 semigroups for j 1,m, and, moreover, let these semigroups commute: exp(tAi )exp(sAj ) = exp(sAj )exp(tAi ), Then problem (1.5) is wel l posed. Theorem 1.1.8. For the Cauchy problem (1.5), let the conditions of Theorem 1.1.7 hold, and let 0 m d (Ai - Aj ) for al l i = j . Then the condition w(t) N for t R implies the relation - Ai i=1 dt m d - Ai , t R. wi (t), where wi (t) N w(t) = dt i=1 Condition (1.8) in Theorem 1.1.7 can b e replaced by a numb er of conditions imp osed on the domains R(Ai - Aj ) for i = j . Let 1,1 ,2 , ..., p 64
-1

t, s R ,

i, j 1,m.

(1.8)

be roots of pth degree of the unity, i.e., k = e

2k p

i

.


Definition 1.1.5. A C0 -function of the Mittag-Leffler type with a parameter p is a function M : C B (E ) having the following prop erties:
p-1

(i)
k,l=0

M(k t + l s) = p2 M(t)M(s) for any t, s R;

(ii) M(0) = I ; (iii) the family of op erators T (t) M(k t + l s), k, l 0,p - 1, with a fixed s R is strongly continuous in t R. For the Mittag-Leffler C0 -function with a parameter p, the p-generator A is defined by the relation Ax = s- lim p!
t0

M(t) - I x tp

for those x at which the limit exists. It is known that the generator of the Mittag-Leffler C0 -function with a parameter p is a linear closed densely defined op erator, and the following relation holds for any x D(A): dp M(t)x = AM(t)x = M(t)Ax, dtp (0) = 0 for k 0,p - 1 .

and, moreover, M(k

)

For the Mittag-Leffler C0 -function with a parameter p, the p erturbation theorems of the Philips­ Miyadera typ e hold (see [17]). Theorem 1.1.9 ([32]). Let A generate a Mittag-Leffler C0 -function with a parameter p, and let M(t) Met , t R. Then for any B B (E ), the operator A + B generates a Mittag-Leffler C0 -function with the parameter p. Proposition 1.1.1. The Mittag-Leffler C0 -function with a parameter p is a C0 -group of operators in the case of p = 1, and in the case of p = 2, it is a C0 -cosine operator-valued function. A Mittag-Leffler C0 -function with a parameter p has a bounded generator A B (E ) for p 3. In the simplest case m = 2, for example, the following theorems hold for problem (1.1). Theorem 1.1.10 ([226]). Let the Cauchy problem (1.1) be uniformly wel l-posed for m = 2, and let P1 (t)E D(A1 ) for t R+ . Then A0 generates a C0 -cosine operator-valued function on E . Theorem 1.1.11 ([226]). Let A1 B (E ). Then the Cauchy problem (1.1) is uniformly wel l posed for m = 2 iff A0 generates a C0 -cosine operator-valued function on E . However, in the general case, even for m = 2, the Cauchy problem (1.1) turns out to b e very complicated. First, in [135], H. O. Fattorini has presented an example of the Cauchy problem (1.1) that has a solution for m = 2, but this solution is not exp onentially b ounded. Second, in contrast to the 65


Cauchy problem for m = 1, the case m = 2 admits more flexibility in the sense of well-p osedness of its statement. As one of the variants, let us present the approach coming back to H. O. Fattorini. The constructions used in proving these theorems practically completely rep eat the techniques used in proving the assertions concerning C0 -cosine and C0 -sine op erator-valued functions (see also [30]). Consider the Cauchy problem u (t)+ Bu (t)+ Au(t) = 0, with A, B C (E ). Definition 1.1.6. We say that the op erators A and B generate M, N -families of operators on E if the following conditions hold: (i) M (t)and BN (t) are strongly continuous in t R+ , and the function N (t)x is strongly continuously differentiable in t R+ for any x E ; ^ (ii) the set E = {x E : M (t)x is strongly differentiable in t R+ , and BM (t)x is continuous in t R+ } is dense in E ; ^ (iii) the op erator A = -M (0) is B -closed, and Bx = -N (0)x for all x E ; (iv) M (0) = N (0) = I and N (0) = 0; ^ (v) M (t + s)x = M (t)M (s)x + N (t)M (s)x for all x E and t, s R+ ; (vi) N (t + s) = M (t)N (s)+ N (t)N (s) for all t, s R+ . Theorem 1.1.12 ([30]). Let A and B generate M, N -families. Then ^ (i) A is closed, D(A) D(B ) E D(B ), and D(A) D(B ) is dense in E ; (ii) the families M and N are uniquely defined by the operators A and B ; (iii) M (0)x = 0 for al l x D(M (0)); (iv) M (t)x = -N (t)Ax for al l x D(A) and t R+ ; ^ (v) N (t)x = M (t)x - N (t)Bx for al l x E and t R+ ; (vi) N (t)x + N (t)Bx + N (t)Ax = 0 for al l x D(A) D(B ) and t R+ ; ^ ^ (vii) for al l x E and t R+ , the element N (t)x E , N (t)x D(A), and N (t)x - x + BN (t)x + AN (t)x = 0; (viii) for al l x E and t R+ , the element A
t 0 t 0

t R+ ,

u(0) = u0 ,

u (0) = u1 ,

(1.9)

N (s)xds D(A) and N (t)x - x + BN (t)x +

N (s)xds = 0; (ix) for al l x E and t R+ , the element
t 0

^ N (s)xds E , M (t)x D(A) and M (t)x + BM (t)x +

AM (t)x = 0; 66


(x) for al l x D(A)
t

^ ^ E and t R+ , the element M (t)x - x E ,

t 0

M (s)xds D(A), and

M x + B (M - I )x + A M (s)xds = 0; (xi) there exist constants C, 0 such that M (t) , N (t) , BN (t) , N (t) Cet , ^ and for al l x E , there exist constants C, 0 such that M (t)x , BM (t)x C (x)et , (xii) the operator 2 + B + A is closable for al l C; (xiii) there exists a constant R+ such that (A, B ) for al l with Re > and ()x := (2 I + B + A)-1 x =
0 0

t R+ ,

t R+ ;

e-t N (t)xdt for

for

x E;

()(I + B )x =
0

e-t M (t)xdt

^ x E;

(xiv) 2 ()x x as for al l x E . The following analog of Theorem 2.1.1 from [17] holds. Theorem 1.1.13 ([294]). Operators A and B generate M, N -families iff the fol lowing conditions hold: (i) the operators A and B are closed, and D(A) D(B ) is dense in E ; (ii) there exist constant C, 0 such that (A, B ), and for Re > , the operator ()A is closable and (())(k) , (B ())(k) , (()B )(k
)



Ck! (Re - )k

+1

for

k N,

Re > ,

(1.10)

where ()B is a bounded extension of the operator ()B with the domain D(A) D(B ) and (·)(k) is the derivative of order k in . In the case where A and B commute, instead of the estimate with the op erator B in (1.10), it can b e, e.g., dk (I - A)() dk (see [30]). If A = 0, then A and B generate M, N -families iff B generates a C0 -semigroup. Let D(B ) D(A), and let (B ) = . If (0 I - B )-1 A has a b ounded extension for a certain p oint 0 (B ), then A and B generate M, N -families iff B generates a C0 -semigroup. 67 Mk! (Re - )k
+1

,kN

0


Proposition 1.1.2 ([294]). Let B be dominated by A with exponent 0 1, i.e., D(A) D(B ) and Bx C x
1-

Ax



for al l x D(A), and let A and B commute. If -A generates a C0 -cosine

operator-valued function and (2 I + A)-1 C ||-2 for Re > , then A and B generate M and N families. Now let us consider an analytic extension of a solution of Eq. (1.9) to the sector ( ) = {z C : z = 0, | arg z | < }. Theorem 1.1.14 ([294]). For given , 0, the fol lowing conditions are equivalent: (i) the Cauchy problem (1.9) is uniformly wel l posed, the families M, N can be analytical ly extended to the sector ( ) = {z C : z = 0, | arg z | < }, for any z we have the embedding N (z )E D(B ), and BN (·) is analytic in ( ). Moreover, for each (0, ), x E , lim N (z )x = 0, lim BN (z )x = 0, lim M (z )x = x, lim N (z )x = 0,
z z 0 z z 0 z z 0

z z 0

and there exists a constant C > 0 such that N (z ) , BN (z ) , M (z ) C e
Re z

for al l

z ( );

(ii) the set D(A) D(B ) is dense in E . For each (0, ), there exists M > 0 such that for ( , ) = C : = , | arg( - )| < + 2 ,

the operator () := (2 + B + A)-1 B (E ) exists, the operator ()A is closable, and () M , | - | B () M , | - | ()B0 M , | - |

where B0 B with D(B0 ) = D(A) D(B ). Moreover, in this case, we have N (z )+ BN (z )+ AN (z ) = 0, M (z )+ BM (z )+ AM (z ) = 0,

where AN (z ) and AM (z ) are analytic in ( ). For each (0, ), lim M (z )x = 0 for any x D(A).

z z 0

The existence and uniqueness of solutions of Eq. (1.9) under certain "hyp erb olic" conditions is considered in [230]. Problem (1.9) in the case of nonlinear B was considered in [188]. 68


1.2. Cauchy Problems for Differential Equations of the 1st and 2nd Orders In this section, we present certain statements of the Cauchy problem for equations of the first and second orders. Equations of the first order were already considered in the pap er [17] but, however, only in connection with C0 -semigroups on the space E . As was already noted, different statements of the Cauchy problem are p ossible. We now present certain arguments that show that a solution is given not by C0 -families on the whole space E . Definition 1.2.1. An integrated solution of the Cauchy problem u (t) = Au(t), u(0) = x, (1.11)

is a continuously differentiable function v (·) : R+ E such that dv (t) = Av (t)+ x, v (0) = 0. (i) v (·) C ([0, ); D(A)) and (ii) dt Definition 1.2.2. Denote by Z (A) the resolving set of the op erator A, i.e., the set of all x E for which the Cauchy problem (1.11) has an integrated solution. Proposition 1.2.1. Let Z (A) be the resolving subspace endowed with the family of seminorms x
a,b

= sup u(t) ,
t[a,b]

a, b R+ .

(1.12)

Then Z (A) is a Frech´ space and T (t)x = u(t) is a local ly equicontinuous semigroup generated by et the operator A|Z
(A)

.

Recall the definition of entire vectors, which is equivalent to [17, Definition 3.1.3]. Definition 1.2.3. Denote by Uc (A) the set of entire vectors of an op erator A, i.e., the set of x D(A ) such that for any t R+ ,


Ak x
k =0

tk < . k!

Proposition 1.2.2 ([117]). Any linear closed operator on E with the resolvent set (A) containing the semiaxis (, ) generates a C0 -semigroup on a certain maximal subspace in E . Proposition 1.2.3 ([117]). For any closed linear operator A, there exists a maximal Frech´ space Z (A) et such that Z (A) E and the Cauchy problem (1.11) is automatical ly wel l posed on Z (A). As is seen from this prop osition, the requirement of existence of a C0 -semigroup is very restrictive. At the same time, namely for C0 -families of op erators, the technical tools for studying approximate methods are most well elab orated. 69


Theorem 1.2.1 ([117]). Let A C (E ). Then Uc (A) = {x E : problem (1.11) has an entire solution} and Uc (A) Z (A). Theorem 1.2.2 ([85]). Let A generate an analytic C0 -semigroup on E . Then Uc (A) = Z (A), and, moreover, the equality holds topological ly and algebraical ly. As is known, a self-adjoint op erator A = A 0 on a Hilb ert space H generates an analytic C0 semigroup, as well as a C0 -cosine op erator function. Moreover, in this case, by the Stone theorem, the op erator iA generates an unitary C0 -group on H . At the same time, the practical problems often require the omitting of the self-adjointness of the op erator and Hilb ert prop erty of the initial space. Therefore, to reveal whether a concrete op erator generates a C0 -semigroup or not is not a simple but often a complicated indep endent problem. Here, we present examples showing when the verification of generation of C0 -families on the Banach space E is p ossible. Theorem 1.2.3 ([43]). For A C (E ) to be a generator of an analytic C0 -semigroup, it is necessary and sufficient that there exist numbers , , and > 1 such that the fol lowing inequality holds for al l M || ; (Re ) +-1 (Re - ) moreover, the fol lowing representation holds for this semigroup: ( I - A)-1 exp(zA) = - for z z C : Im z < Re z cot 2 . 2i
+i -i

Re > 0 :

ezµ µ-1 (µ I - A)-1 dµ


Let Rd b e a certain domain. Denote by C() the space of uniformly continuous b ounded functions on with the norm v (·)
C()

= sup |v (x)|,
x C ()

and let C () = {v (·) : (·)v (·) C(), (t) 0}, v (·)

= (·)v (·)

C ()

.

Example 1.2.1 ([43]). Let = [0, 1], and let Av = v (·) with D(A) = {v (·) : v C (),Av in C ([0, 1]). C (),v (0) = v (1) = 0}. Then A H , 2 At the same time, for the op erator A0 v = v (·) with D(A0 ) = {v (·) : v C ([0, 1]),v(0) = v (1) = 0}, we have A0 H(0, ) on C0 ([0, 1]) = {v (·) : v C ([0, 1]),v(0) = v (1) = 0}. Finally, the op erator A v = v with D(A ) = {v (·) : (·)v (·) C ([0, 1]), A v C ([0, 1])} generates an analytic C0 -semigroup with the estimate exp(tA ) 70
C ([0,1])

e- t ,
2

t R+ .


d

Example 1.2.2 ([43, 268]). The Laplace op erator v =
j =1

generator of an analytic C0 -semigroup on E = W

2,p

(Rd ).

2 v (x) , x Rd , for 1 < p < gives a x2 j

Here, it is appropriate to recall (see [167]) that the op erator i does not generate a C0 -semigroup on Lp (Rd ) for p = 2. Moreover, the op erator generates a C0 -cosine op erator function iff p = 2 or d = 1. Also, we note (see [126]) that the op erator -(i)1/2 does not generate a C0 -semigroup on L1 (R1 ). Example 1.2.3 ([126]). The Laplace op erator on Lp (Rd ), 1 p < , generates an times integrated 11 cosine op erator function for > (d - 1) - . 2p ~ Example 1.2.4 ([125]). Let A b e a strongly elliptic op erator on Rd . Denote by Tr (·) the C0 ~ semigroup generated by the op erators A with the Dirichlet or Neumann conditions on the b oundary in Lr (). Then there exists an analytic C0 -semigroup Tp (·) with the angle /2 in Lp () such that Tp (t)x = Tr (t)x for all x Lp () Lr (). ~ Example 1.2.5 ([126]). Let 1 < p < , let the op erator Ap generate a semigroup Tp (·), and let µ() < ~ . Then Ap generates an times integrated cosine op erator function on Lp (Rd ) for > d1 1 1 - +. 22 p 2

Example 1.2.6 ([35]). Let = R+ . The op erator (Av )(x) = v (x) +

a c v (x) + v (x) generates an x x analytic C0 -semigroup for a, c R and D(A) = {v (·) : v C(R+ ),Av C (R+ )} iff c 0. dv(x) d (1 - x2 ) dx dx with D(A) =

Example 1.2.7 ([43]). Let = [-1, 1]. Then the op erator (Av )(x) = {v (·) : v C(), Av C()} generates a C0 -semigroup.

Example 1.2.8 ([43]). Let (Av)(x) = v (x) + q (x)v (x), x R. Denote by Sp the Banach space of Stepanov functions, i.e., the space of functions on R with the norm v (·)
p,l
1 p

1 = su p l xR

x+l x

|f (s)| ds
p

, l > 0, p 1.

It is known that for different l, the norms are equivalent. For the op erator A H(, ) on C (R), it suffices, and in the case q (x) c > -, it is necessary that q (·) S1 . Denote H
-1 0 -1

2 (z ) =

e-s

2

-2sz

ds and 1 =- 2i
+i

H (t)(µ I - A)
2

H
-i

-1

( )et (2 I - A)-1 d.

71


Theorem 1.2.4 ([35]). For A C (M, ), it is necessary and sufficient that this operator be the generator of an analytic C0 -semigroup, and for each t [0,T ], the estimate H (t)(µ2 I - A)-1 M (t) hold uniformly in (0,), > 0. In this case, C (t, A) = s- lim (H (t)+ H (-t))(µ2 I - A)-1 ,
0

(1.13)

t R+ .

Example 1.2.9 ([35]). Let the op erator A f (x) = we have H (1)(2 I - A)-1 f >

b e given as in Example 1.2.6. Then for the function 0 x-1 2 1 if x [0, 1), if x [1, 1+ 2 ), if x [1+2, )

M , and, therefore, by Theorem 1.2.4, such an op erator A does not generate a C0 -cosine op erator function.


Example 1.2.10 ([43]). Consider the op erator A from Example 1.2.6 but on the space C (R+ ) with (x) = xex , x R+ , R. Then H (t)(2 I - A)-1 v
C

Me|

µt|

v

C

, and, therefore, A C (M, ).

Example 1.2.11 ([43]). Let A b e given as in Example 1.2.8. For A C (M, ) on the space C (R) it is sufficient, and in the case q (x) c > -, it is necessary that q (·) S1 . Consider the problem 2 u(t, x) 2 u(t, x) u(t, x) = xm + xm-1 2 2 t x x where m > 0, x > 0, and initial conditions lim u(t, x) = (x), lim , C
t0 (2) t0

(1.14)

u(t, x) = (x) for any x R+ , where t (R+ ) E , and E is the Banach space of functions C (R+ ) such that lim (x) = lim (x) =
x0 x xR+

0 with the norm = sup |(x)|.

Definition 1.2.4. Problem (1.14) is said to b e uniformly wel l-posed if for any compact set J R+ , we have max |u(t, x)| M ( + ).
tJ

Example 1.2.12 ([43]). For the op erator (Av )(x) = xm v (x)+ xm-1 v (x) with D(A) = {v E : v 2 - (1 + )m < 1. C (2) (R+ ) E, Av E } on the space E just describ ed, condition (1.13) holds for 0 2-m m 2 - (1 + ) For 1, the op erator A does not generate a C0 -cosine op erator function. 2-m 72


2 d 2k 2 defined on the space C (R ), the op erator i=1 xi on C (Rd ) generates a C0 -cosine op erator function iff d 4k +1.
d

Example 1.2.13 ([43]). For the op erator =

+1

Here, in connection with Example 1.2.13, it is relevant to note that for any A C (M, ), every polynomial P (A) = A2m+1 +
2m k =0

ck Ak , ck R, generates an analytic semigroup.

Moreover, in the case of an even m, the op erator (-1)m+1 Am does not necessarily generate a C0 cosine op erator function. Proposition 1.2.4 ([177]). Let A C (M, 0). Then for any k N, the operator (-1)k A2 generates an times integrated cosine operator function for a certain > 0. Moreover, (-1)k A2 H(, /2). Theorem 1.2.5 ([231]). Let A H 0, (-1)m+1 Am + B1 Am-1 + ... + Bm-1 , and let m N. Let Bi B (E ), i 1,m. Then the operator 2 A + Bm generates an analytic C0 -semigroup with the angle . 2
k k

An analogous assertion is not true for C0 -cosine op erator functions! Theorem 1.2.6 ([178]). Let {Aj }m be resolvent commuting operators, and let Aj C (M, 0), j 1,m, j =1
m m

be given on E . Define A0 =
j =1

Aj , D(A0 ) =
j =1

D(Aj ). Then the operator A0 is closable and A0

m-1 . 2 Moreover, this times integrated semigroup satisfies the estimate S (t) M t , t R+ for certain m-1 . M > 0 and 2 generates an times integrated cosine operator function for Theorem 1.2.7 ([177]). Under the conditions and notation of Theorem 1.2.6, the operator iA0 generates a times integrated semigroup for > m/2. Proposition 1.2.5 ([178]). Under the conditions of Theorem 1.2.6 and an additional assumption that the space E = H is a Hilbert space, A0 generates a C0 -cosine operator function. Proposition 1.2.6 ([178]). Let the conditions of Theorem 1.2.6 hold and, additional ly, let E = H be a Hilbert space and a Banach lattice, and, moreover, let C (t, Aj )H+ H+ , t R, j 1,m. Define Ck (t) =
t 0

C (t, A ) 0

(t - s)k-1 C (s, A0 ) ds for k 1, (k - 1)! for k = 0.

Then Ck (·) are positive for k

m . 2 73


Example 1.2.14 ([162]). Let E = Lp (Rd ), 1 < p < . Then the Laplace op erator with D() = 11 W 2,p (Rd ) generates an times integrated cosine op erator function iff (d - 1) - . 2p In [296], concrete differential op erators are studied for revealing whether or not they generate a well-p osed Cauchy problem for a complete second-order equation. 1.3. Resolvent Families For functions k(·) Lp (R+ ) and g(·) W loc
t 1,1

([0,T ]; E ), let us consider the Volterra equation t [0,T ]. (1.15)

u(t) = g(t)+
0

k(t - s)Au(s) ds,

Definition 1.3.1. A strongly continuous family of b ounded linear op erators {R(t) : t R+ } on E is called a resolvent family for (1.15) if it commutes with the op erator A and
t

R(t)x = x +
0

k(t - s)AR(s)xds

for x D(A), t R+ .

If there exists a resolvent family, then any solution of Eq. (1.15) is represented in the form
t

u(t) = R(t)g(0) +
0

k(t - s)g (s) ds,

t [0,T ].

(1.16)

Theorem 1.3.1 ([114, 150, 244]). Let R(·) be a strongly continuous family of operators on R+ such that R(t) Met and |k(t)| Met , t R+ . Then R(·) is a resolvent family iff the fol lowing conditions hold: ^ (i) k () = 0 and (ii) ^ (I - k ()A)-1 x =
0

1 (A) for al l ; ^ k()


et R(t)xdt

^ for al l x E and > , where k(·) is the Laplace transform of the function k(·). In particular, it should b e noted that for k(t) 1, the resolvent family is a C0 -semigroup of op erators, and for k(t) = t, it is a C0 -cosine op erator function. Therefore, the proof of a numb er of assertions on prop erties related to C0 -semigroups and C0 -cosine op erator functions can b e obtained from assertions related to resolvent families. For the kernel k(·) satisfying certain restrictions (p ositivity and b ounded variation) for a resolvent family, many results that hold for C0 -semigroups and C0 -cosine op erator functions were reproved. So, for example, in [196­198], C. Lizama has reproved the assertions on the compactness prop erties, uniform continuity, and p eriodicity. In [170], Jung Chan Chang and S.-Y. Shaw have reproved the theorems on multiplicative and additive p erturbations. 74


1.4. Incomplete Cauchy Problem For a second-order equation, let us consider the so-called incomplete Cauchy problems u (t) = Au(t), u(0) = u0 ,
tR+

t R+ , (1.17)

sup u(t) < ; t R+ ,

u (t) = Au(t),
t0+

lim u(t) = u0 ,

(1.18)
t

lim u(t) = 0; t R+ , (1.19)

u (t) = Au(t), u(0) = u0 ,
t

lim u(t) = 0.

Incomplete Cauchy problems were studied in [117, 137, 140]. Proposition 1.4.1. An operator A has the square root continued to a certain sector containing the semiaxis R+ . Proposition 1.4.2. Let there exist A generating a differentiable C0 -semigroup such that s lim exp(t A) = 0. Then problem (1.18) has a unique solution for any u0 E . A generate a C0 -semigroup such that s- lim exp(t A) = 0. Then problem u0 D(A).
t

A such that exp(t A) is a bounded analytic

C0 -semigroup iff problem (1.17) has a unique solution for each u0 D(A) and this solution is analytical ly

t

Proposition 1.4.3. Let

(1.19) has a unique solution for each

Definition 1.4.1. A C0 -semigroup exp(·A) is called a C0 -semigroup stable in degree q N if st

lim exp(tA)x = 0 for each x D(Aq ). A C0 -semigroup stable in degree 0 is said to b e uniformly

stable. Theorem 1.4.1. Assume that an operator B generates a C0 -semigroup stable in degree 2 and a function v (·) has a continuous second derivative and satisfies the equation v (t) = B2 v (t), t R+ ,

and, moreover, s- lim v (t) = 0. Then v (t) = exp(tB)v (0), t R+ .
t

Theorem 1.4.2. Let A = B2 , where the operator B generates a C0 -semigroup. Then (i) if exp(·B) is stable in degree 2, then problem (1.19) is wel l posed; 75


(ii) if exp(·B) is stable in degree 1, then the fol lowing problem is wel l posed: u (t) = Au(t),
t

t R+ ,
t

u(0) = x, (1.20)

lim u(t) = 0,

lim u (t) = 0;

(iii) if exp(·B) is uniformly stable, then the fol lowing problem is wel l posed: u (t) = Au(t),
t

t R+ ,

u(0) = x, k N0 . (1.21)

lim u(k) (t) = 0,

Proposition 1.4.4. Let (A) = . Then (i) problem (1.19) is wel l posed iff the operator A has the square root stable in degree 2; (ii) problem (1.20) is wel l posed iff the operator A has the square root stable in degree 1;

A generating a C0 -semigroup A generating a C0 -semigroup A generating a stable C0 -

(iii) problem (1.21) is wel l posed iff the operator A has the square root semigroup. Corollary 1.4.1. Any operator A has not more than one square root stable in degree 2.

A generating a C0 -semigroup

Theorem 1.4.3. Let B and C be self-adjoint commuting operators on a Hilbert space H . Then there exist closed complementable subspaces H1 and H2 such that if A = B + iC , then problems (1.19), (1.20), and (1.21) are wel l posed on H1 and the problem u (t) = Au(t), u(0) = u0 , is wel l posed on H2 . Definition 1.4.2. The regularized fractional derivative of order 0 < < 1 of a function u(·) is the function (D where
(D0+ u)(t) = () u)(t) = (D0+ u)(t) -

t R+ , (1.22)

u (0) = u1 .

1 u(0) , (1 - ) t
t 0

1 d (1 - ) dt

u( ) d . (t - )

Consider the Cauchy problem (D 76
()

u)(t) = Au(t),

0 < t T, u(0) = u0 ,

(1.23)


with a closed op erator A. By a solution of problem (1.23) we mean a function u(·) such that (i) u(·) C ([0,T ]; E ); (ii) for t R+ , the values u(t) D(A); t u( ) 1 d is continuously differentiable for t 0, and (iii) the fractional integral (1 - ) 0 (t - ) (iv) the function u(·) satisfies (1.23). Theorem 1.4.4 ([44]). Let there exist the resolvent ( I - A)-1 for > > 0, and let


lim -1/ ln (I - A)-1 = 0.

Then a solution of problem (1.23) is unique. Theorem 1.4.5 ([44]). Let the resolvent ( I - A)-1 exist in the half-plane Re > > 0, and for the same , let ( I - A)-1 C (1 + | Im |)- , 0 < < 1.

Then problem (1.23) has a unique solution. This solution is infinitely differentiable for t > 0, and for each t, its value continuously depends on the initial data u0 .

Chapter 2
COSINE AND SINE OPERATOR FUNCTIONS The existing parallelism b etween the theory of C0 -semigroups of op erators and the theory of C0 -cosine op erator functions has a distinctive character. On one hand, a numb er of definitions and prop erties practically rep eat each other almost literally. On the other hand, for second order equations, by the Kisynski theorem, the main ob ject corresp onding to a C0 -cosine op erator function is a C0 -group, which excludes the app earance of "parab olicity," despite the fact that the generator A of a C0 -cosine op erator function also generates an analytic C0 -semigroup. 2.1. Measurability of Operator Semigroups and Cosine Operator Functions The measurability prop erty of a cosine op erator function profitably differs from that of a semigroup. By the evenness, the measurability of a cosine op erator function implies the strong continuity at zero. We have an analogous situation for p erturbation families. Definition 2.1.1. A function T (·) : R+ B (E ) is called an operator semigroup if it satisfies the conditions T (t + h) = T (t)T (h) for any t, h R+ and T (0) = I . 77


Definition 2.1.2. A function C (·) : R B (E ) is called an operator cosine (or a cosine operator function) if it satisfies the condition C (t + h)+ C (t - h) = 2C (t)C (h) for any t, h R and C (0) = I . Definition 2.1.3. A function S (·) : R B (E ) is called an operator sine (or a sine operator function) if it satisfies the condition S (t + h)+ S (t - h) = 2S (t)C (h) for any t, h R and S (0) = 0. Theorem 2.1.1 ([186]). Let an operator semigroup T (·) be strongly measurable, i.e., the function T (·)x is strongly measurable on R+ for any x E . Then it is strongly continuous on R+ . We stress that in Theorem 2.1.1, the strong continuity is asserted only on R+ but not on R+ ! Proposition 2.1.1 ([186]). Let a function t T (t)x be strongly measurable on R+ . Then it is local ly bounded. Proposition 2.1.2 ([186]). Let an operator cosine C (·) be strongly measurable on R+ . Then it is strongly continuous on R. Proposition 2.1.3 ([186]). Let a function t C (t)x be strongly measurable on R+ . Then it is local ly bounded. Theorem 2.1.2 ([185]). Let a cosine operator function C (·) be such that its restriction to a certain interval J R be weakly Lebesgue measurable, and let the space E be separable and reflexive. Then C (·) is weakly continuous on R. 2.2. Multiplicative and Additive Families. Measurability and Continuity Definition 2.2.1. Let C (·,A) b e a C0 -cosine op erator function. A family {F (t) : t R} of op erators in B (E ) is called a multiplicative perturbation family for C (·,A) if F (0) = 0 and F (t + s) - 2F (t)+ F (t - s) = 2C (t, A)F (s) for t, s R. (2.1)

Definition 2.2.2. A family {G(t) : t R} of op erators in B (E ) is called an additive perturbation family for a C0 -cosine op erator function C (·,A) if G(0) = 0 and G(t + s) - 2G(t)+ G(t - s) = 2G(s)C (t, A) for t, s R. (2.2)

If these families are strongly continuous at zero, then they are called a multiplicative perturbation C0 -family and additive perturbation C0 -family, resp ectively. Clearly, F (·) and G(·) are even functions. The terminology mentioned ab ove is chosen by analogy with the corresp onding definitions of p erturbation families U (·)and V (·)for C0 -semigroups ([17, Sec. 2.2]). 78


Recall that U (·) satisfies the relations U (0) = 0 and U (t + s) - U (t) = T (t)U (s), t, s R+ , and V (·) satisfies the relations V (0) = 0 and V (t + s) - V (t) = V (s)T (t), t, s R+ . The multiplicative and additive p erturbation C0 -families play an imp ortant role in the p erturbation theory of C0 -cosine op erator functions. For example, using the multiplicative and additive p erturbation C0 -families, we can consider wellp osed statements of the Cauchy problem in the form ^ ^ u (t) = A(1 - F ())u(t)+ 3 F ()u(t), t R+ , u(0) = x, u (0) = y.
+

As is known, a C0 -cosine op erator function that is strongly (resp. uniformly) measurable on R tive and additive p erturbation families have the same prop erties.

is

strongly (resp. uniformly) measurable on R (see Sec. 2.1). The following theorem shows that multiplica-

Theorem 2.2.1 ([239]). If a multiplicative perturbation family F (·) is strongly (resp. uniformly) measurable on R+ , then the function F (·) is strongly (resp. uniformly) continuous on R. If an additive perturbation family G(·) is uniformly measurable on R+ , then the function G(·) is uniformly continuous on R. Proof. First of all, the strong continuity of F (·)x on R+ implies the Leb esgue measurability of F (·)x on R
+

(see [76]). Further, let us show that F (·)x is b ounded on any compact subinterval [a, b] R

+

for any x E . Supp ose the contrary. Then there exist x E , a numb er > 0, and a sequence n [a, b] ~ such that n and x F (n )~ n as n .

By the measurability of F (·)~ , there exist a constant c1 and a Leb esgue measurable set [0, ] of x measure 3 µ() > 4 such that sup F (t)~ c1 . x
t

(2.3)

Now, following [102], we set Ak := and A= -, 2 2 B = [0, /2]. 79 k [0,k ] - , Bk := [0,k /2] 2 2 (2.4)


First, µ(AB ) > 0. To prove this, assume that µ(AB ) = 0. Then µ(A)+µ(B ) /2. But µ(A) = µ/2() 3 by the definition of the set A. This means that µ() + 2µ(B ) . Therefore, < µ() - 2µ(B ), i.e., 4 µ(B ) /8. Write = ( [0, /2]) ( [/2, ]) = B D, where µ() = µ(B )+ µ(D) with µ(D) /2. But 3 < µ() = µ(B )+ µ(D) µ(B )+ /2 4 implies µ(B ) > /4, which contradicts (2.5). We have proved that µ(A B ) > 0. Now define the sets E = A B , En = An Bn , and Hn = {n - , En }. Clearly, En E as n , so that µ(Hn ) > /2 for sufficiently large n. For the same n, if En , then and n - 2 b elong to by (2.4). Now, using (2.1) and (2.3), for En , we obtain x x x x n F (n )~ 2 F (n - )~ + F (n - 2)~ +2 C (n - ) F ()~ x 2 F (n - )~ + c1 +2Meb c1 . Therefore, F (t)~ x
n

(2.5)

n - c1 - 2Mc1 e 2


x for t Hn ; denoting lim Hn = H , we have F (t)~ = for t H contradicts the b oundedness of F (t)~ for each t. x

with µ(H ) /2 > 0. This

We now want to prove that the strong measurability, together with the b oundedness, implies the continuity of F (·)x for each t R
+

and each x E . For this purp ose, we choose four p ositive numb ers

, , , and such that < t - and 0 < < < < t. We have from (2.1) that F (t)x = 2F (t - /2)x - F (t - )x +2C (t - /2,A)F (/2)x. 80 (2.6)


The left-hand side, b eing indep endent of , is integrable in , and we have ( - )(F (t ± )x - F (t)x)


=


2 F (t ± - /2) - F (t - /2)x d -




F (t ± - ) - F (t - ) xd

+


2 C (t ± - /2,A) - C (t - /2,A) F (/2)xd .

Therefore, (F (t ± ) - F (t))x 1 - +2
t-/2 t-/2

(F ( ± ) - F ( ))x d +

t- t-

(F ( ± ) - F ( ))x d (2.7)

(C (t ± - /2,A) - C (t - /2,A))F (/2)x d .

By Theorem 3.8.3 from [76],
t-/2 t-/2

0 and

t- t-

0 as

0.

The last summand in (2.7) tends to zero by the Leb esgue theorem on the dominated convergence (see [76, Theorem 3.7.9]). We obtain that F (·)x is continuous for t R+ . Replacing t by t + s in (2.1), we obtain that for all t, s R+ , the function F (t)x = 2C (t + s, A)F (s)x - F (t +2s)x +2F (t + s)x tends to 2C (s, A)F (s)x - F (2s)x +2F (s)x = F (0)x = 0 as t 0+. Therefore, F (·) is strongly continuous on R+ and hence on R, since F (·) is an even function. The proof for the case of uniform measurability is analogous. To prove the assertion for G(·), we can use the following writing of Eq. (2.2): G(n ) = 2G(n - ) - G(n - 2)+ 2G()C (n - , A) in op erating the estimate of form (2.6). The proof is analogous. Theorem 2.2.2 ([239]). A multiplicative perturbation C0 -family and an additive perturbation C0 -family are strongly continuous on R+ for a C0 -cosine operator function C (·,A). Moreover, the uniform continuity at 0 implies the uniform continuity on R+ . 81


Proof. Following [102], we assume the contrary: the multiplicative p erturbation family F (·) is not strongly continuous at a certain p oint t0 R+ , i.e., there exists x0 such that the nonincreasing sequence Kn := sup (F (t) - F (s))x0 : |t - t0 |, |s - t0 | t0 8n

converges to a certain K > 0 as n . We can choose sequences n and n such that |n - t0 | and (F (n ) - F (n ))x0 Kn - Clearly, |n - n | t0 and |2 4n
4n

t0 , 8n

|n - t0 |

t0 , 8n

1 , n

n N.

- 4n - t0 |
4n

t0 , n N. Therefore, 8n - 4n ))x0 Kn , n N.

(F (4n ) - F (2 Using identity (2.1) in the form

2(F (t + h) - F (t)) = (F (t + h) - F (t - h)) + 2C (t, A)F (h) and setting t0 + h = 4n and t0 = 4n , we obtain 2 (F (4n ) - F (4n ))x0 Kn +2Met0 F (4n - 4n )x0 . Therefore, 2 K4n - and, thus, K4n +(K4n - Kn ) 1 +2Met0 F (h)x0 . 2n 1 4n Kn +2Met0 F (4n - 4n )x0 ,

By the convergence F (h)x0 0 as h 0 (recall that h = 4n - 4n ) and K4n - Kn 0 as n we have Kn 0 as n , n N, which is a contradiction to our assumption that Kn K, K > 0. To prove the same assertion for G(·), we can use the identity 2(G(t + h) - G(t)) = (G(t + h) - G(t - h)) + 2G(t)(C (h, A) - I )+ 2G(h), which is obtained from (2.2) and Prop osition 2.4.1 (i).

In the same way as in Prop osition 2.3.2 in [17], we can prove the following assertion. 82


Proposition 2.2.1. Let a multiplicative perturbation family F (·) and a C0 -cosine operator function C (·,A) commute, i.e., F (t)C (t, A) = C (t, A)F (t) for al l t R+ . Then the multiplicative perturbation C0 -family F (·) is an additive perturbation family and it is commutative, i.e., F (t)F (s) = F (s)F (t) for al l s, t R. 2.3. Main Properties of C0 -Cosine and C0 -Sine Operator Functions Definition 2.3.1. A C0 -cosine operator function is defined as a one-parameter family of op erators {C (t), t R}, C (t) B (E ), t R, having the following prop erties: (i) C (t + s)+ C (t - s) = 2C (t)C (s) for any t, s R (d'Alemb ert equation); (ii) C (0) = I is the identity op erator on E ; (iii) s- lim C (h)x = x for any x E .
h 0

With a C0 -cosine op erator function C (·), we associate the C0 -sine operator function
t

S (t)x :=
0

C (s)xds,

x E, t R,

(2.8)

and the lineals E k := {x E : C (·)x C k (R; E )}, k = 1, 2. (2.9)

Definition 2.3.2. A linear op erator A with the domain D(A) consisting of all x for which there exists the limit Ax := s- lim 2
h0+

C (h) - I x h2

(2.10)

is called an infinitesimal generator of a C0 -cosine function C (·). The prop erty that A is a generator of a C0 -cosine op erator function C (·) is written as C (·,A) (and S (·,A) for a C0 -sine op erator function S (·)). Let us present a simplest example of a C0 -cosine op erator function. Example 2.3.1. Let A b e the op erator of multiplication by a complex numb er on the space R. Then A is the generator of the C0 -cosine op erator function (C (t, A)f )(s) = cos(it A)f (s), t R. Proposition 2.3.1 ([88]). Define the operator A1 x := s- lim
h 0

C (2h, A) - 2C (h, A)+ I x h2

with a natural domain (i.e., on those x E at which this limit does exist). Then for x D(A1 ) D(A), we have Ax = A1 x. 83


For a C0 -cosine op erator function C (·,A), we can also define the first generator C x := s- lim with a natural domain. Proposition 2.3.2 ([264]). For a C0 -cosine operator function C (·,A), we have D(A) D(C ) and C x = 0 for any x D(A). Proposition 2.3.3 ([264]). The operators C (t, A),C (s, A),S (t, A), and S (s, A) commute for any t, s R. Proposition 2.3.4. The C0 -sine operator function S (·,A) is continuous in the uniform operator topology. Proposition 2.3.5 ([264, 272]). For al l t, s R, we have the relations (i) (ii) (iii) (iv) (v) (vi) where b0 + b1 z + ... + bn C (t, A) = C (-t, A), S (-t, A) = -S (t, A), S (0,A) = 0;
h0+

C (h, A)x - x h

S (t + s, A)+ S (t - s, A) = 2S (t, A)C (s, A); S (t + s, A) = S (t, A)C (s, A)+ S (s, A)C (t, A); C (t + s, A) - C (t - s, A) = 2AS (t, A)S (s, A); C (2t, A) = 2C (t, A)2 - I, C (t, A)2 - AS (t, A)2 = I ;
+1

C ((n +1)t, A) = b0 I + b1 C (t, A)+ ... + bn
+1

C

n+1

(t, A),

z

n+1

is the Chebyshev polynomial of the first kind of degree n +1.

Proposition 2.3.6 ([264]). For any C0 -cosine operator function C (·,A), there exist constants M 1 and 0 such that for al l t R, we have the estimate C (t, A) M cosh(t), where cosh(t) := 1 t e + e-t is the hyperbolic cosine. 2 t R, (2.11)

Definition 2.3.3. Infimum of the numb ers from (2.11) is called the type of a C0 -cosine op erator function and is denoted by c (A). Proposition 2.3.7 ([207]). The minimum satisfying (2.11) for an appropriate constant M may not exist, i.e., the greatest lower bound of c (A) is not attained in general. 84


Proposition 2.3.8 ([133, 142, 264]). Let an operator A generate a C0 -cosine operator function C (t, A), and let C (t, A) M cosh(t), t R. Then A G (M, 2 ), the C0 -semigroup exp(·A) is analytical ly continued to the right half-plane, and 1 exp(tA) = t


e- 4t C (s, A)ds,
0

s2

t R+ .

(2.12)

Proposition 2.3.9. The representation of the analytic semigroup in Proposition 2.3.8 can be written in the form exp(tA)x = 1 k t(k 2
+1)/2

Pk
0 t

s 2t

e- 4t Ck (s)xds,

s2

t R+ .

Here, Pk is a polynomial of degree k and Ck (t) =
0

(t - s)k-1 C (s, A) ds, where t R+ , k N, and (k - 1)!

C0 (t) = C (t, A). Remark 2.3.1 ([223]). There are examples of analytic C0 -semigroups whose generators do not generate C0 -cosine op erator functions. Proposition 2.3.10 ([179]). Obviously, D(A) E 1 for A C (M, ), and, therefore, the set E 1 is dense in E . Proposition 2.3.11 ([274]). For any x E and t, s R, we have
t

(i)

y :=
s t

S (, A)xd D(A)
s

and

Ay = C (t, A)x - C (s, A)x;

(2.13)

(ii)

z :=
0 0

C (, A)C (, A)xd d D(A) 1 C (t + s, A) - C (t - s, A) x; 2

and

(2.14)

Az = (iii)

(2.15) (2.16)

S (t, A)x E 1 .

Proposition 2.3.12 ([274]). If elements x vary over the whole E , and the numbers t and s vary over R,
t

then the set of elements of the form y =
s

S (, A)xd is dense in E .

Proposition 2.3.13. For any x E , the fol lowing relations hold: s- lim t
t0 -1

S (t, A)x = x

and

s- lim 2t
t0

-2 0

t

S (, A)xd = x.

(2.17)

85


Proposition 2.3.14 ([274]). If x E 1 , then for any t R, (i) (ii) C (t, A)x E 1 ,
0

S (t, A)x D(A) and

and

C (t, A)x = AS (t, A)x;

(2.18) (2.19)

s- lim AS (, A)x = 0

S (t, A)x = AS (t, A)x.

Proposition 2.3.15 ([274]). Let x D(A). Then for al l t R, (i) (ii) C (t, A)x D(A) S (t, A)x D(A) and and C (t, A)x = AC (t, A)x = C (t, A)Ax; S (t, A)x = AS (t, A)x = S (t, A)Ax. (2.20) (2.21)

Proposition 2.3.16 ([272]). For al l t, s R, the fol lowing relations hold: (i) (ii) (iii) (iv) (v) C (2t, A) = C (t, A)2 + C (t, A)S (t, A); C (t, A)S (s, A) = C (s, A)S (t, A); C (t + s, A) - C (t - s, A) = 2C (t, A)S (s, A); (C (t, A) - I )
0 h

(2.22) (2.23) (2.24)
t

S (s, A)ds = (C (h, A) - I )
0

S (s, A)ds;

(2.25) (2.26)

(A - 2 I )
0

t

sinh (t - s) C (s, A)ds = C (t, A) - cosh(t)I ;

here, sinh(·) and cosh(·) are the hyperbolic sine and the hyperbolic cosine, respectively. Proposition 2.3.17 ([135]). The domain of the generator of a C0 -cosine function C (·,A) coincides with E 2 , and for each x D(A), Ax = s- lim C (, A)x.
0

(2.27)

Sometimes the generator of a C0 -cosine op erator function is defined by (2.27). The set of generators of a C0 -cosine op erator function with b ound (2.11) will b e denoted by C (M, ). Proposition 2.3.18 ([264]). Let A, G C (M, ). Then if D(A) D(G) and Ax = Gx for al l x D(A), we have C (t, A) = C (t, G) for al l t R. Theorem 2.3.1 ([113, 131, 264, 269]). For an operator A C (E ) to be a generator of a C0 -cosine operator function, it is necessary and sufficient that for a certain constants M, 0, the resolvent (2 I - A)-1 exists for Re > and the fol lowing inequalities hold: dn (2 I - A)-1 dn 86 Mn! (Re - )n
+1

,

n N0 .

(2.28)


Remark 2.3.2 ([264]). Sometimes, estimate (2.28) is written in the form dn (2 I - A)-1 dn for all Re > , n N0 . In practice, conditions (2.28)­(2.29) turn out to b e difficult to verify, and, therefore, other conditions for generating a C0 -cosine op erator function are of interest. Theorem 2.3.2 ([42]). An operator A C (E ) is a generator of a C0 -cosine operator function iff there exist constants M, > 0, and such that (2 I - A)-1 M ||(Re - ) for al l Re > , (2.30) Mn! 2 1 (Re - )n + 1 (Re + )n (2.29)

+1

+1

and the fol lowing estimate holds uniformly in (0,):
+i

e
-i

2



cosh(t)(2 I - A)-1 xd (t) x ,

t R+ ,

(2.31)

where (·) C (R). Remark 2.3.3. In connection with estimate (2.30), we note (see [57]) that for any fixed > 0, the M condition (2 I - A)-1 , Re > , implies the b oundedness of the sp ectrum (A). ||1+ Proposition 2.3.19 ([210]). In the case where A is a normal operator on a Hilbert space, it generates a C0 -cosine operator function if and only if the conditions for location of the spectrum hold, i.e., {z 2 : Re z > } (A) for a certain . Proposition 2.3.20 ([272]). For Re > c (A), we have 2 (A) and (2 I - A)-1 x =
0

e-t C (t, A)xdt, e-t S (t, A)xdt,

x E; x E.

(2.32) (2.33)

(2 I - A)-1 x =
0



Proposition 2.3.21 ([135]). If x D(A3 ), y D(A), and > c (A), then t4 1 t2 C (t, A)x = x + Ax + A2 x + 2! 4! 2i 1 2i
+i +i

et -3 (2 I - A)-1 A3 xd;
-i

(2.34)

C (t, A)y =

et (2 I - A)-1 yd, t R+ .
-i

(2.35)

87


Writing the inverse Laplace transform in another form, we can obtain other analogous representations of the op erator functions C (·,A) and S (·,A). Proposition 2.3.22 ([222]). Let x D(Ak ) for a certain k N. Then for t R, the fol lowing Taylor formula holds: C (t, A)x = x + t2 t2k-2 Ax + ... + Ak 2! (2k - 2)!
-1 t

x+
0

(t - s)2k-1 C (s, A)Ak xds. (2k - 1)!

Proposition 2.3.23 ([166]). Let A C (M, ), and let r N. Then C (t, A) - I C (t, A) - I for any x E .
r r

= 2-r 2
t 0

r

(-1)r
j =1 t

-j

2r 2 Cr-j C (jt, A)+ (-1)r Cr r I ,

x = Ar

tr

...
0

(t - sj )C (sj ,A)xds1 ds2 ...ds

r

0 j =1

~ Proposition 2.3.24 ([225]). For any A C (M, ), x E0 , and t R the representation


C (t, A)x =
k =0

t2k Ak x/(2k)!

holds, and for each x E0 , the function t C (t, A)~ can be continued in t up to a function analytic on ~~ x the whole complex plane. Proposition 2.3.25 ([133]). The fol lowing Widder­Post formula holds: C (t, A)x = lim
k

(-1)k k!

k t

k +1

dk ((2 I - A)-1 x) dk

=

k t

,

t = 0, x E,

(2.36)

where the convergence is uniform in t from any compact set in R \{0}. Proposition 2.3.26 ([289]). The expression N(, k) := sented in the form (i) (ii) (iii) N(, k) = k! k
+1 2 + Ck+1 k -1 k A + ... + Ck+1Ak/2 (2 I - A)-(k k A + ... + Ck+1 A(k +1)/2 +1)

dk (2 I - A)-1 dk

from (2.36) can be repre-

,

k is even;
+1)

N(, k) = -k! k
k

+1

2 + Ck+1k

-1

(2 I - A)-(k ,

,

k is odd;

N(, k) =
k

(-1)j
j =k/2

(k +1)!j !(2)2j -k 2 ( I - A)-(j (k - j )!(2j - k +1)!

+1)

k is even;

(iv)

N(, k) =
j=
k-1 2

(-1)j
+1

(k +1)!j !(2)2j -k 2 ( I - A)-(j (k - j )!(2j - k +1)!

+1)

,

k is odd.

88


Proposition 2.3.27 ([260]). For a C0 -cosine operator function C (·,A), C0 -sine operator function S (·,A), and any x E and t R, we have
k l 2l C2k l=0 j =0 k l 2l C2k+1 l=0 j =0 m k -(2k -l+j )

(i)

C (t, A)x = lim

k

Clj

(-1)

l-j

t I- 2k I-

2

A t 2k +1
-j 2

x;
-(2k +1-l+j )

(ii)

C (t, A)x = lim

k

Clj

(-1)

l-j

A

x;

(iii)

C (t, A)x = lim e-nt
n

m=0 k =0 j =0 -2

(nt)2m 2k j C C (-1)k (2m)! 2m k

â

I+

nt (I - n 2m - 2k +1
n-1 m

A)-1 (I - n2 A)-(2m-k
j Ck

+j )

x;
2 -(n+m+1-k +j )

(iv)

t S (t, A)x = lim n n

k 2k C2m+1

(-1)

k -j

m=0 k =0 j =0

t I- 2n

A

x;

moreover, in al l the cases, the convergence in t J R is uniform. Here, J is an arbitrary closed interval. Proposition 2.3.28 ([260]). Under the conditions of Proposition 2.3.27, we have the fol lowing uniformly in t [0, 1]:
n m k 2m 2k j C2n C2m Ck (-1)k m=0 k =0 j =0 -(2m-k +j )

C (t, A)x = lim

-j 2m

n

t

(1 - t)2n

-2m-1

â

2n - 2m t(I - (2n)-2 A)-1 (1 - t)+ 2m - 2k +1

x.

In [260], many other relations in an analogous form were also presented. Introduce the following notation:


St(A) := x E :
k =1

Ak x

-

1 2k

<

are the Stieltjes vectors, tR

Up (A) := x E :
k =1

t2k Ak x < for a certain (2k)!

+

are semianalytic vectors, and


Upp (A) := x E :
k =0

t2k Ak x < for all (2k)!

tR

+

are entire vectors.

~ Proposition 2.3.29 ([107]). Let A C (M, ), and let E0 be constructed according to the C0 -semigroup ~ exp(·A). Then E0 Up (A). 89


Proposition 2.3.30 ([107]). Let A C (M, ). We have the embeddings U(A) Up (A) St(A). Proposition 2.3.31 ([107]). Let Upp (A) = E . Then the set of vectors x from D(A ) having the property Ak x
1/k

= o(k) is dense in E .

Proposition 2.3.32 ([107]). Let A C (M, ). Then U(A) Upp (A) = E . Proposition 2.3.33 ([107]). Let Upp (A) = E and let there exist an operator G C (E ) such that (i) (ii) G-1 B (E ); G2 = A, and, moreover, the operators ±G are dissipative.

Then A C (M, ) and C (t, A) = (exp(tG)+ exp(-tG))/2, where G G R(1, 0). Definition 2.3.4. A set of elements S E is said to b e total in E if the set of all its finite linear combinations is dense in E . Proposition 2.3.34 ([107]). Let A1 L(E ) be closed, St(A1 ) be total in E , A2 C (M, ), and let A1 A2 . Then A1 = A2 . Proposition 2.3.35 ([107]). Let A be a closed, symmetric, and semibounded operator on a Hilbert space H . Then the operator A is self-adjoint iff the set St(A) is total in H . In [145], examples of nonlinear cosine op erator functions are presented. However, in contrast to the theory of nonlinear op erator semigroups, there is no general theory of nonlinear cosine op erator functions for now. 2.4. Laplace Transform and Infinitesimal Operators In this section, we present certain basic prop erties of the Laplace transform for C0 -families of mul^ ^ tiplicative p erturbations F (·) and additive p erturbations G(·). Let F (·) and G(·) denote their Laplace transforms, resp ectively. Proposition 2.4.1 ([239]). Let F (·) be a C0 -family of multiplicative perturbations and let G(·) be a family of additive perturbations for a C0 -cosine operator function C (·,A). The fol lowing properties hold: (i) (C (t, A) - I )F (s) = (C (s, A) - I )F (t) and G(s)(C (t, A) - I ) = G(t)(C (s, A) - I ) for t, s R+ ; (ii) the functions F (·) and G(·) d2 (iii) 2 ((2 I - A)-1 F (t)x) = dt d2 G(t)(2 I - A)-1 x = 2 dt2 90 are exponential ly bounded; ^ C (t, A)2 F ()x and ^ G()C (t, A)x for x E, > , and t R+ ;


(iv) F (t)x = (2 I - A)
0 t

^ S (s, A)F ()xds =
0

t

^ ^ S (s, A)3 F ()xds - (C (t, A) - I )F ()x

for x E, t R+ ; (v) ^ G(t)x = G()(2 I - A)
0 t

^ S (s, A)xds = 3 G()
0

t

^ S (s, A)xds - G()(C (t, A) - I )x

for E, t R+ . Proof. Prop erty (i) is easily implied by (2.1) and (2.2). To prove that the C0 -family of multiplicative p erturbations F (·) is exp onentially b ounded, we choose L 1 and R+ such that C (s, A) L and F (s) L for 0 s . Using the relation F (k + s) = 2F (k ) - F (k - s)+ 2C (k , A)F (s) for 0 s , we have F ( + s) 2F ( ) + F ( - s) + 2 Me F (s) 2L + L +2Me L Me 5L Me21 Me1 e( where 5L e1 and 1 , F (2 + s) 2 F (2 ) + F (2 - s) +2Me21 F (s) 2Me21 + Me21 +2LM e21 Me31 1 e(2 By induction, F (k + s) 2 F (k ) + F (k - s) +2 C (k , A) 2Mek
1 +s)1 +s)1

,

.

F (s)

+ Mek

1

+2LM ek

1

5LM ek
+s)1

1

Me(k

+1)1

Me1 e(k

for all s [0, ]. Therefore, F (t) M1 e1 t for M1 = Me1 and all t R+ . To prove (iii), we set (t, ) = (2 - A)-1 F (t) and (t, ) = G(t)(2 - A)-1 , > ,t 0. It follows from (2.1) and (2.2) that t (t, ) = C (t, A) lim 2s
s 0 -2

(2 - A)-1 F (s) = C (t, A)t (0,) 91


if t (0,) exists, and t (t, ) = lim 2s
s 0 -2

(2 - A)-1 G(s)C (t, A) = t (0,)C (t, A)

^ ^ if t (0,) exists. Therefore, it suffices to prove that t (0,) = 2 F () and t (0,) = 2 G(). Taking the Laplace transform in t in (2.1), we have ^ (es - 2+ e-s )F () - es
0 s s 0

e- F ( )d + e-s

e F ( )d = 2(2 - A)-1 F (s) = 2(s, ).

Now, taking the derivative, we obtain ^ 2s (s, ) = (es - e-s )F () - es and differentiating once more, we have ^ 2ss (s, ) = 2 (es + e-s )F () - 2 es
s 0 s 0

e- F ( )d - e-s
0

s

e F ( )d,

e- F ( )d + 2 e-s
0

s

e F ( )d - 2F (s).

^ Therefore, s (0,) = 0 and ss (0,) = 2 F (). Analogously, we can show that s (0,) = 0 and ^ ss (0,) = 2 G(). ^ Integrating tt (t, ) = C (t, A)2 F () twice from zero up to t and using the relations F (0) = 0 and t (0,) = 0, we obtain (2 I - A)-1 F (t)x = (t, )x =
0 t

^ S (s, A)2 F ()xds, x E,

(2.37)

and, therefore, assertion (iv) is proved. Assertion (v) is proved analogously. Remark 2.4.1. If C (·,A) is uniformly continuous, then each C0 -family of multiplicative p erturbations F (·) (resp. each C0 -family of additive p erturbations) of a C0 -cosine op erator function C (·,A) is also uniformly continuous. This follows from formula (iv) (resp. (v)) and Prop osition 2.4.1. Definition 2.4.1. Let F (·) b e a C0 -family of multiplicative p erturbations for a C0 -cosine op erator func2 tion C (·,A). The infinitesimal operator Ws of the family F (·) is defined as Ws x = s- lim 2 F (h)x, h 0 h with a natural domain. The infinitesimal operator As of the pair (C (·,A),F (·)) is defined as As x := s2 lim 2 (C (h, A) + F (h) - I )x, with a natural domain. The infinitesimal operator Wc of a C0 -family of h 0 h additive p erturbations G(·) and the infinitesimal operator Ac of the pair G(·),C (·,A)) are defined in the same way as Wc x = s- lim resp ectively. 92
h 0

2 G(h)x h2

and Ac x := s- lim

h 0

2 C (h, A)+ G(h) - I x, h2


Theorem 2.4.1 ([239]). The operators Ws and As defined above are closed and ^ (i) Ws = (2 - A)F (), Re > ; ^ ^ (ii) As = A(I - F ()) + 3 F (), Re > ; (iii) As = A I - 2 t2
t 0 0 t 0 0

F (s)dsd

+

2 2 t2

^ C (s, A)dsd - (C (t, A) - I ) F (),

where t R+ , Re > . Proof. Let Ah = form 2F (h) 2 x= 2 2 h h Ah x = 2h-2
0 h h 0

2 C (h, A) + F (h) - I . Assertion (iv) of Prop osition 2.4.1 can b e rewritten in the h2 2 ^ ^ S (s, A)3 F ()xds - 2 (C (h, A) - I )F ()x, h

^ ^ S (s, A)3 F ()xds +2h-2 (C (h, A) - I )(I - F ())x.

^ Since the first term in the right-hand side of each of these relations converges to 3 F ()x as h 0 by (2.17), we have ^ D(Ws ) = D(AF ()) and also ^ ^ ^ D(As ) = D A(I - F ()) and As x = 3 F ()x + A(I - F ())x for all x D(As ). ^ and Ws x = (2 - A)F ()x for all x D(Ws ),

^ Since A is closed and F () is b ounded, it is easy to see that Ws and As are closed. Assertions (i) and (ii) are proved. To prove (iii), we use relation (2.1). For all x E and s R+ , we have 2 2 2 C (h, A)+ F (h) - I x = 2 C (h, A) - I x + 2 F (s + h) - 2C (h, A)F (s)+ F (s - h) x 2 h h h = 2 C (h, A) - I h2 = I - F (s) x + 1 F (s + h) - 2F (s)+ F (s - h) x h2

2 2 (C (h, A) - I )(I - F (s))x + 2 C (s, A)F (h)x. 2 h h

Integrating twice for any t R+ , we obtain 2 2 2 (C (h, A)+ F (h) - I )x = 2 (C (h, A) - I ) I - 2 2 h h t + 22 ( I - A) t2
t 0 0 t 0 0

F (s)dsd x 2 F (h)x. h2 93

C (s, A)dsd (2 I - A)-1


Since the last term converges to 22 ( - A) t2
t 0 0

^ C (s, A) dsd F ()x

as h 0+ for all x E (see Prop osition 2.4.1 (iii)), we obtain As in the form (iii). Remark 2.4.2. The definition of the infinitesimal op erator via the limit s- lim h-1 F (h) has no sense.
h0+

Indeed, in this case, using (2.37), we obtain that such an op erator is zero. Generally sp eaking, the domains of the op erators Ws and As are not necessarily dense in E . But under certain conditions on F (·), the op erator As not only has a dense domain but generates a C0 -cosine op erator function. The domains of D(Wc ) and D(Ac ) always contain the dense set D(A). Theorem 2.4.2 ([239]). The infinitesimal operators Wc and Ac have the fol lowing properties for Re > : ^ (i) D(A) D(Wc ) and Wc x = G()(2 - A)x for al l x D(A); (ii) D(A) D(Ac ), and for x D(A), we have ^ ^ Ac x = Ax + Wc x = I - G() Ax + 3 G()x; (iii) D(A) D(Ac ), and for al l x D(A) and t R Ac x = I - 2 t2
t 0 0 + t 0 0

^ G(s)dsd Ax + G()

2 2 t2

C (s, A)dsd - (C (t, A) - I ) x.

^ Moreover, if G(t) is uniformly continuous in t, then Ac is closed, D(Ac ) = D(A), and Ac = (I - G())A+ ^ ^ 3 G() for large Re . If G() is invertible for a certain , then the operator Wc is closed, D(Wc ) = D(A), ^ and Wc = Ac - A = G()(2 I - A). Proof. Let Ah = 2 C (h, A)+ G(h) - I . By (v) of Prop osition 2.4.1, we have h2 2G(h) 2 ^ x = 3 G() 2 h2 h ^ Ah x = 23 G()h-2
0 h h 0

^ S (s, A)xds - G()

2 C (h, A) - I x, h2

^ S (s, A)xds +2(I - G())h-2 (C (h, A) - I )x.

^ The first identity implies D(A) D(Wc ) and Wc x = G()(2 I - A)x for x D(A). ^ ^ The second identity implies D(A) D(Ac ) and Ac x = Ax + Wc x = (I - G())Ax + 3 G()x for x D(A). The proof of (iii) is similar to that of (iii) in Theorem 2.2.2. ^ If G(t) 0 as t 0, then G() 0 as (Prop osition 2.4.2 (ii)). Therefore, the ^ op erator I - G() is invertible for large , and we have D(Ac ) D(A). If {xn } is a sequence in D(A) 94


^ ^ such that xn x and (I - G())Axn y , then Axn (I - G())-1 y , so that x D(A) and ^ Ax = (I - G())-1 y. ^ Therefore, (I - G())A is closed, and hence Ac is also closed. The proofs of the assertions concerning the op erator Wc are going in the line as that for Ac . It follows from (2.1) that if F (t)x = o(t2 )(t 0+) for all x E , then F (t) = 0 for all t R+ , so that F (·) F (0) = 0, and then F (·) F (0) = 0. Similarly, it follows from (2.2) that the condition G(t)x = o(t2 )(t 0+) for all x E implies G(·) 0. Therefore, the rate of convergence to 0 in the case of a nontrivial multiplicative p erturbation family or an additive p erturbation family cannot exceed O(t2 ) for t 0. Proposition 2.4.2 ([239]). We have the fol lowing assertions concerning the rate of convergence to zero: ^ (i) for n = 0, 1, if F (t)x = o(tn ) as t 0+ for al l x E , then n F () = o(1) as and n n n
+1

^ F ()x = o(1) as for al l x E ; ^ ^ G()x = o(1) as for al l x E , and 3 G()x - (Ac - A)x = o(-n ) for al l x D(A);
+1

^ (ii) for n = 0, 1, if G(t)x = o(tn ) as t 0+ for al l x E , then n G() = o(1) as ,
+1

(iii) for n = 0, 1, if F (t) = o(tn ) (resp. G(t) = o(tn )) as t 0+, then n
+1

^ F () = o(1) (resp.

^ G() = o(1)) as ;
+1

(iv) for n = 1, 2, if F (t) = O(tn )(t 0+), then n (Ac - A)x = O(-n ) for al l x D(A);

^ F () = O(1) as ; ^ ^ G() = O(1) as , and 3 G()x -

(v) for n = 1, 2, if G(t) = O(tn ) as t 0+, then n

+1

^ (vi) if F (t) = O(t2 ) as t 0, then w - lim 3 F () x = (A - A )x for any x D(A ). s


Proof. We prove only (ii); the proof of assertions (i), (iii), (iv), and (v) is analogous. For a given choose > 0 such that G(t)x tn for all t [0,]. Then n n
+1 0 +1

> 0,

^ G()x n
+1

+1 0





+


e-t G(t)x dt n+1 -(- e -
)

e-t tn dt + n

e-t Met dt x /n!+ M

x.

This implies n

+1

^ G()x = o(1) as for x E . By the uniform b oundedness principle, we have

^ ^ n G() = o(1) as . Now, we have from (ii) of Theorem 2.4.2 that 3 G()x-(Ac -A)x = o(-n ) for all x D(A).

Chapter 3
95


REDUCTION OF THE CAUCHY PROBLEM FOR A SECOND ORDER EQUATION TO THE CAUCHY PROBLEM FOR A SYSTEM OF FIRST ORDER EQUATIONS As for ordinary differential equations, nth order equations can b e reduced to a set of first order equations by using matrix op erators. The matrix op erator theory is presented in [128] in detail. In the present chapter, we consider only problems of reducing incomplete second order equations to a set of first order equation.

3.1. Kysinski Theorem In a Banach space E , let us consider the following uniformly well-p osed Cauchy problem: u (t) = Au(t), Define the matrix op erator A := 0 I t R; : E 1 â E E 1 â E acting on an element (x, y ) E 1 â E u(0) = u0 , u (0) = u1 . (3.1)

A0 by the formula A(x, y ) = (y, Ax) that is given on the domain D(A) = D(A) â E 1 . In what follows, an x . element (x, y ) E 1 â E in the paragraph formulas will b e written as the vector y

Theorem 3.1.1 ([179]). The space E 1 with the norm x
E
1

:= x + sup

0t1

C (t, A)x

(3.2)

is a Banach space, and the operator A generates the fol lowing C0 -groups of operators on the Banach space E1 â E: exp(tA) x y C (t, A) S (t, A) AS (t, A) C (t, A) x y C (t, A)x + S (t, A)y AS (t, A)x + C (t, A)y , t R. := =

Proposition 3.1.1 ([287]). Let a C0 -cosine operator function C (·,A) be given. Then E 1 coincides with the closure of D(A) in the norm x


:= x +

z>,nN

su p

1 (z - )n n!

+1

dn A(z 2 I - A)-1 x . dz n

(3.3)

Proposition 3.1.2 ([179]). The resolvent of the operator A has the form (I -A)-1 = (2 I- I- A)-1 A)-1 (2 I- A)-1 A)-1 for 2 (A). (3.4)

A(2

(2

I-

96


Proposition 3.1.3 ([179]). Let u(·) be a solution of problem (3.1), and let v (t) := u (t),t R. Then the u(·) is a solution of the fol lowing uniformly wel l-posed Cauchy problem in the Banach space vector v (·) E1 â E: u u u0 u . (t) = A (t), t R; (0) = (3.5) v v u1 v ~ Proposition 3.1.4 ([179]). Let a certain Banach space E 1 be continuously and densely embedded into ~ ~ ~ the Banach space E , and, moreover, let D(A) E 1 for a certain operator A L(E ). Then if in the space ~ E 1 â E , the Cauchy problem u u 0I u ~ (t) = (t) A (t), ~ v v v A0 ~ ~ is uniformly wel l posed and (A) = , we have A C (M, ). Proposition 3.1.5 ([179]). Under the conditions of Proposition 3.1.4, the C0 -semigroup corresponding ~ to problem (3.6) on the space E 1 â E is represented in the G11 (t) G12 (t ~ exp(tA) := G21 (t) G22 (t form ) , ) t R, u v , (3.6)

(0) =

u0 u1

t R+ ,

(3.7)

~ where the family G22 (·) is a C0 -cosine operator function C (·,A) and coincides with G11 (·) on E 1 . Proposition 3.1.6 ([179]). Under the conditions of Proposition 3.1.4, the fol lowing relations hold for ~ x E and y E 1 : G12 (t)x = S (t, A)x and G21 (t)y = C (t, A)y = AS (t, A)y, t R.

~ Proposition 3.1.7 ([179]). The spaces E 1 and E 1 coincide with accuracy up to a norm equivalence. 3.2. Conditions (K) and (F) However, we note that to study problem (3.1) by reducing it to system (3.5) is very inconvenient, since the space E 1 is defined either through the C0 -cosine op erator function C (·,A) or through infinitely many p owers of the resolvent. As a rule, we have only the information ab out the op erator A. Therefore, certain additional conditions that allows us to reduce problem (3.1) to a system without use of the space E 1 are of interest. Proposition 3.2.1 ([274]). Let the space E be Hilbert, and let the operator A be self-adjoint and negativedefinite. Then A C (M, ), and the corresponding space E 1 coincides with D((-A)1/2 ). 97


Let the uniformly well-p osed problem (3.1) have the form u (t) = B2 u(t); where B C (E ). Definition 3.2.1. We say that a solution u(·) of problem (3.8) satisfies Condition (K) if u (·) C [0,T ]; D(B) . Proposition 3.2.2 ([47]). Problem (3.8) has a unique solution satisfying Condition (K) iff the fol lowing Cauchy problem is uniformly wel l posed on the space E â E : u v 0 B B 0 u v (t), t R, u v u0 v0 . (3.9) (t) = (0) = t R, u(0) = u0 , u (0) = u1 , (3.8)

An analog of Condition (K), which allows us to simplify the study of problem (3.1) by using C0 semigroups, is the following Condition (F). Definition 3.2.2. A C0 -cosine op erator function C (·,A) satisfies Condition (F) if the following conditions hold: (i) there exists B C (E ) such that B2 = A, and commuting with A; (ii) S (t, A) maps E into D(B) for any t R; (iii) the function BS (t, A)x is continuous in t R for any fixed x E . Proposition 3.2.3 ([135]). Under Condition (F ), for each t R, we have BS (t, A) B (E ) and D(B) E1 . Proposition 3.2.4 ([135]). There exist a Banach space E and a C0 -cosine operator function (even uniformly bounded) such that Condition (F ) does not hold. Proposition 3.2.5 ([135]). Via the shift Ab := A - b2 I for b > c (A), we can always construct operators Ab and Bb such that B2 = Ab and Bb commutes with any operator from B (E ) commuting with Ab . b Proposition 3.2.6 ([134]). The operator Bb in Proposition 3.2.5 can be constructed, e.g., as fol lows: -i Bb x :=


B commutes with any op erator from B (E )

C (·,A)

-1/2 (I - Ab )-1 (-Ab x)d.
0

98


Theorem 3.2.1 ([272]). Let A and B be operators satisfying condition (i) in Definition 3.2.2, and let 0 (B). The fol lowing conditions are equivalent: (i) the C0 -cosine operator function C (·,A) satisfies Condition (F); (ii) the operator B generates a C0 -semigroup exp(·B) on E ; 0B with the domain D(A) â D(B) generates a C0 -group on E â E ; (iii) the operator B0 0I with the domain D(A) â D(B) generates a C0 -group exp(·A) on (iv) the operator A := A0 D(B) â E , where D(B) is the Banach space of elements D(B) endowed with the graph norm; (v) the embedding D(B) E 1 holds; (vi) D(B) = E 1 . Proposition 3.2.7 ([104]). Let A C (M, 0), and let E be a UMD space. Then Condition (F) holds. Proposition 3.2.8 ([288]). The fol lowing condition is equivalent to conditions (i)­(vi) of Theorem 3.2.1: D(B) is dense in E , and there exist constants M > 0 and 0 such that 2 (A) for any > , the operator functions (2 I - A)-1 and B(2 I - A)-1 are strongly infinitely many times differentiable for > , and the fol lowing estimates hold for any m N0 : ( - )m+1 m! ( - )m+1 m! d d d d
m

(2 I - A)-1
m

M, M.

B(2 I - A)-1

Proposition 3.2.9 ([272]). Under the conditions of Theorem 3.2.1, for t R, we have (i) (ii) exp(tB) = C (t, A)+ BS (t, A), C (t, A) = exp(t B-1 0 0B exp t exp(tA) = 0 I B0 x C (t, A)x + S (t, A)y , exp(tA) = y AS (t, A)x + C (t, A)y B)+ exp(-tB) /2; B0 ; 0I x D(B) â E. y

(iii)

In applications, there often arises the following system of the sp ecial form: u (t) = -A0 u(t)+ Bv(t), u(0) = x, v (t) = Cu(t) - A1 v (t), v(0) = y, 99


on the space H = H0 â H1 with linear op erators A0 : D(A0 ) H0 H0 , B : D(B ) H1 H0 , A1 : D(A1 ) H1 H1 , C : D(C ) H0 H1 .

The corresp onding matrix op erator A is defined as follows: -A0 B A= C -A1 on H with D(A) = D(A0 ) D(C ) â D(A1 ) D(B ) . Theorem 3.2.2 ([195]). Let exp(t, -A0 ) and exp(t, -A1 ) be contractive C0 -semigroups on H0 and H1 , respectively, and let B and C be closed, and, moreover, D(A0 ) D(C ) = H0 and D(A1 ) D(B ) = H1 . Also, let Re{ A0 x, x + A1 y, y - By , x - Cx, y } 0 for any x D(A0 ) D(C ) and y D(A1 ) D(B ). Then the fol lowing conditions are equivalent: (i) A generates a contractive C0 -semigroup on H; (ii) for any > 0, we have (I + A0 - B (I + A1 )-1 C )-1 B (H0 ), (iii) assertions (ii) hold for a certain > 0; (iv) for any > 0, the operators -(A0 - B (I + A1 )-1 C ) and -(A1 - C (I + A0 )-1 B ) generate contractive C0 -semigroups on H0 and H1 , respectively; (v) assertions (iv) hold for a certain > 0. Also, in [195], the conditions under which the op erator A generates an exp onentially stable, differentiable, and analytic C0 -semigroup b elonging to the Gevrey class with > 0 were obtained. (I + A1 - C (I + A0 )-1 B )-1 B (H1 );

Chapter 4
INTERPOLATION The interp olation theory considerably increases the total volume of results in the theory of partial differential equations. We will mostly interest in two global directions: applications to coercive inequalities, which often do not hold in the traditional spaces, and applications to the rate of convergence of approximative methods dep ending on the smoothness of initial data (see the first article in this volume). 100


4.1. Generalities Let X and Y b e two complex Banach spaces continuously emb edded in a Hausdorff top ological vector space E , i.e., X E and Y E . Such Banach spaces X and Y are called an interp olation pair, which is denoted by {X, Y }. Proposition 4.1.1 ([9]). Let {X, Y } be an interpolation pair. Then X + Y and X Y are Banach spaces with the norms x respectively. Obviously, if Y X , then X Y = Y and X + Y = X . In such a case, it is natural to set E = X , which usually holds in applications. Definition 4.1.1. For any t R K-functional K (t, x; X0 ,X1 ) = for any x X0 + X1 . Sometimes, one merely writes K (t, x) if the choice of the spaces X0 and X1 is clear. Definition 4.1.2. The interpolation space (X0 ,X1 ),q , 0 1, 1 q < , constructed according to an interp olation pair {X0 ,X1 } by using the K -method, is the space of all elements x X0 + X1 for which the following norm is finite:

1 q

X Y

= max( x

X

,x

Y

),

x

X +Y

=

x=x0 +x1 x0 X, x1 Y

inf

{ x0

X

, x1

Y

},

+

and an interp olation pair {X0 ,X1 }, we define the so-called Peetre

x=x0 +x1 , x0 X0 ,x1 X1

inf

( x0

X0

+ t x1

X1

)

x

(X0 ,X1 ),

q

=
0

t

-

q

K (t, x)

dt

.

In the case q = , instead of (X0 ,X1 ), , one usually writes (X0 ,X1 ) and defines the norm as x


= su p t
0
-

K (t, x).

The interp olation space with q = is of a sp ecific interest in considering approximations of C0 semigroups of op erators and C0 -cosine op erator functions by using the Favard classes. Along with the K-functional, it is p ossible to use other constructions for constructing interp olation spaces. For more detail, see, e.g., [9, 73]. Definition 4.1.3. We say that a space E K (X0 ,X1 ), if it is continuously emb edded in (X0 ,X1 ) , i.e., K (t, x) ct x
E

for any x E . 101


In connection with Definition 4.1.3, for an interp olation pair {X0 ,X1 }, it is useful to set Jj (X0 ,X1 ) Kj (X0 ,X1 ) = {Xj }, j = 0, 1.

Definition 4.1.4. A Banach space [X0 ,X1 ] constructed by using the complex interp olation method is called the interpolation space corresp onding to an interp olation pair {X0 ,X1 }. Definition 4.1.5. Let b e an op en set in Rd , m N0 , and let 1 q, p . Let = m + , where 0 < 1. We set y f (x) := f (x + y ) - f (x), 2 f (x) := f (x +2y ) - 2f (x + y )+ f (x), and y {x : x + jy for j = 0,k }.
m The Besov space Bp,q (,E ) is defined as the space of all functions f from Wp (; E ), for which the k,y k

:=

( - jy ) =

j =0

seminorm |f |
Bp,q ;E

:=
||=m

|y |- { k x f (x) y

Lp (k,y ;E )

}

Lq (Rd )

is finite for k = 1 or k = 2 when 0 < < 1 or = 1, resp ectively. The norm of the Besov space is defined as follows: f
Bp,q ;E

:= f

Lp ;E

+ |f |

Bp,q ;E

.

Here Lp () is an Lp () space with the measure |x|-d dx, Rd . Theorem 4.1.1 ([219]). Let a, b < . Then (i) for , R, 1 p1 p2 , in the case -
have Bp1 ,q
1

((a, b),E ) Bp2

1 1 1 1 >- or - =- and q1 q2 , we p1 p2 p1 p2 for any m N.

,q

2

((a, b),E );

m m m (ii) Bp,1((a, b),E ) Wp ((a, b),E ) Bp, (a, b),E m In particular, B ,1

(a, b),E C

m

(a, b),E .

4.2. Interpolation in the C0 -Semigroup Theory Recall that by D(Am ) we denote the Banach space of elements x D(Am ) endowed with the norm x
D (Am )

= x + Am x .

Theorem 4.2.1 ([73]). Let m N, 0 < < 1, 1 p < and k, l Z with 0 k < s = m, l > s - k. Then (i) for A G (M, ) and 0 < < , (E, D(Am )),p = x E : x 102
(k,l,) (E,D (Am )),
p

< ,


where x
(k,l,) (E,D (Am )),
p

=x

E

+
0

t ·

-(s-k )

(exp(tA) - I ) A x
l k
p

p E

dt t

1 p

,

and al l these norms are equivalent to the norm

(E,D (Am )),

;

(ii) if < 0, then = is an admissible value in the definition of the norm. Definition 4.2.1. An op erator A C (E ) is said to b e positive if (-, 0] (A) and there exists a numb er C > 0 such that (A - I )-1 C 1+ || for (-, 0].

Note that in the case A G (M, ) with < 0, the op erator -A is p ositive. Theorem 4.2.2 ([73]). Let -A be positive, and let m N, 0 < < 1, 1 p . Then (E, D(Am )),p = moreover, the norm ·


xE: x



=
0

(t

m

Am (tI + A)-m x

E

)p

dt t

1 p

< ;

is equivalent to the norm of the space (E, D(Am )),p .

Theorem 4.2.3 ([73]). Let -A be a positive operator. Then (i) if j, m N and 1 j m, then (E, D(Am ))j/m,1 D(Aj ) (E, D(Am ))j/m, ; (ii) if m N, 0 < < 1, 1 p , and k, l Z, 0 k < s = m , l > s - k, then (E, D(A )),p =
m

xE:
0



t

s-k

A (tI + A) A x

l

-l

k

p E

dt t

1 p

< ;

(iii) if A H(M, ) with < 0, then (E, D(A )),p =
m

xE: x



=
0

t

m-m

A exp(tA)x

m

p E

dt t

1 p

< ,

where

·



is a norm equivalent to the norm of the space (E, D(Am )),p .

Proposition 4.2.1 ([73]). Let A be a positive operator, and let R+ , k, m Z, k 0, 0 < < m. Then for complex numbers -k < Re z - k and x (E, D(Am ))/m,p , the integral Az x =
0

(m) (z + m)(m - k - z )

z

+k -1

Am-k (A + I )-m xd

where (m) :=
0

e-mt t

m-1

dt is the gamma-function, converges. The operator Az is closable and is

independent of . 103


Definition 4.2.2. Let A b e a p ositive op erator, and let z C. The fractional power Az of the op erator A is defined as the closure of the op erator Az . Theorem 4.2.4 ([46]). Let A be a positive operator. Then (i) if m N, Re , Re < m, then A A x = A A x for x D(A2m );

(ii) if Re < 0, then A is a continuous operator and A- A = I ; (iii) if Re , Re > 0, then A A = A+ ; (iv) if m N and 0 < Re < m, then (E, D(Am ))
Re m

,1

D(A ) (E, D(Am ))

Re m

,

;

(v) if 0 < Re < Re < and 1 p , 0 < < 1, then (E, D(A )),p = (E, D(A ))
Re Re

,p

.

Proposition 4.2.2 ([73]). Let A be a positive operator, and let there exist constants and C such that Ait are operators uniformly bounded near zero, i.e., Ait C for - t . If 0 Re < Re < and 0 < < 1, then D(A ), D(A )


= D A(1-

)+

.

Proposition 4.2.3 ([98]). Let A G (M, 0), and let 0 < < 1. Then the fol lowing conditions are equivalent for x E : (i) x D((-A) ); (ii) there exists s- lim
0

1 (-)



t


--1

(exp(tA) - I )xdt.

Proposition 4.2.4 ([98]). Let > 0, A G (M, 0), and let


U ( )x :=
0

( - s)

-1

exp(sA)xds

for 0 < < 1, x E . Then U ( )x D((-A) ). Proposition 4.2.5 ([98]). Let A G (M, 0) be a normal operator on a Hilbert space E = H . Then for
the operator function Ct [exp(·A)], we have Ct [exp(·A)]E D((-A) ) for 0 < < 1, and the operator (-A) Ct [exp(·A)] is strongly continuous.

104


Theorem 4.2.5 (reiteration theorem, [73]). Let A be a positive operator satisfying the conditions of Proposition 4.2.2, and let Re > 0. Then for 1 p < , 0 < 0 < 1 < 1, and 0 < < 1, (E, D(A )0 ,p , (E, D(A )1
,p

= E, D(A

(1-)0 +1 ,p

.

As was already noted, if A H(, ) with 0, then the op erator -A is p ositive. At the same time, the construction of fractional p owers is simplified in this case. The location of the sp ectrum of the op erator A H(, ) is as follows:

Fig. 1 and (I - A)-1 M for . Assume that = 0. Then we can set | - | (-A)- = 1 2i - (I + A)-1 d, 0 < < ,


where the contour in the integral is going around upward; see Fig. 1. The op erators (-A)- are b ounded, and for integer = m N, we have (-A) = (-A)-n . Moreover, the op erators (-A)- (-A)- = (-A)-(+ ) form a semigroup, (-A)- const, 0 Re 1, and this semigroup is strongly continuous at zero, i.e., (-A)- x x as 0 for any x E . Complex p owers are defined by the formula (-A)z = 1 2i -z (I + A)-1 d,
z 0

Re z < 0.

(4.1)

Proposition 4.2.6 ([73]). Let A H(, ). Then {(-A)z }Re moreover, D((-A) ) = E .

is a C0 -semigroup analytic in the open

left half-plane. As the inverses to bounded operators, the operators (-A) , 0 < < , are closed and,

Proposition 4.2.7 ([47] (momentum inequality)). Let A be positive. Then for any < < , we have A x C (, , ) A x
- -

· A x

- -

for

x D(A ). 105


In [90, 211], fractional p owers of a p ositive op erator A are defined as follows: sin( ) -1 if A is b ounded, then A := µ (µI + A)-1 xdµ, 0 < Re < 1; 0 if A is unb ounded and 0 (A), then A := [(A-1 ) ]-1 ; if A is unb ounded and 0 (A), then A x := lim (A + I ) x on those x at which the limit exists.
0+

With such a definition, it is easy to see that A1 = A, and thecasewhere D(A) = E and 0 (A) turns out to b e appropriate. As was shown in [211], it is easy to prove the relations A A = A A = A+ , (A ) = A , and the integral expansions for the expression A x - are binomial coefficients. Moreover, the example showing that D and t1 = t2 was presented in this work. Proposition 4.2.8 ([253]). The fol lowing Landau inequalities hold for A G (M, 0): 72 x A3 x 2 , Ax 25
2 Cp (-1)p p (A + p=0 (As+it1 ) \D(As+it2 ) n I )-p x, where Cp

= for any s > 0

A2 x A3 x

3 4

3 x A3 x 2 , A2 x C x A4 x 3 , A2 x

3


4

3


4

81 x 40

2 3

A3 x , A4 x .

C x

A4 x 2 , Ax

C x

Theorem 4.2.6 ([47]). Let A be positive. Then the operators -A generate analytic C0 -semigroups for 1 . 2 Theorem 4.2.7 ([47]). Let A H(, ) with < 0. Then the operator -(-A) is a generator of an analytic C0 -semigroup for any 0 1. Theorem 4.2.8 ([126]). Let > 0, and let A H(, /2); moreover, let |z | Re z


exp(zA) M

for

Re z > 0.

Then the operator -(-A)1/2 generates an analytic C0 -semigroup analytic in the right half-plane and
1 2

exp(-z (-A)

1/2

) M

|z | Re z

+

for

Re z > 0.

1 Moreover, A generates a times integrated cosine operator function for > + . 2 106


Proposition 4.2.9 ([65]). Let A G (M, 0). Then -(-A)1/2 generates an analytic C0 -semigroup, and the fol lowing representations hold for t > 0: t exp(-t(-A)1/2 ) = 2 t exp(-t(-A)1/2 ) = 2 t exp(-t(-A)1/2 ) = 2
1/2 1/2 0 0 0

e- e- e-

t2 4s

exp(sA)

ds , s3/2 ds , s3/2 ds . s1/2

t 4s

exp(tsA)

ts 4

exp(tA/s)

Imaginary p owers of an op erator -A H(, ) with the prop erty 0 (A) can b e defined, e.g., as (A)is = gs (A)(A + I )2 A-1 , where gs () = is 1 and gs (A) = 2 (1 + ) 2i gs ()(I - A)-1 d (see [188]).


(4.2)

4.3. Interpolation in the Theory of C0 -Cosine Operator Functions As is known, an op erator A C (M, 0) also defines an analytic C0 -semigroup, and, therefore, following the previous section, we can define its fractional p owers Az . However, we present certain concrete relations that take into account the sp ecific character of a cosine op erator functions. So, by (2.12), we can follow the previous section, and expressing the resolvent through the cosine op erator function (see (2.32)), for b > c (A) we obtain (see [131]) (b2 I - A)- x = 23/2- b1/2- ()


s
0

-1/2

K-1/2 (bs)C (s, A)xds

(4.3)

for > 0, where K is the Mcdonald's function, which is represented through the Bessel function I (t) as follows: I- (t) - I (t) for = ±, ±2 , .... 2 sin( ) Let A C (M, 0). Then for k N and k - 1 < < k, we have the following relation useful in the K (t) = interp olation theory: (-A) x =


1 C,

t
k 0

-2

(C (t, A) - I )k x

dt , t

x D(Ak ),

where C,k =
0

t

-2

(cos(t) - 1)k

dt . t

Proposition 4.3.1 ([166]). Let r N,A C (M, 0), and let 0 < < r . An element x D((-A) ) iff there exists the limit s- lim in this case, this limit is -(-A) x. 107
0+

1 C,



t
r

-2

(C (t, A) - I )r x

dt ; t


Proposition 4.3.2 ([166]). Let r N,A C (M, 0), and let 0 < < r. An element x D((-A) ) belongs to D((-A) ) ) iff x E and there exists the limit 1 C,


w - lim in this case, this limit is -(-A ) x .

0+

t
r

-2

(C (t, A ) - I )r x

dt ; t

In [131], Fattorini has studied the relation b etween the domains of fractional p owers of op erators with a set of elements on which C0 -semigroups have fractional derivatives. Recall that a C0 -semigroup of op erators has a continuous fractional derivative of order 0 for t 0 iff there exist > (A) and a continuous function f (·), with the function s e-t exp(tA)x =


f (s) integrable in s 0, and, moreover,
t

ei ()

(s - t)-1 f (s) ds,

t 0.

(4.4)

Denote by E, the set of elements x E satisfying (4.4), and by F the set D((bI - A) ) for b (A). Proposition 4.3.3 ([131]). Let 0, and let A G (M, ). Then E, = E , > . For C0 -groups of op erators, the case of the previous prop osition is complemented by one more relation
- - - - E, = F , > , where F = D((bI + A) ) and E, corresp onds to exp(·A).

Proposition 4.3.4 ([131]). Let 0,x E , and let 0 < < 1. Then D((-A) ).

0

( - s)2

-1

C (s, A)xds

As Fattorini has shown, it is not p ossible to set = = 1/2 in the last prop osition. This forces the app earance of Condition (F); see p. 98. For a cosine op erator function C (·,A), let us define the modulus of continuity as follows: r (tr ,x) := sup (C (s, A) - I )r x
|s|t E

,

x E,

where r N. Also, we set K (t, x; E, U ) = inf { x - g
g U

E

+ t|g|U }.

Proposition 4.3.5 ([166]). Let r N, and let 0 < t < . Then there exist constants C1 ,C2 > 0 such that C1 K (t2r ,x; E, D(Ar )) r (tr ,x)+ min(1,t2r ) x where K is the Peetre functional. If A C (M, 0), we can set = . 108
E

C2 K (t2r ,x; E, D(Ar )),


As for semigroups of op erators, define (E, D(A)),q as the space with the norm


x

(E,D (A)),

q

:=
0

(t

-

1/q

K (t ,x) dt

r

q

.

In the case q = , the norm is given by x
(E,D (A)),


=x

E

+ sup (t
tR+

-2

1 (t, x)).

Theorem 4.3.1 ([166]). Let 0 < < r , r N, 1 q < (or resp. 0 r , q = ). Then the intermediate spaces (E, D(Ar )),q with = /r and 0 < < have the fol lowing equivalent norms:


(i)
0

t x x

-/r

K (t, x; E ; D(Ar ))q


dt t

1/q

;
1/q

(ii) (iii)

E

+
0

(t (t
0

-2

r (tr ,x))q

dt t

; dt t
1/q

E

+

-2

(C (t, A) - I )r x )q

.

If A C (M, 0), then we can set = in the previous theorem. Corollary 4.3.1 ([166]). Under the conditions of the previous theorem, we have (i) C (t, A)(E, D(Ar )),q (E, D(Ar )),q , t R+ , 0 < < r , 1 q < (or 0 r , q = ); (ii) S (t, A)(E, D(Ar )) r - 1/2, q = ).
r

,q

(E, D(Ar ))

+1/2 r

,q

, t R+ , 0 < < r - 1/2, 1 q < ( or 0

Proposition 4.3.6 ([166]). Let A C (M, 0). Then (i) (E, D(Ar ))
r

,1

D((-A) ) (E, D(Ar )) (E, D(Ar ))
r

r

,

if r N, 0 < < r ;

(ii) (E, D(A )),

q)

,q

if r N, 0 < < r, 1 q , = /r , since the Favard

class (E, D((-A) ))1,

consists of elements with the norm x +sup
>0

1 C,



t
r
r

-2

(C (t, A) - I )r

dt t

.
E

In particular, D((-A) ) is dense in (E, D(Ar )) Let (E, D(Ar ))o
r

,q

for 0 < < r, 1 q < .
r

,q

denote the closure of D(Ar ) in (E, D(Ar ))

,q

.

Corollary 4.3.2 ([166]). An element x (E, D(Ar ))o ,q , 0 r, r N, iff lim (C (t, A) -
r

t0+

I )x

(E,D

(Ar

))

,q r

= 0. 109


Proposition 4.3.7. Let A C (M, 0). Then (i) if x (E, D(Ar ))
r

,q

with 0 < < r, 1 q < ( or 0 r, q = ), r N, then (C (t, A) - I )r x
E

= O(t2 ),
r

t 0+ .

(4.5)

Conversely, if (4.5) holds, then x (E, D(Ar ))

,

;
r

(ii) for 0 < < r, r N, an element x (E, D(Ar ))o (C (t, A) - I )r x
E

,

iff t 0+ . (4.6)

= o(t2 ),

In [131], it was proved that for A C (M, 0), x D(A1/2 ) implies sin( At)x D(A ), 0 < < 1/2, and A sin( At)x ct
1-2

Ax .

Proposition 4.3.8. Let A C (M, 0), and let E be reflexive. Then S (t, A)(E, D(A))1/2,


D(A),

t R+ .

- + As in the previous section, denote by E, ,E, the subpaces related to C (·,A) as for C0 -groups

early, i.e., the spaces of fractional derivatives. Proposition 4.3.9 ([131]). Let 0, = k +1/2, k N0 . Then
- + E2, ,E2, D((bI - A) ),

, b > c (A).

(4.7)

In the case = k +1/2, k N0 , inclusion (4.7) can violate. Theorem 4.3.2 ([131]). Let E = Lp (X, ,µ) with 1 < p < ,A C (M, ), and let u0 D((bI - A) ), u1 D((bI - A) ), = max{ - 1/2, 0}. Then for a solution of problem (3.1), we have (i) if 0 1, then u(t) - u(0) = O(t2 ), t 0+; (ii) if 1/2 1, then u(·) is continuously differentiable and u (t) - u (0) = O(t
2-1

), t 0+.

To obtain the assertion of the theorem in the general Banach space E , we need an additional smoothness, i.e., u0 D((bI - A)+ ), u1 D((bI - A) ), = max{ + - 1/2, 0} for a certain > 0. Definition 4.3.1. For an op erator A C (E ), we set D(A) := {x E : {xn } D(A) such that xn
E E D (A)

M and lim xn - x
n

E

= 0}.

In the op erator semigroup theory, D(A) is the Favard class (saturation class). 110


Proposition 4.3.10. Let A C (E ). Then for t 0+, O(t) for x D(A)E , K (t, x; E, D(A)) = o(t) for x N (A). Moreover, if E is reflexive, then D(A) = D(A) .
We set Ct [C (·,A)] = E

t

t 0

(t - s)

-1

C (s, A) ds.

Proposition 4.3.11. Let A C (M, ). Then the fol lowing conditions are equivalent for 0 < 2: (i) (ii)
Ct [C (·,A)]x - x E

= O(t ),

t 0;

K (t ,x,E ; D(A)) = O(t ),

t 0.
E

Moreover, the condition x N (A) is equivalent to Ct [C (·,A)]x - x

= o(t2 ), t 0.

With the notation u = (A - c2 I )u(t) = B 2 u(t), u(0) = x, u (0) = y for c = 0, we have the following assertion. Theorem 4.3.3 ([122]). Let x E be such that C (t, A)x - x = o(t2 ) as t 0. Then x D(B 2 ) and B 2 x = 0. The saturation C (t, A)x - x = O(t2 ), t 0+, holds iff x D(B 2 ) . If E is reflexive, then D(B 2 ) = D(B 2 ). Denote V (t)x = 1 t
t E E

C (s, A)xds.
0

Theorem 4.3.4 ([122]). Let x E be such that V (t)x - x = o(t2 ), t 0+. Then x D(B 2 ) and B 2 x = 0. The saturation V (t)x - x = O(t2 ), t 0+, holds iff x D(B 2 ) . If c = 0, then we have the op erator B 2 + c2 I in the saturation theorem. For example, the following theorem holds. Theorem 4.3.5 ([122]). Let x and y be such that t iff x D(B 2 ), (B 2 + c2 I )x = 0, and y D(B 2 ) . Proposition 4.3.12 ([252]). For A C (M, 0), we have the Landau inequalities A2 x
4 E -1 E

(u(t) - x) - y = o(t2 ), t 0. Then x D(B 2 ),
-1

y D(B 2 ), (B 2 + c2 I )x = (B 2 + c2 I )y = 0 and u(t) = x + ty . Moreover, t

(u(t) - x) - y = O(t2 ), t 0



1024 x 315

3

A4 x , A2 x

4



400 x 49

2

A4 x 2 , A3 x

4



2880 x A4 x 3 . 343 111


Chapter 5
SPECTRAL PROPERTIES OF C0 -COSINE OPERATOR FUNCTIONS In the same way as for a C0 -semigroup, necessary and sufficient conditions for A to generate a C0 -cosine op erator function are formulated in terms of conditions on the location of the sp ectrum and estimates for the resolvent; see [17]. For a narrow class of C0 -cosine op erator functions on a Hilb ert space, these conditions are essentially based on the location of the sp ectrum; see [210]. 5.1. Location of the Spectrum Proposition 5.1.1 ([221]). Let a C0 -cosine operator function C (·,A) be given. Then (i) (ii) (iii) cosh(t (A)) (C (t, A)), t R; t R; t R.

cosh(t P (A)) = P (C (t, A)), cosh(t R (A)) R (C (t, A)),
N

Proposition 5.1.2 ([5, 222]). If µ R (C (t, A)) and {n }n µ P (C (t, A) ).

is the set of roots of the equation µ =

cosh(n t), then 2 0 R (A) for a certain n0 N, and 2 P (A) does not hold for any n N n n

Proposition 5.1.3 ([5, 221]). If µ C (C (t, A)) and n are from Proposition 5.1.2, then 2 C (A) n (A). The case where 2 (A) for al l n N is possible. n Proposition 5.1.4 ([139, 181]). If E = H is Hilbert and A C (M, 0) or C (·,A) is a family of normal operators, then (C (t, A)) = cosh(t (A)), t R.

Proposition 5.1.5 ([105]). Let a C0 -cosine operator function C (·,A) satisfy Condition (F), and let E = H be Hilbert. Then µ (C (t, A)) iff {z 2 : cosh(zt) = µ} (A) and sup{ z R(z 2 ; A) : cosh(zt) = µ} < . Proposition 5.1.6 ([206]). Let A C (M, 0). Then (i) (A) R- ; (ii) if E = {0}, then (A) = ; (iii) the spectrum (A) is bounded iff A B (E ). A Banach space E is said to b e hereditarily indecomposable (in brief, an H.I. space) if whenever X1 and X2 are closed infinite-dimensional subspaces of E and > 0, then there exist unit vectors x1 X1 ,x2 X2 112


such that x1 - x2 < . In other words, this prop erty can b e reformulated as follows [147]: for any two infinite-dimensional subspaces X1 ,X2 E such that X1 X2 = {0}, the subspace X1 + X2 is not closed. Proposition 5.1.7 ([248]). Let E be an H.I. space, and let A C (M, ). Then (A) is either a finite set (possibly empty) in C or consists of a sequence {µn } that either converges to a certain point of C n=1 or satisfies lim Re µn = -.
n

Proposition 5.1.8 ([248]). Let E be an H.I. space, and let C (·,A) be a non-quasi-analytic C0 -cosine operator family. Then (A) C = . Proposition 5.1.9. Let A G R(M, ). Then (i) the spectrum of A lies in the strip - < Re z < (see Fig. 2);

Fig. 2 (ii) the operator A2 generates a C0 -cosine operator function by the formula C (t, A2 ) = 1 exp(tA)+ exp(-tA) , t R. 2

Proposition 5.1.10 ([15]). For A C (M, ) and the corresponding matrix differential operator A = 0I arising in reducing the Cauchy problem (3.1) to system (3.5), the fol lowing relation holds: A0 {2 : (A)} = (A). Proposition 5.1.11 ([221]). Let A C (M, ). Then the spectrum (A) lies on a certain parabola whose branches are directed to the left; see Fig. 3.

113


Fig. 3 Proposition 5.1.12 ([5]). There exist a C0 -cosine operator function C (·,A) and a Banach space E such that the sets r1 := {t : 0 (C (t, A))}, r2 := {t : 0 P (C (t, A))}, and r3 := {t : 0 C (C (t, A))}, are dense in R, and, moreover, R = r1 r2 r3 . Proposition 5.1.13 ([86]). For the C0 -cosine operator function C (t, A) = a Banach algebra B with unity, we have (i) (ii) R(2 ,A) =
0 k =0

t2k k A , A B , given on (2k)!

0 c (A) < ; e-t C (t, A) dt

for

Re > c (A);

(iii) R(2 ,A) =
0

e-t S (t, A) dt

for

Re > c (A);

(iv) C (t, A) = (v) S (t, A) = 1 2i et (2 I - A)-1 d,


1 2i

et (2 I - A)-1 d,


t R;

t R,

where is a certain contour enclosing the spectrum of the operator A B . Proposition 5.1.14 ([86]). Under the conditions of Proposition 5.1.13, we have c (A)2 = sup (|| +Re )/2.
(A)

Theorem 5.1.1 ([106]). Let A C (M, ). The fol lowing conditions are equivalent: (i) 1 (C (2, A)); 114


(ii) -N2 (A) and the sequences 0 RN and S are bounded in B (E ); (iii) -N2 (A) and there exist the limits 0 Rx := s- lim RN x
N N

1 = N

N -1

n

(-k2 I - A)-1

n=0 k =-n n

=

1 N

N -1

A(-k2 I - A)-1

n=0 k =-n

and

Sx := s- lim SN x
N

(5.1)

for al l x E . Theorem 5.1.2 ([106]). Let A C (M, ). In a Hilbert space E = H , the fol lowing condition are equivalent: (i) 1 (C (2 ; A)); (ii) -N2 (A) and supk 0
Z

k(-k2 I - A)-1 < .

Chapter 6
UNIFORMLY BOUNDED C0 -COSINE OPERATOR FUNCTIONS In this chapter, we collect assertions that, in one way or another, are related to the b oundedness of cosine op erator functions, although we consider C0 -cosine op erator functions of p olynomial and sometimes exp onential growth. The matter is that the asymptotic b ehavior of resolving families for second order 1 equations differs from that of op erator semigroups, and the representation C (t, A) = (exp(t A) + 2 exp(-t A)) does not always hold. 6.1. Behavior of C0 -Cosine Operator Functions at Infinity As was already noted, the b oundedness of a C0 -cosine op erator function is a prop erty that is not obtained by a shift of the generator Ab = A + bI . Proposition 6.1.1 ([144]). There exist operators A C (M, ), such that for any number b R, the operator A + bI does not generate a bounded C0 -cosine operator function. With the cosine equation (i) (see p. 83) one associates the hyp erb olic cosine cosh(t) of exp onential growth, as well as the b ounded ordinary function cos(t). In the general case, for a C0 -cosine op erator function, a p olynomial growth in t is also p ossible. So, for example, we have the following. 115


Example 6.1.1 ([207]). Let E = R2 . Then C (t) = 1
t2 2

0 , t R, is a C0 -cosine op erator function

1 on E (i.e., (i) on p. 83 holds). If the Euclidean norm is given on E , then C (t) = 1+ t2 t4 +, 2 4 t R.

Under the conditions of the previous example, for any > 0, there exists M 1 such that C (t) M cosh(t), t R, but C (·) is not b ounded on R. Note by the way that Example 6.1.1 describ es exactly the case where there arises the problem on the representation of C (·) as the half-sum of two exp onential functions [109]. Example 6.1.2 ([119]). Let A be the operator on the space C 1 ([a, b]) defined by the formula (Af )(s) := sf (s), s [a, b].

Then A is a generator of a C0 -family of a cosine op erator function (C (t, A)f )(s) = h(s, t)f (s), s [a, b], t R, where h(s, t) = cos(t -s), s [a, b], t R. The norm of this C0 -cosine op erator function is equal to C (t, A) = max and admits the estimate C (tn ,A) c(1 + |tn |) with certain {tn }, lim tn = , and a constant c > 0.
n s[a,b]

sup |h(s, t)|, sup

s[a,b]

d h(s, t) ds

Therefore, the sp ectrum of the op erator A coincides with the closed interval [a, b], and for any a, b < 0, the norm C (t, A) is not b ounded on R. Example 6.1.3 ([119]). Let A be the operator on the space L1 (R) defined by the formula (Af )(s) := d ds
2

f (s),

s R. 1 f (t + s)+ f (t - s) , s, t R. 2

Then A is a generator of the C0 -cosine op erator function (C (t, A)f )(s) = Moreover, the C0 -semigroup generated by A admits the estimate exp(zA) |z | Re(z )
1 2

for any z with Re z 0.

Example 6.1.4 ([119]). Let A be the operator on the space Lp (Rd ) defined by the formula (Af )(s) := f (s), 116 s Rd , d N.


Then A is a generator of a C0 -cosine op erator function C (·,A) only in the case p = 2 in general. Moreover, the C0 -semigroup generated by the op erator A admits the estimate exp(zA) |z | Re(z )
1 d| p - 1 | 2

for any z

with

Re z 0.

Theorem 6.1.1 ([119]). The fol lowing implications of conditions hold: (i) = (ii) = (iii). (i) A function C (·,A)x is of exponential type less than or equal to for al l x E ; (ii) {z 2 : Re z > } (A), and for any > , there exists a constant M = M ( ) such that z (z 2 I - A)-1 M whenever Re z > ;

(iii) the function C (·,A)x is of exponential type not exceeding for al l x D(A). Theorem 6.1.2 ([119]). The fol lowing implications of conditions hold: (i) = (ii) = (iii). (i) a C0 -cosine operator function C (·,A) is bounded; (ii) there exists a constant M1 such that exp(zA) M1 |z | Re(z )
1 2

for any z with Re z 0;

(iii) there exists a constant M2 such that C (t, A)x M2 ( x + t2 Ax ), t R, x D(A).

An attempt to give necessary and sufficient conditions for the uniform b oundedness of C0 -cosine op erator functions was undertaken in [108], but, as K. Bo jadzhiev showed, the proof contains inaccuracies. 6.2. Uniformly Bounded C0 -Cosine Operator Functions For all x E and a R+ , let us define the op erator 2 Fa x :=
0

sin(at) t

2

C (2t, A)xdt,

which, obviously, is b ounded for A C (M, 0). Proposition 6.2.1 ([245]). Let 0 a b. Then
a

Fa Fb x = Fb Fa x = 2
0

Fu xdu +(b - a)Fa x,

x E.

Proposition 6.2.2 ([245]). For certain 0 a b, let the fol lowing relation hold:
b

2
a

Ft xdt = (b - a)(Fa x + Fb x)

for any x E . Then the open interval (-b2 , -a2 ) (A). 117


Proposition 6.2.3 ([245]). For the operator Fa , the fol lowing relations hold for a R+ :
k Fa = (k - 1)k a 0

(a - t)k

-2

Ft dt,

k = 2, 3, ..., exp(itFa ) = I + itFa - t

2 0

a

eit(a-s) Fs ds.

Let a C0 -cosine op erator function C (·,A) b e such that the op erator
+ia

Ea x := s- lim Ea, x := s- lim
0+

0+ +i0

(2 I - A)-1 + R( I - A)-1 xd

2

(6.1)

(where = + i ) is linear and continuous for all a R+ . Proposition 6.2.4 ([245]). There are examples of uniformly bounded C0 -cosine operator functions for which the family {Ea } from (6.1) is not defined. Proposition 6.2.5 ([245]). For a C0 -cosine operator function C (·,A), let family (6.1) be defined, and let Ea = Eb for certain 0 < a < b. Then (-b2 , -a2 ) R (A) P (A) = . Proposition 6.2.6 ([245]). Let the function Ea, x be bounded for a [0, a] and [0, ] with any a, 0, = 0. Then for al l x E and a [0, a], there exists Ea x, the operator Ea is bounded, and for al l 0 a b, the relation Ea Eb = Eb Ea = iEa holds; moreover, for almost al l a R+ , we have the 2 sin(at) C (t, A)xdt in the case where the integral converges. relation Ea x = 0 t Further, for 0 a b denote := (-b2 , -a2 ) and E := Eb - Ea . Proposition 6.2.7 ([245]). Under conditions of Proposition 6.2.6, for any two intervals 1 and 2 , we have E1 E2 = E2 E1 = E
1



2

.

Proposition 6.2.8 ([245]). Let E, x const for any R+ and R+ . Moreover, let Ea = Eb for certain 0 a b. Then (-b2 , -a2 ) (A). Proposition 6.2.9 ([245]). Under the conditions of Proposition 6.2.6, let the function Ea x be continuous at the point a0 R+ for any x E . Then -a2 R (A). 0/ Proposition 6.2.10 ([245]). Let the space E be reflexive and strongly convex with the Gateauxdifferentiable norm, A C (M, 0), and let the operator C (t, A) have a real spectrum for any t R. Then R (A) = . 118


Proposition 6.2.11 ([14]). Let A C (1, 0), and let A be the operator from Theorem 3.2.1. Then for t [ln 2, ), C (t, A)
B (E 1 ,E )

t · ln 2, S (t, A)

B (E,E 1 )

t +1,

and the resolvent (I -A)-1 satisfies the estimate R(, A) (Re - ln 2)-1 for Re > ln 2.

Proposition 6.2.12 ([6, 7]). Let 0 (A), and let A C (M, 0). Then sup S (t, A)
tR

dist(0, 2

(A)) sup C (t, A) .
tR

Proposition 6.2.13 ([6, 7]). For A C (M, 0), the set {x E : sup S (t, A)x < } is dense in E iff
tR
+

one of the fol lowing conditions holds: (i) s- lim n R(n ,A)x = 0 for any x E and a certain sequence n R+ such that lim n = 0; (ii) the set R(A) is dense in E ; (iii) N (A ) = {0}. Proposition 6.2.14 ([108]). An operator A C (M, 0) satisfies Condition (F) iff the fol lowing condition holds for each closed interval [a, b]: sup exp(tG|D
(G,µ) n n

) : µ N,t [a, b] < .

Proposition 6.2.15 ([177]). Let A C (M, 0). Then the operator iA generates an times integrated 1 group for > . 2 Proposition 6.2.16 ([139]). Let A C (M, 0), and let E = H . Then there exist a self-adjoint operator Q and a constant M > 0 such that ( 3(2M + 1))-1 I Q MI and the operator QC (t, A)Q-1 is self-adjoint for each t R. Moreover, C (t, A) = Q-1 cos(tL)Q and L = L 0, where L := QAQ-1 . Proposition 6.2.17 ([146]). For A C (M, 0) and x D(A), the fol lowing inequality holds: sup S (t, A)Ax
tR
+

2

4 sup C (t, A)Ax · sup C (t, A)x .
tR
+

tR+

Proposition 6.2.18 ([206]). Let A C (1, 0), and let
m

Cm (t) :=
j =0

2j C2m

t 2m

2j

A

j

2m t

4m

2m t

2

-2m

I -A

, (Cm (t) -

2j where C2m are binomial coefficients. Then lim Cm (t)x = C (t, A)x for al l x E and m C (t, A))x t2 Ax / m for al l x D(A), m 2.

119


Proposition 6.2.19 ([221]). The functions C (·,A)x and S (·,A)x are uniformly bounded for any x E iff there exists a constant M 1 such that for Re z , dk 2 (z I - A)-1 , dz k z dk Mk! dk-1 R(z 2 I - A)-1 + k k-1 (z 2 I - A)-1 k+1 , k dz dz |z | k N.

Proposition 6.2.20 ([177]). Let an operator A generate a C0 -cosine operator function such that for the corresponding C0 -sine operator function S (·,A) the estimate S (t, A) Mt, t R 3 operator iA generates an times integrated semigroup for > . 2 6.3. Asymptotics of the Functions F (·) and G(·) In this section, we study the asymptotic b ehavior of the C0 -families F (t) and G(t) as t . Let us consider the case under the assumption that there exist a real 0 and a nonzero b ounded op erator P B (E ) such that
t +

holds. Then the

lim 2e-0 t C (t, A)x = Px for all

x E.

(6.2)

Clearly, in this case, there exists a constant M1 1 such that C (t, A) M1 e0 By the identity 2e-0 2t (C (2t, A)+ I ) = 2e-0 t C (t, A)2e-0 t C (t, A), in case (6.2), the numb er 0 cannot b e negative, since then e-0 convergence. In the case 0 = 0, we have from (6.4) that P +2I = P 2 . On the other hand, setting t and s in the cosine equation (see (i) on p. 83), we have 2P = P 2 . Therefore, C (t, A) P/2 = I as t . Setting t in relation (i) on p. 83, we obtain C (s, A) = I for all s R. In connection with these simple arguments, we note that in [81], the assumption on the convergence of C (t, A) to P as t , which has no meaning, was made. We note by the way that in the case C (t, A) P as t , it follows from (iv) of Prop osition 2.4.1 ^ ^ that F (t)x = 2-1 t2 3 F ()x (for any > ), which never converges as t if x N (F ()). The same situation takes place for G(·). Therefore, the case 0 = 0 is not interesting, and in what follows, we will assume that 0 > 0. It is clear from (6.4) that the op erator P is a pro jection. It is known that a generator of a C0 -cosine op erator function C (·,A) also generates a C0 -semigroup exp(·A) defined by formula (2.12). 120
2t t

for t R+ .

(6.3)

(6.4)

as t , and there is no


As will b e shown in the following theorem, the convergence of 2e-0 t C (t, A) to P as t implies the convergence of e-0 t exp(tA) to P in the same top ology.
2

Theorem 6.3.1. Let condition (6.2) hold with 0 R+ . Then P is a projection with the range R(P ) = N (2 I - A) and the kernel N (P ) = R(2 I - A). If, moreover, P is of finite rank and 2e-0 t C (t, A) - 0 0 P 0 as t , then 2 > E (A), B (A) = {2 }, and 2 is a simple pole of the resolvent (I - A)-1 . 0 0 0 Proof. We prove that the C0 -semigroup e-0 t exp(tA) strongly converges to P as t . Then it
2

follows from the ergodic theorem (see [76]) that P is a pro jection with R(P ) = N (2 I - A) and N (P ) = 0 R(2 I - A). 0 We have e
-2 t 0

e-0 t exp(tA)x = 2 t
2

0

e

-

s2 4t

e

0 s

(2e

-0 s

e-0 t C (s, A) - P )xds + 2 t
2

0

e- 4t e0 s dsP x.

s2

The first term Q1 (t) on the right-hand side converges to zero, and the second term Q2 (t) converges to Px as t . Indeed, e-0 t 2 t
2

0

s2 1 e- 4t e0 s ds = 2 t

0

e-

(s-20 t)2 4t

1 ds =

-0 t

e-u du
2

1 -u2 e du = 1 as t . Therefore, Q2 (t) converges to Px as t . converges to - Let > 0 b e sufficiently small, and let R+ be so large that 2e-0 s C (s, A)x - Px s . Then the quantity e-0 t Q1 (t) 2 t
2

for all

0

e

-

s2 4t

e

0 s

(2e

-0 s

e-0 t C (s, A) - P )x ds + 2 t
2

0

e- 4t e0 s ds x ,

s2

and hence is b ounded by the constant 2 as t . That is, Q1 (t) 0 as t . If 2e-0 t C (t, A) - P 0 as t , then in a similar way, we prove that e-0 t exp(tA) - P 0
2

as t . When P is of finite rank, the semigroup exp(·A) attains the limit with the growth exp onent 2 . It follows from the theorem in [291] that 2 > E (A), B (A) = {2 }, and 2 is a simple p ole of the 0 0 0 0 resolvent (I - A)-1 . We need the following assertion. Lemma 6.3.1 (see [17, Lemma 7.3.1]). If a strongly continuous function f (·) : R
t +

E is such that

lim f (t) = , E , then for any with Re > 0, we have e-t
0 t

es f (s) ds /

for

t .

(6.5)

121


Proposition 6.3.1. Let a C0 -cosine operator function C (·,A) satisfy (6.2) with 0 > 0. Then 2e-0 t S (t, A) P/0 and 2e-0
t t 0

S (s, A)ds P/2 strongly as t . 0

Proof. We have the following relations: 2e-0 t S (t, A) = e-0 and 2e-0
t 0 t t 0 t

e0 s 2e-0 s C (s, A) ds

S (s, A)ds = e-0

t 0

t

e0 s e-0

s 0

s

e0 2e-0 C (, A) d ds.

Now the assertion follows from Lemma 6.3.1. Theorem 6.3.2 ([239]). Let a C0 -cosine operator function C (·,A) satisfy condition (6.2) with 0 R+ . Then for each > 0 and each x E , we have ^ s- lim 2e-0 t F (t)x = (2 /2 - 1)P F ()x, 0
t

1 ^ (As - A)x if x D(A) and F ()x N (2 I - A)-1 , simultaneously, and is equal to 0 2 0 ^ zero if F ()x R(2 I - A)-1 . 0 which is equal to Proof. For 0 R+ , let us write the relation 2e-0 t F (t)x = 2e-0
t 0 t

^ ^ S (s, A)3 F ()xds - 2e-0 t (C (t, A) - I )F ()x.

Using Prop ositions 6.3.1 and 2.4.1 (iv) and setting t , we obtain the required result. Analogously, using Prop ositions 6.3.1 and 2.4.1 (v), we obtain the following assertion. Theorem 6.3.3 ([239]). Let a C0 -cosine operator function C (·,A) satisfy condition (6.2) with 0 R+ . Then for each > 0 and each x E , we have
t

^ lim 2e-0 t G(t)x = G()(2 /2 - 1)Px, 0

which is equal to

1 (Ac - A)x if x N (2 I - A)-1 and is equal to zero if x R(2 I - A)-1 . 0 0 2 0

As was mentioned ab ove, if C (t, A) strongly converges as t , then C (·,A) I , and F (·) and G(·) grow. In what follows, we will consider the b ehavior of F (·) and G(·) under the assumption that sup t
t>0 -2 0 t 0 s

C (u, A) duds < and t

-2

C (t, A) 0

(6.6)

strongly as t . We need the following assertion. 122


Proposition 6.3.2 ([258]). Under assumption (6.6), we have (i) the mapping P : x lim 2t
t -2 ts 00 tsuv 0000

C (u, A)xduds is a projection with R(P ) = N (A), N (P ) =

R(A), and D(P ) = N (A) R(A); (ii) there exists x := - lim 2t
t -2

C (, A)yd dv duds iff y A(D(A) R(A)) (= R(A) in the

case where C (·,A) is (C, 2)-ergodic, i.e., D(P ) = E ). Moreover, this element x is a unique solution of ~ ~ the equation Ax = y in R(A), i.e., x = A-1 y , where A = A|R(A) . Using Prop osition 2.4.1 (iv) and the prop osition mentioned ab ove, we obtain the following theorem. Theorem 6.3.4 ([239]). Under assumption (6.6), the fol lowing assertions hold: (i) there exists the limit y = lim 2t
t -2

^ F (t)x iff F ()x N (A) R(A) for a certain (and al l) > .
-2 ts 00

^ When the limit exists, y = 3 P F ()x and is independent of ; ^ (ii) for F ()x N (A) R(A), z = lim 2t
t

^ F ( )xd ds does exist iff F ()x A(D(A) R(A))

~^ for a certain (and al l) > . In this case, z = -(2 I - A)A-1 F ()x, which is independent of . Proof. We have from (iv) of Prop osition 2.4.1 that 2 2 2 ^ F (t)x = 2 (I - C (t, A))F ()x + 2 t2 t t and 2 t2
t 0 0 s t 0 0 s

^ C (, A)3 F ()xd ds

(6.7)

F ( )xd ds = -

2 t2 2 t2

t 0 t 0

s 0 s 0 0

u 0

v

^ C (, A)3 F ()xd dv duds (6.8)

^ ^ C (, A)F ()xd ds + F ()x.

Then assertions (i) and (ii) follow from (6.7) and (6.8), resp ectively, as a consequence of Prop osition 6.3.2. By Prop ositions 2.4.1 (v), 6.3.2, and 2.4.2 (ii), we have the following theorem. Theorem 6.3.5. Under assumption (6.6), we have the fol lowing assertions: (i) if x N (A) R(A), then s- lim 2t
t -2

G(t)x = Ac Px;
-2 ts 00

(ii) if x A(D(A) R(A)), then s- lim 2t
t

~ ~ G( )xd ds = -(Ac - A)A-1 x = x - Ac A-1 x, where

~ A = A|R(A) .

Chapter 7
123


ERGODIC PROPERTIES Ergodic prop erties of op erator C0 -semigroups were considered, e.g., in [17, 20, 66, 76].

7.1. Standard Limits Proposition 7.1.1 ([146]). Let A C (M, 0). For any x = y +z R(A) N (A),
T

1 T
0 T

C (t, A)xdt - z = O(|T |-1 ),

1 T
0

S (t, A)xdt -

T z = O |T |-1 2

as

T .

Proposition 7.1.2 ([146]). In the case where A C (M, 0) and E is reflexive, we have E = R(A) N (A), 1T and, moreover, for each x E , we have the strong convergence of C (t, A)xdt to x as T . T0 The following definition for C0 -cosine op erator functions is analogous to Definition 7.1.8 in [17, p. 69] for C0 -semigroups. Definition 7.1.1. A C0 -semigroup exp(·A) is said to b e weakly (strongly, uniformly) (C, ) ergod ic at infinity if the op erator Ct [C (·,A)]x := t -t 0e Ct [C (·,A)]x dt t - t 0

(t - s)-1 C (s, A)xds does exist for all t > 0;

< for all x E and > max(0, (A)), and if the limit (C, )-

lim C (·,A) := lim Ct [C (·,A)] exists in the weak (strong, uniform) op erator top ology. This is the so-called

Cesaro limit. Definition 7.1.2. A C0 -cosine op erator function C (·,A) is said to b e weakly (strongly, uniformly) ergodic in the Abel sense if the limit


(A)- lim C (t, A) := lim
t 0+ 0

e-t C (t, A)dt lim 2 R(2 ,A)
0+

(7.1)

exists in the corresp onding op erator top ology. Setting t 0+ instead of t or instead of 0+, we obtain the definition of ergodicity at zero. Theorem 7.1.1 ([76]). If for a fixed 0, there exists the limit (C, )- lim x( ) = y E, then for , there exist the limits (C, )- lim x( ) = A- lim x( ) = y.


124


Proposition 7.1.3 ([259]). A C0 -cosine operator function C (·,A) given on the Grothendieck space E is strongly (C, 1)-ergodic iff the fol lowing conditions hold: (i) S (t, A) = O(t) (ii) s- lim t (iii) as t ; = 0 for al l s R+ ;
-1 C (t, A)S (s, A) t w - cl(R(A )) = R(A ).

Proposition 7.1.4 ([259]). Under the conditions of Proposition 7.1.3 with the space E having the Dunford­Pettis property, a C0 -cosine operator-valued C (·,A) is uniformly (C, 1)-ergodic iff condition (i) of Proposition 7.1.3 holds, w - cl(R(A )) = R(A ). We set T (t, A) := (t - s)C (s, A)ds and define Q2 as on the page 148. w
0 t

C (t, A)S (s, A)

= O(t) as t for each s R+ , and, final ly,

Proposition 7.1.5 ([259]). Let E be a Grothendieck space, and let K (t, Q) be a w -continuous C0 -cosine operator function. If T (t, A) = O(t2 ) as t and w - lim t then R(Q2 ) = N (Q), N (Q2 ) = R(Q), and D(Q2 ) = E . w w w Moreover, if s- lim t
t -2 t -2

K (t, Q)T (s) x = 0 for al l s R+ ,

K (t, Q)T (s) = 0 for al l s R+ then the C0 -cosine operator function K (·,Q)

is strongly (C, 2)-ergodic. Proposition 7.1.6 ([259]). A C0 -cosine operator function C (·,A) on the Grothendieck space E is strongly (C, 2)-ergodic iff (i) T (t, A) = O(t2 ) as t ; (ii) s- lim t (iii)
-2 C (t, A)T (s, A) t w - cl(R(A )) = R(A ).

= 0 for al l s R+ ;

Proposition 7.1.7 ([259]). Under the conditions of Proposition 7.1.6, let the space E have the Dunford­ Pettis property. In this case, a C0 -cosine operator function C (·,A) is uniformly (C, 2)-ergodic iff T (t) = O(t2 ), C (t, A)T (s, A) = O(t2 ) as t and s R+ , and also w -cl(R(A )) = R(A ). For any x E , we set Pc x := s- lim


t

1 S (t, A)x, t
n-1

Pa x := s- lim
0+ 0

e

-t

C (t, A)xdt,

and Pt x := s- lim

n

1 n

C (kt, A)x .
k =0

125


Proposition 7.1.8 ([257]). For A C (M, 0), the operators Pc and Pa coincide and are projections. We have the relations R(Pc ) = N (A) =
s>0

N (C (s, A) - I ),

N (Pc ) = R(A) =
s>0

R(C (s, A) - I ),

D(Pc ) =
s>0

N C (s, A) - I
s>0

R C (s, A) - I
n

= {x E : {tn },tn ,

w- lim (S (tn ,A)x)/t
n

does exist}.

Proposition 7.1.9 ([257]). Let A C (M, 0). For each t R+ , the operator Pt is a projection, and R(Pt ) = N (C (t, A) - I ), N (Pt ) = R(C (t, A) - I ),

D(Pt ) = N (C (t, A) - I ) R(C (t, A) - I ) = x E : {nk },nk , w- lim 1 nk
nk -1

k

C (lt, A)x
l=0

does exist .

Proposition 7.1.10 ([257]). Let A C (M, 0). Let there exist > 0 such that the operator C (t, A)+ I is invertible for t (0,) (in particular, this holds if C (t, A) - I < 2 for t (0,)). Then Pt = Pc for al l t (0, 2). Consider the op erator
Ht [C (·,A)]x =

t

t

s
0

-1

C (s, A)xds,

t > 0, x X.

(7.2)

Theorem 7.1.2 ([176]). Let C (·,A) be a bounded C0 -cosine operator function on a Banach space E . The fol lowing assertions are equivalent: (i) 0 C (A) (A);
(ii) the function C (·,A) is Ht -stable for al l > 0; (iii) lim Htn0 [C (·,A)] = 0 in the weak operator topology for a certain 0 > 0 and a certain positive n

sequence {tn } converging to . It is clear from the proof of this theorem that the b oundedness of C (·,A) is not necessary for the implication (iii) = (i). It is well known that a generalized solution of the abstract Cauchy problem u (t) = Au(t), t (-, ), u(0) = x, u (0) = y,

is given by the formula u(t) = C (t, A)x + S (t, A)y. 126


Theorem 7.1.3 ([176]). Let C (·,A) be a bounded C0 -cosine operator function on a Banach space E , and
assume that 0 C (A) (A). Then a generalized solution u(t) is Ht -stable for al l > 0, for al l x E ,

and for al l y from a certain dense subset of E. Theorem 7.1.4 ([257]). Let C (·,A) be a bounded cosine operator function on a Banach space E , and assume that 0 (A) C (A). Then the fol lowing conditions are equivalent: (i) y A(D(A) R(A)); (ii) x := - lim 2t
t -2 tsuv

C (, A)yd dv duds does exist;
0000 tm -2 m tm s u v

(iii) for a certain sequence {tm }, the weak limit x := -w- lim 2t exist.

C (, A)yd dv duds does
0 000

~ ~ Such an x is a unique solution of the equation Ax = y in R(A), i.e., x = (A)-1 y, where A = A|R(A) . Theorem 7.1.5 ([176]). Let C (·,A) be a bounded cosine operator function on a Banach space E . The fol lowing conditions are equivalent: (i) 0 (A) C (A);
(ii) for al l > 0, we have s- lim Ct [C (·,A)] = 0;

(iii) for a certain 0 > 0 and a certain positive sequence {tn } converging to as n , we have
w- lim Ctn0 [C (·,A)] = 0. n The assertion that lim Ct [u(·)] = 0, which is similar to Theorem 7.1.3 (for a generalized solution

t

u(·) of a second order equation) can b e proved in the same way. 7.2. Tauberian Theorem As was noted, for a C0 -cosine op erator function C (t, A), the notion of stability is vacuous b ecause the convergence C (t, A) P B (E ) as t implies C (t, A) I. It is clear that integrated semigroups or cosine functions do not have asymptotic convergence prop erties b ecause they naturally increase p olynomially [116]. On the other hand, one can consider the Cesaro averaging of cosine op erator functions. The necessary and sufficient condition for a b ounded C0 -cosine op erator function to b e (C, )-stable for any > 0, i.e., 1/t
t 0 -1

t

(t - s)

C (s, A)ds 0 strongly as t , is that 0 (A) C (A), as was seen in the previous

section. Therefore, the b ehavior of (C, )-averages is defined just by the p oint 0. It is known that for at least p olynomially b ounded C0 -semigroups, the b ehavior of some Cesaro typ e averages is determined by the b ehavior of the resolvent in a neighb orhood of zero [168]. We are going to show here that for C0 -cosine op erator functions, the situation is very similar. 127


To see that the b ehavior of Cesaro typ e averages is closely connected with the b ehavior of the resolvent of a p olynomially b ounded cosine op erator function in a neighb orhood of zero, we consider the basic example of n â n nilp otent matrix Q= 00 10 01 0 0 0 ... ... .. . 0 .. . 0 ... ... .. . .. . .. . 1 0 0 0 . 0 0 0

00 1 . . .. .. . .. 00 0

t2 t2(n-1) ; therefore, the C0 -cosine op erator Then Qn = 0 and cosh(t Q) = I + Q + ··· + Qn-1 2! (2n - 2)! function cosh(t Q), t 0, is certainly p olynomially b ounded. The resolvent of Q is given by (2 I - Q)-1 = -2 I - Hence, for = 2n - 2, we have
0+

Q 2

-1

= -2 I +

Q Q2 Qn-1 + 4 + ··· + 2(n-1) 2

.

lim 2+ (2 I - Q)-1 = Qn

-1

and 2+ (2 I - Q)-1 n for all || 1. From the p oint of view of Cesaro convergence, we have
t t+1

lim

1
0

t

cosh(s Q) ds =

Qn-1 , (2n - 1)!

and, more generally,
t t+m

lim

1
0

t

(t - s)m-1 cosh(s Q) ds =

(m) Qn ( + m +1)

-1

(7.3)

for all m = 1, 2,... . Let A b e the generator of a p olynomially b ounded cosine op erator function acting on a Banach space E , i.e., there exist numb ers M > 0 and 0 such that C (t, A) M (1 + t) Let P B (E ) b e a b ounded linear op erator. Theorem 7.2.1 ([169]). Let > 0, and let (7.4) hold. Then the conditions (i) +2 (2 I - A)-1 P in the strong operator topology as 0+ in R; 128 for all t 0. (7.4)


(ii) there exist C > 0,N 0 and 0 > 0 such that 2+ (2 ei2 - A)-1 C , cosN ()
t 0

0 < 0 , (-/2, /2),

are necessary and sufficient for the existence of a positive integer m such that
t

lim

(m + +1) (m)tm+

(t - s)m-1 C (s, A)xds = Px

(7.5)

for each x E . Remark 7.2.1. Since for an op erator A that generates a p olynomially b ounded distribution cosine, we can find the corresp onding p olynomially b ounded and strongly continuous m-times integrated cosine [80, 214, 215], the same theorem ought to b e valid for the distribution case. Remark 7.2.2. Assume that 2 (2 I - A)-1 P as 0+, i.e., (i) holds with = 0. Then we obtain from the Hilb ert identity that 2 µ2 (2 I - A)-1 (µ2 I - A)-1 = Now setting = 2 µ2 ((µ2 I - A)-1 - (2 I - A)-1 ). 2 - µ2 (7.6)

2µ and then passing to the limit as µ 0+, we obtain from (7.6) that P 2 = P, i.e., P

is a pro jection. In the case where > 0 in (i), the op erator P is no longer a pro jection. It follows from (7.6) that P has the prop erty P 2 = 0.

Chapter 8
UNIFORMLY BOUNDED C0 -COSINE OPERATOR FUNCTIONS As in the case of op erator semigroups, the norm-continuity is a very restrictive requirement for cosine op erator functions, since it implies the b oundedness of the generator. Conditions for generation of a C0 cosine op erator function for an infinitesimal op erator A are more restrictive than those for generation of op erator semigroups. 8.1. Norm-Continuity It is very natural that the b oundedness of A in the case of a C0 -cosine op erator function follows under weaker additional assumptions than in the case of op erator semigroups. So, for example, the condition tA exp(-tA) C implies the b oundedness of A for C = 1/e (see [17]), and in the case of cosine op erator functions, for b oundedness of A, the b oundedness AS (t, A) const with any constant is sufficient. Definition 8.1.1. A C0 -cosine op erator function C (·,A) is continuous in the uniform op erator top ology (norm-continuous) if the function C (·,A) : R B (E ) is continuous in the op erator norm. 129


Proposition 8.1.1 ([273]). Let a C0 -cosine operator function C (·,A) be continuous in the uniform operator topology. Then A B (E ) and


C (t, A) =
k =0

t2k Ak /(2k)!,

t R,

(8.1)

and, moreover, the series uniformly converges in t on each finite closed interval [0,T ]. Sometimes, in the literature, series (8.1) is written as cosh(t A) analogously to the scalar case. We note that a C0 -sine op erator function S (·,A) is always uniformly continuous in t R, as follows from its definition. Proposition 8.1.2 ([272]). Let A B (E ). Then series (8.1) is a C0 -cosine operator function whose generator is A. Theorem 8.1.1 ([87]). Each of the fol lowing conditions is equivalent to the norm-continuity of C (·,A): (i) lim C (t, A) - I = 0;
t0 -1

(ii) lim t
t0

S (t, A) - I = 0;

(iii) the generator A is bounded; (iv) R(C (t, A)) E 1 for al l t (, ) with certain < ; (v) the inclusion R(S (t, A)) D(A) and the strong continuity of the function t AS (t, A) hold for al l t (, ) with certain < . Proposition 8.1.3 ([7]). The generator A of a C0 -cosine operator function with a non-quasi-analytic weight is bounded iff one of the fol lowing conditions holds: (i) for a certain > 0, we have sup
0
C (t, A) - I < 1;

(ii) the C0 -cosine operator function C (·,A) is the restriction to R of an entire operator function ~ C (·) : C B (E ) of exponential type (equal to r (A)). Proposition 8.1.4 ([273]). Let a C0 -cosine operator function C (·,A) be twice strongly differentiable at zero. Then A B (E ). Proposition 8.1.5 ([206]). Let A C (M, 0). Then A B (E ) iff the spectrum (A) is bounded. Proposition 8.1.6. Let a C0 -cosine operator function C (·,A) be norm-continuous. Then 2t (i) lim 2 S (s, A)ds - I = 0; t0 t 0
h

(ii) for sufficiently smal l h, the operator
0

S (s, A)ds has a bounded inverse;

130


(iii) for sufficiently smal l h, we have the relation
h -1

A = (C (h, A) - I )
0

S (s, A)ds

.


Definition 8.1.2. A Grothendieck space is a Banach space in which every w -convergent sequence in E is w-convergent.

Definition 8.1.3. We say that a Banach space E has the Dunford­Pettis property if xn ,x 0 whenn ever xn weakly converges to zero in E and x weakly converges to zero in E . n Proposition 8.1.7 ([259]). Any C0 -cosine operator function C (·,A) given on a Grothendieck space with the Dunford­Pettis property (for example, E = L is such a space) is norm-continuous, i.e., A B (E ). Before formulating the next assertion, we recall [147] that if E is an H.I. space and B B (E ), then there exists a unique p oint B (B ) such that the op erator B - B I is strictly singular. Moreover, B - B I is a Riesz op erator. Proposition 8.1.8 ([248]). Let E be an H.I. Banach space, and let C (·,A) satisfy condition (7.4). Then A B (E ), and there is a positive integer m such that (A - A I )m is a compact operator. Proposition 8.1.9 ([7]). If B B (E ) and a C0 -cosine operator function C (·,B ) is uniformly bounded in t R, then the fol lowing Bernshtein inequality holds: B r (B ) · sup C (t, B ) ,
tR

where r (B ) is the spectral radius of the operator B . Proposition 8.1.10 ([27, 242]). Let B B (E ). Then the functions C (t, -B 2 )x and S (t, -B 2 )y are uniformly bounded in t R for any x, y E iff there exists an equivalent norm exp(itB )


·



on E such that

1 for t R (such operators B are said to be Hermitian-equivalent on E ).

Proposition 8.1.11 ([27, 242]). An operator B B (E ) is Hermitian-equivalent on E iff there exists a constant C > 0 such that sin(tB ) C, t R. 131


Proposition 8.1.12 ([86]). In a Banach algebra B with unit, let a C0 -cosine operator function C (·,A), A B be given. Then for 2 > sup (|| +Re )/2, we have
(A) +i

1 C (t, A) = 2i

et R(2 ,A)d,
-i +i

t R+ ,

1 S (t, A) = 2i

et R(2 ,A)d,
-i

t R+ .

Proposition 8.1.13 ([100]). If for a C0 -cosine operator function R(S (t, A)) D(A) for t R and AS (t, A) const for t [a, b], a < b, then A B (E ). Proposition 8.1.14 ([100]). If for a C0 -cosine operator function SV(C (·,A),t) const for a certain t R+ , then A B (E ). 8.2. Positivity of Perturbation Families Definition 8.2.1. In the case where E is a Banach lattice with a p ositive cone E + , we say that a function L(·) is positive on E if for each t R+ , the op erator L(t) is p ositive (we write L(t) L(t)E + E + . In the case where E is a Hilb ert space with the inner product (·, ·), we say that L(·) is p ositive (we write L(t) 0) if for each t R+ , the op erator L(t) is p ositive in the sense that (L(t)x, x) 0 for all x E. Proposition 8.2.1 ([197]). A C0 -cosine operator function C (·,A) dominates I , i.e., C (·,A)- I is positive in the sense of a Banach lattice or in the sense of a Hilbert space iff the generator A is bounded and positive. The following prop ositions are a reformulation of prop erties of a C0 -family of multiplicative p erturbations and a C0 -family of additive p erturbations.
µ Let FB (·) and Gµ (·) b e functions defined for B B (E ) by the formulas B µ FB (t)x := (A - µI ) t

0) in the sense that

S (s, A)Bx ds,
0 t

x E, t R+ , (8.2) x E, t R+ .

Gµ B

(t)x := B (A - µI )
0

S (s, A)xds,

µ Then FB (·) is a C0 -family of multiplicative p erturbations and Gµ (·) is a C0 -family of additive p erturbaB

tions. 132


Proposition 8.2.2 ([239]). Let E be a Banach lattice. Each C0 -family of multiplicative perturbations
µ FB (·) for a C0 -cosine operator function C (·,A) on E defined in (8.2) with µ 0 and B

0 is positive iff

the operator A is positive. The same holds for a C0 -family of additive perturbations.
µ Proposition 8.2.3 ([239]). Let E be a Hilbert space. Each C0 -family of multiplicative perturbations FB (·)

for a C0 -cosine operator function C (·,A) on E defined in (8.2) with µ 0 and B 0 that commutes with C (·,A) is positive iff the operator A is positive. The same holds for a C0 -family of additive perturbations.

Chapter 9
ALMOST-PERIODIC C0 -COSINE OPERATOR FUNCTIONS Let us recall in brief the definition of almost p eriodicity of op erator functions. Definition 9.0.1. A function f (·) : R exists l R
+

E is said to b e almost-periodic if for each

> 0, the set

J (f, ) = { > 0 : f (t + ) - f (t)
+

for all t R+ } is relatively dense in R+ . That is, there
+

such that each subinterval from R

of length l intersects J (f, ). An op erator function

Q(·) : R+ B (E ) is said to b e almost-periodic if for each x E , the function Q(·)x is almost p eriodic. 9.1. Almost Periodicity of the Basic Families Definition 9.1.1. A C0 -cosine op erator function or a C0 -sine op erator function are said to b e almostperiodic (a.-p.) or uniformly a.-p. if for any x E , the functions C (·,A)x or S (·,A)x are a.-p. (uniformly a.-p.). Proposition 9.1.1 ([103]). If E is weakly sequential ly complete, then a weakly a.-p. C0 -cosine operator function is almost-periodic. Theorem 9.1.1 ([62]). A C0 -cosine operator function C (·,A) is almost-periodic iff the fol lowing three conditions hold: (i) the C0 -cosine operator function C (·,A) is uniformly bounded; (ii) the spectrum (A) R- ; (iii) the set of eigenvectors of the generator A is total on the space E . If, moreover, µ (A) is an isolated point of the spectrum, then µ is a simple pole of the resolvent (I - A)-1 and E = R(µI - A) N (µI - A). Theorem 9.1.2 ([62]). The Cauchy problem (3.1) has an a.-p. generalized solution for any u0 ,u1 E iff conditions (i)­(iii) of Theorem 9.1.1 hold and 0 (A). 133


Theorem 9.1.3 ([62]). A C0 -cosine operator function C (·,A) and a C0 -sine operator function S (·,A) are uniformly a.-p. iff the fol lowing three conditions hold: (i) the C0 -cosine operator function C (·,A) and the C0 -sine operator function S (·,A) are uniformly bounded in t R; (ii) the set (A) is a harmonic subset in R- and 0 (A); (iii) the set of eigenvectors of the generator A is total in the space E . Proposition 9.1.2 ([62]). If a C0 -cosine operator function C (·,A) is uniformly a.-p., then (A) consists of simple poles of the resolvent (I - A)-1 . In this case, (A) = P (A). Proposition 9.1.3 ([161]). The fol lowing conditions are equivalent: (i) a C0 -cosine operator function C (·,A) is periodic as an operator function; (ii) the C0 -cosine operator function C (·,A) is strongly periodic; (iii) the C0 -cosine operator function C (·,A) is weakly periodic. Theorem 9.1.4 ([141, 205]). A uniformly bounded C0 -cosine operator function C (·,A) is periodic with period 2 iff the fol lowing three conditions hold: (i) the spectrum (A) {l : l = -k2 ,k Z}; (ii) the spectrum (A) consists of simple poles of the resolvent; (iii) the set of eigenvectors of the generator A is total in the space E . Under conditions (i)­(iii), the Riesz projections are given by the formulas 2 1 cos(ks)C (s, A)xds 0 P (-k2 )x = 1 2 C (s, A)xds 2 0 and, moreover, for x D(A), we have the relation


for for

k = 0, k=0

C (t, A)x =
k =0

cos(kt)P (-k2 )x,

(9.1)

where the series converges uniformly in t R. Proposition 9.1.4 ([141]). In the case where E = H and C (·,A) - 2 is -periodic, relation (9.1) holds for al l x E and the convergence of the series is uniform in t R. Theorem 9.1.5 ([63]). The function C (t, A)u0 + S (t, A)u1 is 2 -periodic for any u0 ,u1 E iff conditions (i)­(iii) of Theorem 9.1.4 hold and 0 (A). 134


Proposition 9.1.5 ([141]). A C0 -cosine operator function C (·,A) is periodic with period T iff the func~ tion F (z ) := (1 - e-Tz )zR(z 2 ,A) can be analytical ly continued up to an entire function F (z ) such that the fol lowing estimate holds for |z | > r : ~ F (z ) Me(q
|z |
2-

)

,

where

q, M , r, > 0.

(9.2)

Proposition 9.1.6 ([221]). The uniformly wel l-posed Cauchy problem (3.1) has periodic solutions with period T iff A C (M, 0) and the function F (z )/z can be analytical ly continued up to an entire function Q(z ) such that estimate (9.2) for Q(z ) holds for |z | > r . Proposition 9.1.7 ([139]). Let a C0 -cosine operator function C (·,A) be given on a Hilbert space H , and let C (·,A) be weakly a.-p. Then we have the relation C (t, A) = Q-1 C (t, V )Q, where V is a self-adjoint operator given by V :=
0

P () and P () is a set of mutual ly orthogonal projections.

Proposition 9.1.8 ([205]). The periodicity of a C0 -cosine operator function C (·,A)x for each x D(A) implies the periodicity of C (·,A). Proposition 9.1.9 ([7]). Let (-A) R
+

be a not more than countable set. Then al l solutions of

problem (3.1) are almost-periodic iff the fol lowing conditions hold: (i) the C0 -cosine operator function C (·,A) is uniformly bounded in t R; (ii) 0 (A); (iii) for each limit point 0
n

(A), there is a sequence n R converging to zero as n such

that s- lim n (n + i0 )((n + i0 ) · I - A)-1 x = 0 for each x E . Proposition 9.1.10 ([7]). A C0 -cosine operator function C (·,A) is a.-p. in the uniform operator topology iff it is uniformly bounded on R and (A) is a harmonic subset of iR. (-A) have no limit points in R+ . Then:

Proposition 9.1.11 ([7]). Let A C (M, ), and let

(i) the linear span of eigenvectors and root vectors of A is dense in E if there exists a function (t) such that C (t, A) (t)
t

and

(t) C (1 + |t|)

for

t R, 0;

(9.3)

(ii) under the condition lim (t)/t = 0, where (·) is the function from (9.3), the C0 -cosine operator function C (·,A) is periodic with period 1 iff (A) {-(2k)2 : k N}. In [160], asymptotic almost-p eriodic in the sense of Stepanov op erator semigroups and cosine op erator functions were considered. 135


9.2. Almost-Periodicity of the Families F (·) and G(·) Proposition 9.2.1. If a continuous function f (·) : R+ E converges to a certain element E as t , then 2t
-2 0 t

sf (s) ds as t .

(9.4)

Proof. It is clear that as in the case of Lemma 6.3.1 (see also [17, Lemma 7.3.1]), it suffices to consider the case = 0. We set t = + and write 2 t2 Since 2 ( + )2
+ t

sf (s) ds =
0

2 ( + )2



sf (s) ds +
0

2 ( + )2

+

sf (s) ds.


(9.5)

sf (s)ds sup f (t)
t

for all and and f (t) 0 as t , we can choose so large that the second term in (9.5) b ecomes less than a certain > 0. Then we can choose so large that the first term in (9.5) also b ecomes less than . This proves (9.4) with = 0. The next theorem yields necessary and sufficient conditions for each C0 -family of multiplicative p erturbations (or each C0 -family of additive p erturbations) to b e almost-p eriodic. Theorem 9.2.1 ([239]). Each C0 -family of multiplicative perturbations F (·) for C (·,A) is almost periodic iff C (·,A) is almost periodic and 0 (A). The same assertion holds for a C0 -family of additive perturbations. Proof. Let C (·,A) b e almost-p eriodic. Then the condition 0 (A) implies the almost-p eriodicity of the function
t t

S (s, A) ds =
0 0

S (s, A)AA-1 ds = (C (t, A) - I )A-1 , t R.

Therefore, Prop osition 2.4.1 (iv) implies the almost p eriodicity of F (·). Conversely, if each C0 -family of multiplicative p erturbations is almost-p eriodic, then two particular C0 -families of multiplicative p erturbations C (t, A) - I and x N (A), then x = C (s, A)x -
s 0 t

S (s, A)ds are almost-p eriodic functions. If
0 -2 t 0

S (u, A)Axdu = C (s, A)x for all s R+ and x = 2t

S (s, A)xds 0

as t , since an almost-p eriodic function is b ounded. Therefore, A is injective. Then since an almost1su p eriodic function is ergodic (see, e.g., [78, p. 21]), the limit S (v, A)xdv du does exist as s for s0 0 each x E. By Prop osition 9.2.1, the limit 2 t2 136
t

s
0

1 s

s 0 0

u 0

v

C (, A)xd dv duds


exists as t for any x E . Since C (·,A) is uniformly b ounded, we have from Prop osition 6.3.2 (i.e., [240, Theorem 3.7]) that R(A) = E . Therefore, 0 (A). Remark 9.2.1 ([62]). The assumptions that C (·,A) is almost p eriodic and 0 (A) are equivalent to the condition that each mild solution of the Cauchy problem (3.1) is almost-p eriodic. We can deduce the following theorem from Theorem 9.2.1. Theorem 9.2.2. Each C0 -family of multiplicative perturbations F (·) for C (·,A) is periodic iff C (·,A) is periodic and 0 (A). In this case, F (·) and C (·,A) have the same period. The same assertion holds for a C0 -family of additive perturbations.

Chapter 10
COMPACTNESS IN THE THEORY OF C0 -COSINE OPERATOR FUNCTIONS Compactness prop erties are widely used in various asp ects of the theory of resolving families. We denote by B0 (E ) (or B0 (E, F ) in the case of distinct spaces) the set of compact op erators acting on E . 10.1. Compact Basic Families Definition 10.1.1. A C0 -cosine op erator function C (·,A) is said to b e compact (we write C (·,A) B0 (E )) if the op erator C (t, A) B0 (E ) for any t R+ . A C0 -sine op erator function S (·,A) is said to b e compact if the op erator S (t, A) B0 (E ) for any t R. Proposition 10.1.1 ([273]). If an operator C (t, A) B0 (E ) for each t (, ) for certain < , then the C0 -cosine operator function C (·,A) B0 (E ) and the C0 -sine operator function S (·,A) B0 (E ). Proposition 10.1.2 ([273]). If an operator S (t, A) B0 (E ) for each t (, ) and for certain < , then the C0 -sine operator function S (·,A) B0 (E ), t R. Proposition 10.1.3 ([64]). If dim E = , then for no t0 > 0, the operators C (t0 ,A) and C (2t0 ,A) can be compact simultaneously. Proposition 10.1.4 ([273]). Under the condition of Proposition 10.1.1, we necessarily have dim E < . Theorem 10.1.1 ([273]). The fol lowing conditions are equivalent: (i) a C0 -sine operator function S (·,A) B0 (E ); (ii) the resolvent (2 I - A)-1 B0 (E ) for any with Re > c (A). 137


Theorem 10.1.2 ([64]). The fol lowing conditions are equivalent: (i) a generator A B0 (E ); (ii) the operator 2 (2 I - A)-1 - I B0 (E ) for each > c (A); (iii) the operator S (t, A) - tI B0 (E ) for any t R; (iv) the operator C (t, A) - I B0 (E ) for any t R. Proposition 10.1.5 ([159]). Let C (t, A) - I B0 (E ) for any t R. Then (I - A)-1 - µ(µI - A)-1 B0 (E ) for al l , µ (A) such that Re , Re µ > c (A). Proposition 10.1.6 ([159]). Let C (t, A) - I B0 (E ) for each t (, ) and for certain < . Then C (t, A) - I B0 (E ) for any t R. Proposition 10.1.7 ([159]). If S (t, A) - tI B0 (E ) for each t (, ) and for certain < , then S (t, A) - tI B0 (E ) for any t R. Proposition 10.1.8 ([61]). Let a C0 -sine operator function S (t, A) and the operator function C (t, A) - I be compact for each t R. Then the space E is necessarily finite-dimensional. Proof. By our assumptions, it follows from Theorems 10.1.1 and 10.1.2 that the resolvent (I - A)-1 and the op erator (I - A)-1 - I are compact for certain = 0. This implies that I is a compact op erator, i.e., E is finite-dimensional. Let us define the sets NB I0 = {t > 0 : the op erator C (t, A) has no b ounded inverse}, NB I1 = {t > 0 : the op erator S (t, A) has no b ounded inverse}. Proposition 10.1.9 ([159]). Let C (t, A) - I B0 (E ) for al l t R. Then the sets NB I0 and NB I1 either are empty, simultaneously, or are infinite of continuum cardinality, and there exist constants 0 ,1 > 0 such that NB Ij (j , ),j = 0, 1. 10.2. Compactness of the Difference of Cosines Many linear distributed parameter control systems can b e reduced to the form v (t) = Av (t)+ Bu(t), v(0) = v0 , t R+ , (10.1)

where A generating a C0 -semigroup on the state Hilb ert or Banach space E and B is the control op erator acting from control space to the state space. When we design a feedback control u(t) = Fv (t) for some 138


feedback op erator from the state space to the control space, the closed-loop system b ecomes v (t) = (A + BF )v (t), v(0) = v0 , t R+ . (10.2)

In the context of the stabilization theory, we want to choose a feedback op erator F in order to force the closed-loop system to p ossess stability prop erties that are not enjoyed by the original system. One imp ortant class in physical applications is that of op erators F such that BF is compact on the state space. When BF is compact, it was first proved in [290] that the difference of the semigroups exp(A + BF )t and exp(tA) is compact for any p ositive t. Hence E (A) = E (A + BF ), (10.3)

where E stands for the essential growth rate of the associated semigroup. Prop erty (10.3) holds for any two C0 -semigroups whenever their difference is compact for some t > 0 (see Theorem 3.52 in [202]). This is the basis of the compactness method that was used in studying the stabilization of elastic systems (see [247]) and the sp ectral prop erty of the transp ort equation (see [283]). The compactness method was first formulated in [276] for Hilb ert spaces, and later on, it was generalized to Banach spaces in [151]; it says that a compact p erturbation cannot make the system exp onentially stable if it is asymptotically but not exp onentially stable. This leads to the general study of necessary and sufficient conditions for compactness of the difference of two C0 -semigroups. A recent result in [190] says that exp(tA) - exp(tB ) is compact for some t > 0 iff R(; A) - R(; B ) is compact under the norm-continuity assumption. On the other hand, it is more convenient to write most of control hyp erb olic systems in the form of a second-order system instead of a first-order evolution equation in an abstract space (see [20, 251]): v (t) = Av(t)+ Bu(t), v(0) = v0 , v (0) = v1 , t R+ . (10.4)

System (10.4) can b e transferred into the first order equation (3.5); however, there are some problems, since A does not, in general, generate a C0 -semigroup on E â E . In this connection, the problem of compactness of the difference of two C0 -cosine op erator functions is of interest. Let C (t, A) and C (t, B ) b e two cosine functions on a Banach space E , satisfying C (t, A) , C (t, B ) Mew|t| , t R, for some constants M, w 0. Denote A,B (t) = C (t, A) - C (t, B ) for all t R. Theorem 10.2.1 ([191]). Let
A,B

(t) be norm-continuous for t > 0. Then for al l > w2 , the operator (t) is compact for t 0.

R(; A) - R(; B ) is compact iff

A,B

We can characterize the norm-continuity in Hilb ert space analogously to Theorem 2.5 in [190] and the proof is just a simple modification. 139


Proposition 10.2.1. Let A and B generate cosine functions C (t, A) and C (t, B ), respectively, on a Hilbert space H , and let C (t, A) , C (t, B ) Met for some constants M 1, R. Then C (t, A) - C (t, B ) is norm-continuous for t > 0 iff for every > ,
|r |

lim

( + ir )(R(( + ir )2 ,A) - R(( + ir )2 ,B )) = 0

and
n n n n

lim

( ± ir )(R(( ± ir )2 ,A) - R(( ± ir )2 ,B ))x 2 dr = 0,

lim

( ± ir )(R(( ± ir )2 ,A ) - R(( ± ir )2 ,B ))y 2 dr = 0

uniformly for x H, y H with x , y 1. Theorem 10.2.2 ([191]). Let S (t, A) and S (t, B ) be the sine functions of C (t, A) and C (t, B ), respectively. Then S (t, A) - S (t, B ) is compact for t > 0 iff R(; A) - R(; B ) is compact for > w2 . Now we prove a similar result for the cosine as in [190, Prop osition 2.7]. Proposition 10.2.2 ([191]). Suppose that Then
h 0 A,B

(t) is compact for t > 0 and norm-continuous at t = 0.

lim A,B (t + h) - 2

A,B

(t)+ A,B (t - h) = 0 for any t 0.

(10.5)

Proof. We have =
A,B

(t + h)+ A,B (t - h) - 2

A,B

(t)
A,B

C (t + h, A)+ C (t - h, A) - C (t + h, B )+ C (t - h, B ) - 2
A,B

(t)

= 2C (t, A)C (h, A) - 2C (t, B )C (h, B ) - 2

(t)

= 2 C (t, A)C (h, A) - C (h, A)C (t, B ) +2 C (h, A)C (t, B ) - C (t, B )C (h, B ) - 2A,B (t) = 2C (h, A)A,B (t)+ 2A,B (h)C (t, B ) - 2
A,B

(t) as h 0.

= 2[C (h, A) - I ]A,B (t)+ 2A,B (h)C (t, B ) 0

Remark 10.2.1. In the proof of Theorem 10.2.1, we actually used (10.5), but not the norm-continuity of A,B (·). Theorem 10.2.3 ([191]). Let
A,B

(t) be norm-continuous in t at 0. Then
2

A,B

(t) is compact for t > 0

iff R(; A) - R(; B ) is compact for > w and (10.5) holds. 140


Proposition 10.2.3 ([191]). Suppose that the assumptions of Theorem 12.2.1 (resp. Theorem 12.3.1) below hold. A,
(I +B )A

Then

A,A(I +B )

(t) (resp.

A,

(I +B )A

(t)) is compact for t > 0 iff

A,A(I +B )

(t) (resp.

(t)) satisfies (10.5) and R(; A) - R(; A(I + B )) (resp. R(; A) - R(;(I + B )A)) is compact

for large enough. We complete this section by comparing the results on the difference of semigroups and cosine op erator functions. We first consider b ounded p erturbations. It is well known that if A generates a C0 -semigroup exp(tA), then A + B , B B (E ), also generates a C0 -semigroup exp(t(A + B )). It was shown in [302] that exp(t(A + B )) - exp(tA) is norm-continuous for t > 0 if it is compact for t > 0. As for the cosine case, the compactness hyp othesis can b e removed. Theorem 10.2.4 ([191]). Let A be a generator of a cosine function C (t, A), and let B B (E ). Then
A+B,A

(t) is norm-continuous in t R.

Combining Theorems 10.2.1 and 10.2.4, we have the following. Theorem 10.2.5 ([191]). Let B B (E ), and let A generate a cosine function. Then compact for t > 0 iff R(; A + B ) - R(; A) is compact for large enough. Proposition 10.2.4 ([191]). Suppose that C0 -semigroups exp(tA) and exp(tB ) commute and D(B ) D(A). Assume that exp(tB ) is a C0 -group. If (t) := exp(tA) - exp(tB ) is compact for al l t > 0, then A = B + K , where the operator K is compact. Proof. One can write exp(-tB )(t) = exp(-tB )exp(tA) - I, t R+ . By the assumption, the op erator exp(-tB )exp(tA) - I is compact for any t > 0, and, moreover, exp(-tB )exp(tA) is a C0 -semigroup with the generator A - B . It follows from [111] that the op erator A - B is compact.
A+B,A

(t) is

For cosines with b ounded generators, we have the following characterization. Proposition 10.2.5 ([191]). Let B B (E ). Then Proof. The compactness of
A,B A,B

(t) is compact iff A - B is compact.

(t) for any t > 0 implies (see [284]) that R(µ, A) - R(µ, B ) is compact

for some µ. In such a case, the op erator I - (µI - B )R(µ, A) is compact. This means that (µI - B )R(µ, A) is a Fredholm op erator of index 0, i.e., it has a closed range R((µI - B )R(µ, A)) = E. Since µI - B is one-to-one on E , we obtain R(R(µ, A)) = E. By the Banach theorem, µI - A is b ounded. Since the t 2t op erator A is b ounded, we have 2 S (s, A) ds - I 0 as t 0. Hence the op erator S (s, A) ds is t0 0 141


invertible. If A,B (t) is compact, then B (S (s, B ) - S (s, A)) ds is also compact. Now from
0

t

A,B (t) = (A - B )
0

t

S (s, A) ds - B
0

t

(S (s, B ) - S (s, A)) ds

it follows that the difference A - B is a compact op erator. Conversely, if A - B is a compact op erator, then A is b ounded and R(, A) - R(; B ) = R(; A)(B - A)R(, B ) is compact; therefore, the compactness of
A,B

(t) follows from Theorem 10.2.1.

In the following example b oth A and B generate C0 -semigroups and C0 -cosine functions. The op erator exp(tA) - exp(tB ) is compact for all t > 0, but C (t, A) - C (t, B ) is not compact. Example 10.2.1 ([191]). Let E = l1 and {en } b e the standard basis for it, i.e. en = (0, ..., 0, 1, 0, ..., 0,...), where 1 is in the nth coordinate. Let


Ax :=
n=1

-n(x, en )en ,

B x :=
n=1 i=1

-(n + n2 )(x, en )en , |xi |. Then the C0 -semigroups generated by


where x = (x1 ,x2 , ..., xn , ...) and (x, en ) = xn , x them are exp(tA)x =
n=1

l

1

=

e-nt (x, en )en ,

exp(tB )x =
n=1

e-(n

+n2 )t

(x, en )en

and the C0 -cosine functions are given by the formulas


C (t, A)x =
n=1

cos(nt)(x, en )en ,

C (t, B )x =
n=1

cos(n + n2 )t(x, en )en .

Since e-nt - e-(n

+n2 )t

0 as n , the op erator exp(tA) - exp(tB ) can b e approximated in norm by a
N

sequence of op erators SN (t)x =

(e-nt - e-(n
+1

+n2 )t

)(x, en )en with ranges of finite dimension. Therefore,

the op erator exp(tA) - exp(tB ) is compact for t 0. However, C (t, A) - C (t, B ) is not compact. Indeed,
2 take t = /2 and choose {yk } := {(nk j )} l1 , where i is the Kronecker delta. Now, for n = 2k +1, we

n=1

obtain nt = k + /2 and (n + n2 )t = 2k2 +3k + . Therefore, we have cos(nt) - cos((n + n2 )t) = ±1; if k can b e divided exactly by 2, then we choose +; otherwise, we choose -. Thus,
2 (C (/2,A) - C (/2,B ))yk = {[cos(nt) - cos(n + n2 )t]nk +1 2 } = {±nk +1

},

which means [C (/2,A)-C (/2,B )](yk -ym )

l

1

= 2 for k = m. Therefore, we cannot choose a convergent

subsequence from the sequence {[C (/2,A) - C (/2,B )]yk }. Also, we can see that C (t, A) - C (t, B ) is 142


not norm-continuous in t. Indeed, for each t > 0, define sk = t + [C (t, A) - C (t, B )] - [C (sk ,A) - C (sk ,B )] =

1 . Then sk t as k and k + k2

B (l1 )


{[cos(nt) - cos(nsk )] - [cos(n + n2 )t - cos(n + n2 )sk ]} n=1 k (sk + t) sin 2 k (sk + t) sin 2 k (sk - t) - sin 2 1 2(1 + k)

l

2 sin = 2 sin It is clear that sin

k + k2 (sk + t) sin 2 1 . 2

k + k2 (sk - t) 2

- sin((k + k2 )t +1/2) sin

1 0 as k . But sin((k + k2 )t +1/2) does not converge to 0 as k 2(1 + k) for every t > 0! To prove this, we supp ose the contrary: sin((k + k2 )t + 1/2) 0 as k . Then sin((k +1 +(k +1)2 )t +1/2) 0 as k . Now, since sin((k +1 +(k +1)2 )t +1/2) = sin((k + k2 )t +1/2+2(k +1)t) = sin((k + k2 )t +1/2)) cos(2(k +1)t)+ cos((k + k2 )t +1/2)) sin(2(k +1)t),

we obtain sin(2(k +1)t) 0 as k , since cos((k + k2 )t +1/2) cannot converge to 0 according to the relation sin2 x + cos2 x = 1. Hence sin(2(k + 1 + 1)t) 0 as k . Thus, from sin(2(k + 1 + 1)t) = sin(2(k +1)t)cos (2t)+ cos(2(k +1)t)sin (2t) we obtain sin(2t) 0 as k . Therefore, we have t = n /2 for some n N. But for such t, we find that sin((k + k2 )t + 1/2)) = ± sin(1/2), which contradicts our assumption of the convergence to 0. This means that C (t, A) - C (t, B ) is not norm-continuous. The converse does not hold: we have the following prop osition. Proposition 10.2.6 ([191]). Suppose that A and B generate cosine functions C (t, A) and C (t, B ). If C (t, A) - C (t, B ) is compact for t > 0, then exp(tA) - exp(tB ) is compact. Moreover, if C (t, A) - C (t, B ) is norm-continuous for t > 0, then C (t, A) - C (t, B ) is compact for t > 0 iff exp(tA) - exp(tB ) is compact. Proof. Since C (t, A) - C (t, B ) is compact, it follows from [284] that S (t, A) - S (t, B ) =
t 0

(C (s, A) -

C (s, B ))ds is also compact, which implies that R(; A) - R(; B ) is compact by Theorem 10.2.2. Moreover, since b oth exp(tA) and exp(tB ) are analytic, exp(tA) - exp(tB ) is norm-continuous, and the compactness of exp(tA) - exp(tB ) follows from Theorem 2.3 of [190]. If, in addition, C (t, A) - C (t, B ) is normcontinuous, then the compactness of exp(tA) - exp(tB ) implies that R(; A) - R(; B ) is compact. Now the compactness of C (t, A) - C (t, B ) follows from Theorem 10.2.1.

The following prop osition extends Prop osition 2.7 from [190]. 143


Proposition 10.2.7 ([191]). Suppose that D(A) D(B ), where A generates an analytic C0 -semigroup exp(tA) and B is a generator of a C0 -semigroup exp(tB ). If (t) := exp(tA) - exp(tB ) is compact for t > 0, then (t) is norm-continuous for t 0. 10.3. Compactness of the Families F (·) and G(·) Definition 10.3.1. A C0 -family of multiplicative p erturbations F (·) (resp. a C0 -family of additive p erturbations G(·)) is said to b e compact if the op erators F (t) (resp. G(t)) are compact for each t R+ . Proposition 10.3.1. If a C0 -sine operator function S (·,A) is compact and if a C0 -family of multiplicative perturbations F (·) is norm-continuous at zero, then F (·) is compact. The same is true for a C0 -family of additive perturbations. Proof. Integrating relation (2.1) in t from 0 up to , we obtain
+h

F ()xd -



F ()xd = 2S (, A)F (h)x.
-h

(10.6)

The compactness of S (·,A) implies the compactness of the left-hand side of (10.6) for any , h R+ . Since F (·) is uniformly continuous, we can take in (10.6) the derivative in without loss of the compactness prop erty, since the obtained left-hand side of (10.6) remains a compact op erator. Using the condition F (0) = 0 and the uniform continuity of F (·) and tending to zero, we obtain that F (h) is compact for each h R+ . The requirement of the uniform continuity in Prop osition 10.3.1 is also necessary, as the following prop osition shows. Proposition 10.3.2 ([239]). If a family of multiplicative perturbations F (·) is compact, then the family F (·) is uniformly continuous on R+ . ^ Proof. Since F (·) is compact, the Laplace transform F (·) is also compact (see [284]). By formula (iv) of Prop osition 2.4.1, the assertion is proved, since the strong convergence b ecomes uniform after p ostmultiplying by a compact op erator. Proposition 10.3.3 ([239]). Let a C0 -cosine operator function C (·,A) on a Banach space E be such that each of the families of multiplicative perturbations F (·) (or the C0 -families of additive perturbations G(·)) for C (·,A) is compact. Then E is finite-dimensional. Proof. By assumption, two particular families of multiplicative p erturbations F1 (t) = C (t, A) - I 144
t

and F2 (t) =
0

S (s, A) ds


are compact. Then since we know (see Theorem 10.1.2) that the op erator C (t, A) - I is compact for all t R
+

iff the generator A is compact, the family C (·,A) is norm-continuous on R+ . Therefore, the
-2

op erator C (0,A) = I, b eing the limit in norm of compact op erators 2t implies that E is finite-dimensional.

F2 (t) as t 0 is compact. This

Chapter 11
ADJOINT COSINE OPERATOR FUNCTIONS Cosine op erator functions adjoint in the sense of Phillips were little considered in the literature: on one hand, b ecause of close analogies with the theory of C0 -semigroups of op erators, and on the other hand, b ecause of the absence of very valuable applications. But we essentially use the prop erties of C (·,A) in the p erturbation theory when considering the lifting theorems. 11.1. C0 -Cosine Operator Functions Adjoint in the Sense of Phillips Denote by C (t, A) , t R, the restriction C (t, A) |E , t R, where E which the adjoint family C (·,A) is strongly continuous at zero. Proposition 11.1.1 ([222]). For a C0 -cosine operator function C (·,A) given on E , we have: (i) if x D(A ), then for any t R, C (t, A) x D(A ) and A C (t, A) x = C (t, A) A x , E is the subspace on

and the fol lowing relation holds for any x E : x, (C (t, A) - I )x =
0 t

(t - s) x, C (s, A) A x ds;

(ii) the inclusion

x

D

(A

) holds iff there exists the limit and, moreover, A x = y .

w - lim (2/s2 )(C (s, A) - I )x = y ,
s0+

Theorem 11.1.1 ([222]). In the notation of Proposition 11.1.1, we have the fol lowing: (i) the subspace E = D(A ), where the closure is understood in the strong topology of the space E ; (ii) the subspace E is invariant with respect to C (t, A) , and C (t, A) , t R, is a C0 -cosine operator function on E ; (iii) the generator A of the C0 -cosine operator function C (·,A) is maximal among the restrictions of the operator A to E (i.e., A is a part of the operator A on E ); (iv) if E is reflexive, then E = E and A = A ; (v) for each t R+ , the operator C (t, A) is the w -closure of the operator C (t, A) . 145


Proposition 11.1.2 ([222]). For any x D(A ), we have (C (t, A) - I )x (t2 /2)) A x · sup 11.2. Adjoint Families Definition 11.2.1. An op erator function K (t), t R, given on the space E and satisfying the conditions K (0) = I on it and the functional cosine equation (see (i) on p. 83), is said to b e a w -continuous C0 cosine op erator function if for each t R+ , the op erator K (t) is continuous on the space E in the w -top ology (we write w -w -continuous), and for any x E , the function t K (t)x is w -continuous on E in t R. Proposition 11.2.1 ([259]). An operator Q L(E ) is w -w -closed iff it is adjoint to a densely defined closed operator Q C (E ). Moreover, D(Q) = E iff D(Q) = E , and in this case, both Q and Q are bounded, and, moreover, Q = Q . Theorem 11.2.1 ([259]). An operator function K (t) on E is a w -continuous C0 -cosine operator function iff K (·) = C (·,A) , where C (·,A) is a certain C0 -cosine operator function. If Q is a generator of K (t), t R (in the sense of the w -topology) and A is a generator of a C0 -cosine operator function C (·,A), then Q = A . For a w -continuous C0 -cosine op erator function with a generator Q, we accept the notation K (t, Q). Proposition 11.2.2 ([259]). In the notation of Theorem 11.2.1, the fol lowing conditions are equivalent: (i) an element x D(Q); (ii) K (t, Q)x - x = O(t2 ) as t 0; (iii) lim t
t0+ -2 0st

C (s, A) .

K (t, Q)x - x < .

Proposition 11.2.3 ([259]). Let Q L(E ). An operator Q generates a w -continuous C0 -cosine operator function iff it is w -densely defined, w -w -closed, and there exist constants M > 0 and > 0 such that for Re > , the point 2 (Q) and dm Mm! ((2 I - Q)-1 ) , dm ( - )m+1 m N.

Proposition 11.2.4 ([259]). If a set D D(Q) is w -dense in D(Q) and is invariant with respect to a w -continuous C0 -cosine operator function K (·,Q), then D is the w -core of the operator Q. Proposition 11.2.5 ([259]). The fol lowing conditions are equivalent for w -continuous C0 -cosine operator functions K (t, Q1 ) and K (t, Q2 ): 146


(i) the domains D(Q1 ) D(Q2 ); (ii) (K (t, Q1 ) - K (t, Q2 ))x = O(t2 ) as t 0 for any x D(Q1 ). Proposition 11.2.6 ([259]). The fol lowing conditions are equivalent: (i) K (t, Q1 ) - K (t, Q2 ) = O(t2 ) as t 0; (ii) D(Q1 ) = D(Q2 ) and Q2 - Q1 is a bounded operator on D(Q1 ); (iii) D(Q1 ) D(Q2 ) and Q2 - Q1 is a bounded operator on D(Q1 ); (iv) D(Q2 ) D(Q1 ) and Q2 - Q1 is a bounded operator on D(Q2 ). Moreover, in these cases, Q2 - Q1 lim 2t
t0 -2

K (t, Q1 ) - K (t, Q2 ) sup 2t
tR+

-2

K (t, Q1 ) - K (t, Q2 ) ,

and, moreover, the equalities are attained, e.g., at contractive w -continuous C0 -cosine operator functions. Proposition 11.2.7 ([259]). If K (t, Q1 ) - K (t, Q2 ) = o(t2 ) as t 0, then K (t, Q1 ) = K (t, Q2 ), t R. Proposition 11.2.8 ([259]). For a w -continuous C0 -cosine operator function K (t, Q), t R, we have N (Q) =
t>0

N K (t, Q) - I



,

and w - cl(R(Q)) is w - cl

t>0

R(K (t, Q) - I ) .

If E is a Grothendieck space, then R(Q) = w- cl
t>0

R K (t, Q) - I



= span{R(K (t, Q) - I ) : t R+ },

where the closure is understood in the strong topology of E . Introduce the following notation: Q1 := s- lim t s
t -1

S (t, A) ,

Q1 := w- lim t w
t

-1

S (t, A) ,

Q1 := w - lim t w
t

-1

S (t, A) .

Proposition 11.2.9 ([259]). Assume that for Q = A , the fol lowing conditions hold: (a) S (t, A) = O(t) (b) w - lim t
t -1

as

t ;

((K (t + s, Q) - K (t - s, Q))S (s, A) x = 0 for al l x E and s R+ .

Then: (i) Q1 Q1 Q1 are projections, and, moreover, s w w Q1 w


lim t
t

-1

S (t, A) ,

and

D(Q1 ) D(Q1 ) s w

and

D(Q1 ) w 147


are strongly closed; (ii) R(Q1 ) = R(Q1 ) = R(Q1 ) = N (Q), N (Q1 ) N (Q1 ) R(Q), and s w w s w Sp := span{R(K (t, Q) - I ) : t > 0} N (Q1 ) w - cl(R(Q)). w If we replace condition (b) by a stronger condition (b ) s- lim for al l s R+ , then Sp N (Q1 ). s
t

1 (K (t + s, Q) - K (t - s, Q))S (s, A) = 0 t

Proposition 11.2.10 ([259]). Let conditions (a) and (b) of Proposition 11.2.9 hold, and let E be a Grothendieck space. Then R(Q1 ) = N (Q), w N (Q1 ) = R(Q) w and D(Q1 ) = N (Q) R(Q) = E . w

If, in addition, the condition (b ) holds, then K (·,Q) is strongly (C, 1) ergodic, i.e., D(Q1 ) = E . s Definition 11.2.2. Denote by Q2 ,Q2 , and Q2 the Cesaro (C, 2)-averagings of the Cesaro (C, 1)s w w averagings Q1 ,Q1 , and Q1 defined in the corresp onding way. s w w Proposition 11.2.11 ([259]). Let T (t, A) = O(t2 ) as t , and let s- lim t x D(Q). Then Q2 = Q2 Q2 are bounded projections such that s w w Q2 w


-2

t

K (t, Q)x = 0 for al l

lim 2t
t

-2

T (t, A) ,

R(Q2 ) = R(Q2 ) = N (Q), s w

R(Q) = N (Q2 ) N (Q2 ) w - cl(R(Q)). s w The subspaces D(Q2 ) and D(Q2 ) are strongly closed in E , and s w D(Q2 ) = N (Q) R(Q) = {x E : tn : lim 2t s
n -2 n

T (tn )x exists}.

Proposition 11.2.12 ([259]). Let Gx := s- lim t
t0+ -1

S (t, A) x

for those x E for which the limit exists. Then for a w -continuous C0 -cosine operator function K (·,Q) with Q = A , we have D(G) =
t>0

R(S (t, A) ) = {x E : tn : w - lim t .

-1 n n

S (tn ,A) x exists}.

Moreover, G = ID 148

(G)


Chapter 12
PERTURBATIONS OF C0 -COSINE OPERATOR FUNCTIONS The p erturbation theory of C0 -cosine op erator functions differs from that of C0 -semigroups in an interesting way. On one hand, the generator of a C0 -cosine op erator function lies in a more narrow class of op erators than G (M, ); for example, it always generates an analytic C0 -semigroup, and, therefore, its fractional p owers (-A) , 0 1, are defined in a sufficiently simple way. On the other hand, it is not clear up to now whether the M. Watanab e p erturbation is the strongest p erturbation or not, and what happ ens with SV of a family of multiplicative p erturbations if SV(F (·),t) = O(t ), t 0+, for a certain 0 < < 1. 12.1. General Multiplicative Theorems We quote the following theorem (see [192, Theorem 3.10]) for C (·,A), which is convenient for practical applications and which will b e rep eatedly used b elow. Theorem 12.1.1. A C0 -cosine operator function C (·,A) satisfying the estimate C (t, A) Met for al l t 0 is a C0 -cosine operator function with a generator A iff the condition > implies 2 (A) and we have (2 I - A)-1 =
0

e-t C (t, A) dt.

Proposition 12.1.1 ([240]). Let A be a densely defined closed linear operator on a Banach space E , and let B (X ). The fol lowing assertions hold: (i) if the operator ~ A generates a cosine operator function C (·,A), then the operator A also gen-

erates a C0 -cosine operator function; (ii) if the operator A ^ generates a cosine operator function C (·,A), and for a certain real , the A also generates a C0 -cosine operator function; generates a cosine operator function C (·,A) and D((A ) ) = D(A ), then operator - A is invertible, then the operator (iii) if the operator A the operator

A also generates a C0 -cosine operator function. B (E ) b elongs to the class M 1(A) of multiplicative -I

Definition 12.1.1. We say that an op erator

p erturbations of the generator A of a C0 -cosine op erator function C (·,A) if the op erator B = satisfies the following Condition (M 1): for all continuous functions f C ([0,t]; E )
t

(M 1a )
0

S (t - s, A)Bf (s)ds D(A),
t

(M 1b )

A
0

S (t - s, A)Bf (s)ds MB (t) f

[0,t]

, 149


where B : [0, ) [0, ) is some continuous nondecreasing function with B (0) = 0 and f
0st

[0,t]

=

sup

f (s) .

Remark 12.1.1. If B + I M 1(A), then C (t + h, A)B - C (t, A)B 0 as h 0 for any t. To see this, we first set f (t) = x for t 0 in (M 1b ). It follows that (C (h, A) - I )B 0 as h 0. This, together with the fact that (C (·,A) - I )B is a C0 -family of multiplicative p erturbations, proves the assertion. Theorem 12.1.2 ([240]). Let A be the infinitesimal generator of a C0 -cosine operator function C (·,A) on E . If an operator O(B (t)) (t 0+ belongs to M 1(A), then both A and A are generators of C0 -cosine operator satisfies (t) - C (t, A) = functions. Moreover, the C0 -cosine operator function ). (·) generated by A

Remark 12.1.2. (i) If (M 1a ) and (M 1b ) hold for all functions in a dense subset of C ([0,t]; E ), then b ecause of the closedness of A, we easily see that they actually hold for all f in C ([0,t]; E ). Hence since (M 1a ) holds for all f in C 1 ([0,t]; E ), which is dense in C ([0,t]; E ), Condition (M 1) can b e replaced by the equivalent condition:
t

A
0

S (t - s, A)Bf (s)ds MB (t) f

[0,t]

for all f C 1 ([0,t]; E ).

(12.1)

Thus, we only need to verify Condition (M 1 ) in practical applications. (ii) If (M 1) holds with some B (t) = o(t2 ), then (·) C (·,A) (see [240, Corollary 3.6]), so that
t 0

A(I + B ) = A and AB =0. Conversely, the latter condition implies C (·,A)B = B , and hence A S (t - s, A)Bf (s)ds 0 for all f C ([0,t],X ). Thus, (M 1) holds with some B (t) = o(t2 ) iff AB = 0, and in this case, (M 1) actually holds with B (·) 0. Let (Z, |· |) b e a Banach space satisfying Condition (Z ) with resp ect to C (·,A): (Za ) Z is continuously emb edded in E , (Zb ) for all continuous functions C ([0,t],Z ),
t 0 t

S (t - s, A)(s)ds D(A),

(Zc )

A
0

S (t - s, A)(s)ds (t) sup |(s)|Z ,
0st

where (·) : [0, ) [0, ) is a continuous nondecreasing function with (0) = 0. It is easy to verify Condition (Z ) for the spaces D(A) and the Favard class (Fav Indeed, if Z = D(A), then (Z ) holds with (t) = O(t2 ) as t 0+ . Corollary 12.1.1. If Z is a Banach space satisfying Condition (Z ), then I + B (E, Z ) M 1(A), so that for every B B (E, Z ), both A(I + B ) and (I + B )A are generators of C0 -cosine operator functions. 150
C (·,A)

, | · |Fav

C (·,A)

).


Definition 12.1.2. We say that an op erator

B (E ) b elongs to the class M 2(A) of multiplicative -I

p erturbations of the generator A of a C0 -cosine op erator function C (·,A) if the op erator B = satisfies
t

B (t) := sup
0

BS (s, A)Ax ds : x D(A), x 1 0 as t 0+ .

(12.2)
2-1

Remark 12.1.3. As was shown by Fattorini [131], in the case of E = Lp ,we have AS (t, A)x = O(t as t 0 for 1/2 1 and x D((A - cI ) ). Therefore, 1/2.

)

M 2(A), e.g., if B = (A - cI )- for

Theorem 12.1.3 ([240]). Let A be the infinitesimal generator of a C0 -cosine operator function C (·,A) on E . If an operator 0+ ). 12.2. Perturbations by the Family F (·) For any fixed and an op erator B B (E ), let FB, (·) and GB, (·) b e the functions defined by FB, (t)x := (2 I - A)
0 t

belongs to M 2(A), then both

A and A

are generators of C0 -cosine operator

functions. Moreover, the cosine function C1 (·) generated by A satisfies C1 (t)- C (t, A) = O(B (t)) (t

S (s, A)Bx ds = 2
0 t 0

t

S (s, A)Bxds - (C (t, A) - I )Bx, x E, t 0,
t 0

(12.3) (12.4)

GB, (t)x := B (2 I - A)

S (s, A)xds = 2 B

S (s, A)xds - B (C (t, A) - I )x, x E, t 0.

Definition 12.2.1. Op erator function f (·) is called a function with a local ly bounded semivariation if for some t > 0,
n

SV(f (·),t) := sup
j =1

[f (tj ) - f (t

j -1

)]xj : xj E, xj 1

< ,

where the supremum is taken over all partitions of the interval [0,t]. Op erator function f (·) is called a function with a local ly bounded strong variation if for some t > 0 and all x E ,
n

Var(f (·)x, t) := sup
j =1

(f (tj ) - f (t

j -1

))x : 0 = t0 < t1 < ··· < tn = t, n 1

< .

Finally, op erator function f (·) is called a function with a local ly bounded uniform variation if for some t > 0,
n

Var(f (·),t) := sup
j =1

(f (tj ) - f (t

j -1

)) : 0 = t0 < t1 < ··· < tn = t, n 1

< .

The next theorem gives a characterization of M 1(A) in terms of the semivariation of FB, (·). 151


Theorem 12.2.1 ([240]). An operator

B (E ) belongs to M 1(A), i.e., B =

- I satisfies Condition

(M 1), iff SV(FB, (·),t) = o(1) (t 0+ ) for some (and al l) > . Moreover, in Condition (M 1b ), one can choose B (t) = SV(FB, (·),t) in the case SV(FB, (·),t) = O(t2 ), and B (t) = O(t2 ) in the case SV (FB, (·),t) = o(t2 ). Now, from Theorem 12.2.1, we can deduce the following theorem on an additive p erturbation. Theorem 12.2.2 ([240]). Let A be the generator of a C0 -cosine operator function C (·,A) on E . If P B (D(A),E ) is such that
t 0 t

S (t - s, A)Pg(s)ds D(A), g(s)
D (A)

(12.5) (12.6)

A
0

S (t - s, A)Pg(s)ds P (t) sup

0st

for al l g C ([0,t], D(A)) and for some function P (·) with P (t) = o(1) (t 0+ ), then the operators A + P and A +(A - I )P (A - I )-1 ( > ) are generators of C0 -cosine operator functions. Proof. Without loss of generality, we may assume that A is invertible, so that A + P = (I + PA-1 )A. In view of Theorem 12.2.1, we only have to verify Condition (M 1) for the op erator B = PA-1 . Indeed, if f C ([0,t]; E ), then A-1 f C ([0,t], D(A)), so that, setting g = A-1 f in (12.5) and (12.6), we have
t 0

S (t - s, A)Bf (s)ds =
0

t

S (t - s, A)P (A-1 f )(s)ds D(A)

and
t

A
0

S (t - s, A)Bf (s)ds P (t) sup

0st

A-1 f (s)

D (A)

P (t)( A-1 +1) f

[0,t]

.

Corollary 12.2.1. Let A be the generator of a C0 -cosine operator function C (·) on E . If P is a continuous operator acting from D(A) to Z (where Z is a Banach space satisfying condition (Z )), then A + P and A +(A - )P (A - )-1 ( > ) are generators of C0 -cosine operator functions. Proof. Let P B (D(A),Z ), and let g C ([0,t],D(A)). Then we have Pg C ([0,t]; Z ), and by
t

Condition (Z ),
0 t

S (t - s, A)Pg(s)ds D(A) and

A
0

S (t - s)Pg(s)ds P (t) sup |Pg(s)|Z P (t) P
0st

B (D (A),Z )

0st

sup

g(s)

D (A)

.

The conclusion now follows from Theorem 12.2.2. 152


12.3. Perturbations by the Family G(·): Additive Perturbations The following theorem gives a characterization of M 2(A) in terms of the strong variation of GB, (·) for x D(A); more precisely, (GB, (·),t) := sup{Var(GB, (·)x, t); x D(A), x 1}. Theorem 12.3.1 ([240]). An operator B (E ) belongs to M 2(A), i.e., B = (12.7) - I satisfies Condi-

tion (M 2), iff (GB, (·),t) = o(1) (t 0+ ) for some (and al l) > . Moreover, (GB, (·),t) and the function B (t) in Condition (M 2) has the same order of convergence at zero whenever the order does not exceed O(t2 ). Proof. From (12.4), we see that Var(GB, (·)x, t) = Var(BC (·,A)x, t)+ 2 B
0 t

S (s, A)x ds

if the variation exists. Since for x D(A), Var(BC (·,A)x, t) =
0 t

d BC (s, A)x ds = ds
t 0

t

BS (s, A)Ax ds,
0

we have (GB, (·),t) - B (t) 2 B S (s, A) ds 2 B Met t2 .

Hence B (t) tends to 0 as t 0+ iff (GB, (·),t) does. They have the same order of convergence at zero whenever one of them has the order less than or equal to O(t2 ). In general, M 1(A) and M 2(A) are prop er subsets of I + B (E ). Each of the conditions M 1(A) = I + B (E ) and M 2(A) = I + B (E ) is equivalent to the condition that A is b ounded. Indeed, if for every B B (E ), the op erator (I + B )A generates a cosine op erator function, then for B = -2I , we have that -A also generates a C0 -cosine op erator function. Hence b oth A and -A generate analytic C0 -semigroups and, consequently, A is b ounded. From Theorem 12.3.1, we deduce the following additive p erturbation theorem. Theorem 12.3.2 ([240]). Let A be the generator of a cosine operator function C (·) on E . If P is an operator satisfying the fol lowing conditions: D(A) D(P ) and P (2 I - A)-1 B (E ) for some > ;
t

(12.8) (12.9)

P (t) := sup
0

PS (s)x ds; x D(A), x 1

< 1 for some t > 0,

then the operators A + P and A +(A - I )P (A - I )-1 are generators of C0 -cosine operator functions. Moreover, the C0 -cosine function C1 (·) generated by A + P satisfies C1 (t) - C (t, A) = O(P (t)) (t 0+ ). 153


Proof. We may assume that A is invertible, so that A + P = (I + PA-1 )A. We set B = PA-1 . Then
t 0

BS (s)Ax ds
0

t

PS (t)x ds

for all x D(A). Hence (12.9) implies B (t) P (t) < 1 for some t > 0, and the conclusion follows from Theorem 12.3.1. From Theorem 12.2.2, we can deduce the following p erturbation theorem of Watanab e ([287, Theorem 2]). Corollary 12.3.1 ([240]). Let A be the generator of a C0 -cosine operator function C (·,A) on E . If P B (E 1 ,E ), then A + P and A +(A - I )P (A - I )-1 ( > ) are generators of C0 -cosine operator functions. Moreover, the cosine function C1 (·) generated by A + P satisfies C1 (t) - C (t, A) = O(t) (t 0+ ). Proof. It is proved in [255] that P B (E 1 ,E ) implies (12.8). To show (12.9), let x D(A). Then for t [0, 1], we have PS (t, A)x P
B (E 1 ,E )

S (t, A)x

E

P

B (E 1 ,E )

S (t, A)x + sup

01

AS (, A)S (t, A)x

K x .

Therefore, P (t) = O(t) (t 0+ ), and hence the conclusion follows from Theorem 12.3.2. From Theorem 12.3.2, we can also deduce the following corollary: when a = , it is Theorem A in [262] (see also [255, Corollary 2.1]), and when a < , it is Corollary 2.2 in [255], which contains Theorem 3.2 in [262]. Corollary 12.3.2 ([240]). Let A be the generator of a C0 -cosine operator function on E . Let P be an operator satisfying conditions (12.8), and let
a

L() := sup
0

e-s PS (s, A)x ds; x D(A), x 1


<

(12.10)

for some a (0, ] and > . Let L() := lim L(). Then for each with || < L()-1 , A + P and A + (A - I )P (A - I )-1 ( > ) are generators of C0 -cosine operator functions. Proof. Choose numb ers 0 < µ < µ1 < µ2 < 1 such that || = µL()-1 . Fix so large that L()/L() < µ2 µ1 , and then fix t (0,a) so small that et < . Then for all x D(A) with x 1, we have µ µ1
t 0

P S (s, A)x ds ||et
0

t

e-s PS (s, A)x ds ||et L() = et µL()/L()

µ2 µ1 = µ2 < 1, µ µ1 µ

i.e., P (t) < 1. Now the conclusion follows from Theorem 12.3.2. 154


From this corollary, Shimizu and Miyadera were able to deduce a generalization ([262, Corollary 2.2]) of the p erturbation theorem of Fattorini [138] and Travis and Webb [275]. The latter theorem states that if a closed op erator P satisfies D(A) D(P ) and PS (·,A)x C ([0, 1]; E ) for every x E , then A + P is the generator of a C0 -cosine op erator function. This is also an immediate consequence of Theorem 12.3.2, since it is clear that P (t) = O(t) (t 0+ ).

Next, we consider mixed-typ e p erturbations induced by a C0 -family of multiplicative p erturbations and a C0 -family of additive p erturbations. Thus, the following two theorems follow immediately from Theorems 12.2.1, 12.3.1, and Corollary 12.3.1. Theorem 12.3.3 ([240]). If a C0 -family of multiplicative perturbations F (·) for C (·,A) is local ly of ^ ^ bounded semivariation and if SV(F (·),t) = o(1) (t 0+ ), then the operator A1 := A(I - F ()) + 3 F (), > , is the infinitesimal generator of some cosine operator function C1 (·). Theorem 12.3.4 ([240]). If a C0 -family of additive perturbations G(·) for C (·,A) is local ly of bounded ^ ^ strong variation and if (F (·),t) = o(1) (t 0+ ), then the operator A2 := (I - G())A + 3 G(), > , is the infinitesimal generator of some C0 -cosine operator function C2 (·). 12.4. Comparison of Cosine Operator Functions In this section, we give some characterizations of the prop erty that C1 (t, A1 ) - C (t, A) = O(t2 ) (t 0+ ). Theorem 12.4.1 ([240]). Let C (·,A) be a C0 -cosine operator function with a generator A, and let A1 be a linear operator. The fol lowing statements are equivalent: (i) A1 generates a C0 -cosine operator function C1 (·,A1 ) that satisfies C1 (t, A1 ) - C (t, A) = O(t2 ) (t 0+ ); (ii) there exists B B (E, Fav
C (·,A)

) such that A1 = A(I - B )+ 2 B for some > ;

(iii) there exists B B (E ) such that the function F (·) FB, (·) defined in (12.3) is square Lipschitz ^ ^ continuous and A1 = A(I - F ()) + 3 F (); (iv) A1 generates a C0 -cosine operator function C (·,A1 ), D(A ) = D(A ), and A - A is a bounded 1 1 operator acting from D(A ) to E ; (v) A1 generates a C0 -cosine operator function C (·,A1 ) and (2 I - A1 )-1 - (2 I - A)-1 = O(-4 ) ( ). 155


Proof. The proof of (i) (ii) is similar to the proof for C0 -semigroups. Let B be the operator defined by Bx := x - (2 - A)-1 (2 - A1 )x for x D(A1 ). Since for all x D(A1 ), Bx = lim x - 2 (2 I - A)-1 x +
0

= lim x - 2 (2 I - A)-1
0

22 ( I - A)-1 (C (, A1 )x - x) 2 2 x + 2 (2 I - A)-1 (C (, A)x - x)

-

22 ( I - A)-1 (C (, A)x - C (, A1 )x) 2 2 lim (2 I - A)-1 C (, A) - C (, A1 ) 0 2

x K (2 I - A)-1

x,

where B is b ounded and can b e extended to a b ounded op erator (still denoted by B ) on the whole space E . To show that B maps E to Fav lim sup
0 C (·,A)

continuously, we set y = Bx for x D(A1 ). Then (2 2 - C (, A)+ I )(x + y ) - (2 2 - C1 (, A1 )+ I )x

2 2 C (, A)y - y lim sup 2 2 0

+ C (, A)x - C1 (, A1 )x so that |Bx|Fav
C (·,A)

+22 y K x +22 y K (1 + 22 (2 - A)-1 ) x ,

K (1 + (1 + 22 ) (2 - A)-1 ) x for all x D(A1 ) (and hence for all x E ). satisfies Condition (Z ), Corollary 12.1.1 implies that A(I - B )+ 2 B

From (2 - A)(x - Bx) = (2 - A1 )x, we have A1 x = A(I - B )x + 2 Bx for all x D(A1 ), i.e., A1 A(I - B )+ 2 B . Since Fav
C (·,A)

is the generator of a C0 -cosine op erator function and hence coincides with A1 . ^ Taking the Laplace transform of F (·), we obtain F (µ) = 2 µ-1 (µ2 - A)-1 B - µ(µ2 - A)-1 B + µ-1 B , ^ so that B = F (). Further, using (12.3), for all x E , we have F (t)x - (C (t, A) - I )Bx which implies that B B (E, Fav
C (·,A) t 0

2

S (s, A)Bxds = O(t2 )(t 0+ ),

) iff F (t) = O(t2 )(t 0+ ). Hence (ii) and (iii) are equivalent.

(iii) (i). In view of Theorem 12.3.3, we need only to show that if F (t) = O(t2 )(t 0+ ), then Var(F (·),t) = O(t2 )(t 0+ ). But, b ecause of (12.3), this is equivalent to showing that (C (t, A)-I )B = O(t2 ) (t 0+ ) implies Var(C (·,A)B, t) = O(t2 )(t 0+ ). Therefore, we supp ose that (C (s, A) - I )B Ks2 for 0 s . For any sub division {t0 ,t1 , ··· ,tn } of [0,t] [0, 1] with hi = ti - t the largest integer such that ni hi ti . One has C (ti ,A) - C (t and, therefore, C (ti ,A) - C (t
i-1 i-1 i-1

, let ni be

,A) = C (t

i-1

,A) - C (t

i-1,A

- C (ti - t

i-1

)) + 2C (t

i-1

,A)(C (ti - t

i-1

,A) - I ),

,A) C (ti - ni hi ,A) - C ((ni +1)hi - ti ,A) +2KM et ni h2 i 2KM et (ni +2)h2 4KM et thi . i

156


Therefore,
n n

C (ti ,A) - C (t
i=1

i-1

,A) 4KM et t
i=1

hi 4KM et t2 .

Hence Var(C (·,A)B, t) = O(t2 )(t 0+ ). (i) (iv) is proved in ([238, Theorem 3.5]). To prove (iv) (v), we write (2 I - A1 )-1 - (2 I - A)-1 = (2 I - A )-1 (A - A )(2 I - A )-1 1 1 (2 I - A1 )-1 A - A 1 (2 I - A)-1 = O 1 4 .

Finally, to prove (v) (i), we write (see [238, Theorem 3.9]) C (t, A1 )x - C (t, A)x = lim 4 (2 I - A1 )-1 (C (t, A1 ) - C (t, A))(2 I - A)-1 x


lim

t 0



S (t - s, A1 )4 ((2 I - A1 )-1 - (2 I - A)-1 )C (s, A)xds Kt2 x .
C (·,A)

Hence mixed typ e p erturbations of the form A1 = A(I - B ) + 2 B with B B (E, Fav

)

characterize those C0 -cosine functions C1 (·,A1 ) which satisfy C (t, A1 ) - C (t, A) = O(t2 ) (t 0+ ). In this case, although D(A ) = D(A ), the domain of A1 may not contain the domain of A. What kind of 1 C0 -cosine op erator functions C (·,A1 ) have the prop erty that D(A) D(A1 ) and C (t, A1 ) - C (t, A) = O(t2 )(t 0+ )? It is clear that additive p erturbations of A by b ounded op erators generate cosine functions with this prop erty. Theorem 12.4.2 ([240]). Let C (·,A) be a cosine operator function with the generator A. For any operator A1 , the fol lowing statements are equivalent: (i) D(A) D(A1 ), and A1 generates a C0 -cosine operator function C1 (·) such that C1 (t, A1 ) - C (t, A) = O(t2 )(t 0+ ); (ii) there exists an operator B B (E ) such that R(B ) Fav some > ; (iii) there exists B B (E ) such that the function G(·) GB, (·) defined in (12.4) is square Lipschitz ^ ^ continuous at 0 and A1 = (I - G())A + 3 G(); (iv) A1 = A + Q for some Q B (E ). Proof. (iv) (1) is obvious. We first prove (i) (ii)+(iv). Since D(A) D(A1 ), we can define the b ounded op erator B := I - (2 I - A1 )(2 I - A)-1 = (A1 - A)(2 I - A)-1 for > . Then we have 157
C (·,A)

and A1 = (I - B )A + 2 B for


A1 x = (I - B )Ax + 2 Bx for x D(A). Using (12.4), we can write G(t)x = (A1 - A)
0 t

S (s, A)xds, x E, t 0.

Since (A1 - A)x lim

2

b ounded extension Q B (E ). Thus (iv) holds. This also implies G(t)x Var(G(·)x, t) Q
0 t

t0+ t2

(C (t, A1 ) - C (t, A))x K x for all x D(A), the op erator A1 - A has a

S (s, A1 )x ds Q Met t2 x , x E,

so that G(t) (GB, (t) = O(t2 ). It follows from Theorems 12.2.1 and 12.1.3 that (I - B )A + 2 B is a generator of cosine function, and hence it coincides with A1 . Further, using (12.4) for all x E , we have G (t)x - (C (t, A) - I )B x Hence G(t) = O(t )(t
2 t 0

2
C (·,A)

S (s, A)B x ds = O(t2 ) (t 0+ ).

0+

) iff R(B ) Fav



. In particular, this completes the proof of (ii).

^ To prove (ii) (iii), it remains to show that B = G(). This can b e done by taking Laplace transform of GB, (·) in (12.4). (iii) (i). It suffices to show that R(B ) Fav
C (·,A)

implies the b oundedness of BA. Indeed, for

all x D(A) with x 1 and for all x E , we have | BAx, x | = | lim 2t
t0 -2

(C (t, A) - I )x, B x | lim2t
t0

-2

(C (t, A) - I )B x .

The uniform b oundedness principle implies that {BAx : x D(A), x 1} is b ounded. Hence the op erator BA is b ounded on D(A). 12.5. Preservation of Properties under Additive Perturbations This section contains, as a rule, known facts. However, they have app eared historically first and, moreover, in this section some useful relations are formulated explicitly. Proposition 12.5.1 ([222, 287]). Let A C (M, ), and let B B (E ). Then the operator A+ B generates a C0 -cosine operator function and C (t, A + B ) - C (t, A) 0 as B 0 uniformly on any compact set in R. Proposition 12.5.2 ([221]). Under conditions of Proposition 12.5.1, if > + ~ ~ ~ a number M = M ( ) such that ~~ C (t, A + B ) Me |t| , 158 t R. M B , then there exists

(12.11)


Proposition 12.5.3 ([132]). Let A C (M, ). Then for x E , t I1 ( t2 - s2 ) 2 C (t, I + A)x = C (t, A)x + t C (s, A)xds, t2 - s 2 0 where I1 is the Bessel function and C (t, A + 2 I ) M cosh( 2 + 2 t) for C.

(12.12)

In [132, 265, 266], other more precise estimates of the expression C (t, A ± 2 I ) are presented for Banach and Hilb ert spaces. Proposition 12.5.4 ([255, 288]). Let A C (M, ), and let D(A) D(G). If there exist 0 and M 1 such that (A) contains the set {z : z > } and the function G(z 2 I - A)-1 is infinitely many times differentiable, and, moreover, 1 (z - )n n!
+1

d dz

n

G(z 2 I - A)-1 x M x

for x E and any n N and z > , then A + G generates a C0 -cosine operator function. Proposition 12.5.5 ([144]). If A, G C (M, ), then the operator A + G (or its closure) may not generate a C0 -cosine operator function in general, even in the case where C (·,A) and C (·,G) commute. However, we always have A + B H(, /2). Proposition 12.5.6 ([21]). Let A, G C (M, ), and let D1 := D(A) D(G) be dense in E . Then on D1 , the fol lowing "generalized" cosine function (in the sense of fulfil lment of conditions (i)­(ii) of Definition 2.3.1) is defined: t2 ~ C (t, A + G)x = C (t, A)x + 2 where j1 (t, A) :=
1 0

j1 (t 1 - s2 ,A)C (ts, G)xds,

4 1 1 - s2 C (ts, A)xds, x D1 . If A + G generates a C0 -cosine operator function 0 ~ C (t, A + G), then C (t, A + G) = C (t, A + G), t R. Proposition 12.5.7 ([275]). Let A C (M, ), and let the operator G C (E ) be such that (i) for the C0 -sine operator valued function, R(S (t, A)) D(G) for al l t R; (ii) the function GS (t, A) is strongly continuous in t R. Then the operator A + G generates a C0 -cosine operator function. Moreover,


C (t, A + G)x =
k =0

^ Ck (t)

and

S (t, A + G)x =
k =0

^ Sk (t),

(12.13)

159


^ where C0 (t) := C (t, A), ^ Ck (t) :=
0 t

^ C (t - s, A)GSk

-1

(s) ds,

^ and S0 (t) := S (t, A), ^ Sk (t) :=
0 t

^ S (t - s, A)GSk

-1

(s) ds,

and series (12.13) converges absolutely in B (E ). Proposition 12.5.8 ([275]). Under conditions of Proposition 12.5.7, we have the relation (I - A - G)-1 = (I - A)-1
k =0

(G(I - A)-1 )k .

Proposition 12.5.9 ([273]). Under conditions of Proposition 12.5.7, if the C0 -sine operator function S (·,A) is compact, then the C0 -sine operator function S (·,A + G) is also compact. Proposition 12.5.10 ([269]). Let A1 ,A2 C (M, ), and let D(A1 ) D(A2 ). Then C (t, A1 )x - C (t, A2 )x =
0 t

S (t - s, A2 )(A1 - A2 )S (s, A1 )xds

for al l x D(A1 ). Theorem 12.5.1 ([269, 273]). Let A C (M, ), and let G C (E ) be such that (i) D(A) D(G); (ii) there exists a continuous function K (t) having the property GS (t, A)x K (t) x for al l x D(A).

Then the operator A + G generates a C0 -cosine operator function. Proposition 12.5.11 ([273]). Let an operator A C (M, ) satisfy Condition (F) with an operator G C (E ), and, moreover, for a certain operator Q C (E ), let the condition D(G) D(Q) hold. Then A + Q generates a C0 -cosine operator function. Proposition 12.5.12 ([273]). In Proposition 12.5.11, let the condition of inclusion of domains be replaced by the condition D(A) D(Q). If the operator Q is G-bounded, then the operator A + Q generates a C0 -cosine operator function. Proposition 12.5.13 ([262, 270]). Let A C (M, ), G M(C (t, A)), G(I - A)-1 B (E ), and let || <
- K1 . Then the operator A + G generates a C0 -cosine operator function, lim C (t, A + G) - C (t, A) = 0 0

160


uniformly on any compact set from R, and


C (t, A + G) =
k =0 t

k C k (t),

where C 0 (t) := C (t, A); Ck (t) :=
0

C

k -1

(t - s)GS (s, A)xds, x D(A), C k (t) is a continuous extension

of Ck (t) to the whole E and M is the class of perturbations in the sense of Miyadera. Proposition 12.5.14 ([208]). Let C0 -cosine operator functions C (t, A1 ) and C (t, A2 ) be such that C (t, A1 ) Met and C (t, A2 ) Net for t R and (A1 - A2 )x a x + b A1 x for x D(A1 ). Then for z > , we have C (t, A1 ) - C (t, A2 ) (z 2 I - A1 )-1 (a + b 2 )M . z2 - 2 MN Qt sinh(t), 2 MN Q cosh(t) - cosh(t) , 2 - 2

where Q := (1 + M )b +

Proposition 12.5.15 ([208]). Under conditions of Proposition 12.5.14, if a, b 0, then s- lim C (t, A2 )x = C (t, A1 )x
a,b0

for

x D(A1 ).

Proposition 12.5.16 ([259]). Let K (t, A) be a w -continuous cosine operator function, and let B be a w -w -continuous operator on E . Then A + B is the w -generator of a w -continuous cosine-function K (t, A + B ) and lim
B 0

K (t, A + B ) - K (t, A) = 0 uniformly on any compact set t [0,T ]. 12.6. An Integral Operator on Lp ([0,T ]; E )

For C0 -groups of op erators and C0 -cosine op erator functions, we can formulate sp ecific p erturbation theorems, which assume certain "hyp erb olicity" conditions. Let J R b e a certain interval. Denote by (J ; E ) the vector space of all linear combinations of mappings of the form j x, where x E and j is the characteristic function of the interval T J , i.e., (J ; E ) is the space of step functions. Let {L(t)} - b e a strongly continuous family of b ounded op erators on E , and let A C (E ). t= Assume that the following hyp otheses hold: H1. for each x E and t R, the integral continuous as a mapping from R into E ;
t 0

L(s)xds D(A) and the mapping t A L(s)xds is
0

t

H2. there exist a subset D D(A ) and a constant M 1 such that (a) for any D , the mapping t L (t)A is continuous as a mapping from R into E ; (b) for each x E , there exists D such that M and x = x, . 161


By H1, for any f (·) (R; E ), the mapping t A L(t - s)f (s) ds is continuous as a mapping from R into E , and we can define the op erator K : (R; E ) C (R; E ) by the formula
t 0

t

(Kf )(t) := A
0

L(t - s)f (s) ds,

t R,

f (·) (J ; E ). : (R; D ) C (R; E ) by the

By H2, for each g (·) (R; D ), we can analogously define K formula (K g )(s) :=
s

L (t - s)A g (t) dt,

s R,

g (·) (R; D ).

Let T > 0 b e finite. Then the op erators K and K
t

induce the op erators t [0,T ], f (·) (J ; E ) (12.14)

(KT f )(t) := A
0

L(t - s)f (s) ds,

and
T

(KT g )(s) := Let p, q R+ , and let

L (t - s)A g (s) ds,

s [0,T ];

g (·) (J ; D ).

s

11 + = 1. For f (·) Lp (J ; E ) and g (·) Lq (J ; E ), we set pq f, g =
J

f (s),g (s) ds,

and by the Fubini theorem, we have KT f, g Now we note that if |||KT |||p
1

=

f, KT g

,

f (·) ([0; T ]; E ), g (·) ([0; T ]; D ).

,p

2

:= sup

KT f

Lp2 ([0;T ];E )

: f (·) ([0; T ]; E ), f

Lp1 ([0,T ];E )

=1

< ,

then by the denseness of ([0; T ]; E ) in Lp1 ([0,T ]; E ) and the closedness of the op erator A, for any f (·) Lp1 ([0,T ]; E ), the element KT f (·) Lp2 ([0,T ]; E ) and is defined for almost all t [0,T ] by expression (12.14). We call attention to that if p2 = , then the range of KT , in fact, lies in C ([0,T ]; E ), and KT : Lp1 ([0; T ]; E ) C ([0; T ]; E ) is continuous. Analogous arguments lead to the continuity of the mapping KT : Lq2 ([0; T ]; E ) C ([0; T ]; E ), where E H2 (b), we obtain |||KT |||q where 1 1 1 1 + = 1, + = 1, and p1 q 1 p2 q 2 |||KT |||q 162
2 2

is the strong closure of D . Moreover, using

,q

1

|||KT |||p

1

,p

2

M |||KT |||q

2

,q

1

,

(12.15)

,q

1

:= sup ||KT g ||Lq1

([0;T ];E )

: g ([0; T ]; D ), g

Lq2 ([0;T ];E )

=1 .


12.7. Lifting Theorem for C0 -Groups Recall that for a generator of a C0 -semigroup exp(·A), the domain D(A ) is a w -dense set in E , and A is a closed op erator on E = D(A ). In this section, we consider the op erator KT defined by the formula
t

(KT f )(t) := A
0

exp((t - s)A)Bf (s) ds,

t [0,T ],

(12.16)

where B B (E ). Theorem 12.7.1 ([232]). Let A G R(M, ), and let KT B Lp1 ([0,T ]; E ),Lp2 ([0,T ]; E ) for certain p1 ,p2 R such that KT f
C ([0, ];E ) +

and T R+ . Then KT B Lp1 ([0,T ]; E ),C ([0,T ]; E )

and there exists a constant C > 0

C f

Lp1 ([0, ];E )

for any

[0,T ].

(12.17)

Definition 12.7.1. We say that the op erator KT for the C0 -group exp(·A) satisfies Condition HG if there exist constants C, > 0 and T > 0 such that for any x E , there exists a function hx (·) C ([-, T ]; E ) having the prop erties (HG1) (HG2) hx (·)
C ([-,T ];E )

C x

E

;

Bhx (t) = exp(tA)Bx,

- t T - .

Condition HG hor C0 -groups holds, e.g., in the following two cases: (i) B commutes with exp(·A); in this case, hx (t) = exp(tA)x, - t T ; (ii) B has a b ounded inverse; in this case, hx (t) = B
-1

exp(tA)Bx, t [-, T ].

Theorem 12.7.2 ([232]). Let conditions of Theorem 12.7.1 hold, and, additional ly, let Condition HG hold. KT f Then KT B L1 ([0,T ]; E ),C ([0,T ]; E )
C ([0, ];E )

and there exists a constant C > 0 such that

C f

L1 ([0, ];E )

for any 0 T .

Corollary 12.7.1 ([232]). Let conditions of Theorem 12.7.1 hold. Then there exist constants L, > 0 such that SV(KT ,T ) LeT T
1/p
1

, T > 0.

Corollary 12.7.2 ([232]). Let E = H be Hilbert, and let conditions of Theorem 12.7.1 hold. Then there exist constants L, > 0 such that SV(KT ,T ) LeT T
1/2

, T > 0.

12.8. Lifting Theorem for C0 -Cosine Operator Functions Let E b e a Banach space, and let A : D(A) E E b e a generator of a C0 -cosine op erator function on E . Also, let us consider the adjoint family {C (·,A) }. It is well known that it is also a cosine family 163


of linear b ounded op erators on the dual space, which, however, can b e not strongly continuous. Recall that the space E is defined as E = { x E : s - lim C (t, A) x = x },
t0

where the limit is understood in the strong top ology of the space E . On the other hand, C0 -cosine op erator functions can b e studied by using C0 -groups. More precisely, introducing the Kysinski space E 1 , we can reduce the consideration to a C0 -group of op erators {exp(tA)}+ on E 1 â E defined as t=- C (t, A) S (t, A) AS (t, A) C (t, A) , t R. Its generator is A : D(A) â E 1 E 1 â E E 1 â E ; it is given by A = 0 I (12.18)

exp(tA) =

. A0 Let B B (E ) b e a linear continuous op erator on E . It is easy to verify that the family L(t) := In this section, we consider the continuity of the corresp onding convolution op erator
t

S (t, A)B , t R, satisfies all the conditions presented in Sec. 12.6 with resp ect to D = D(A ).

(KT f )(t) := A
0

S (t - s, A)Bf (s) ds,

t [0,T ] R+ ,

(12.19)

in Lp ([0,T ]; E ) norms, p [1, +]. Note that in the case of the C0 -cosine op erator functions considered, one can show that KT
B (Lp1 ([0,T ];E ),L([0,T ];E ))

grows exp onentially in T . This prop erty is stated in the lifting theorem

(Theorem 12.8.1). Further, it is natural to try to find a new convolution op erator KT constructed according to the group U that reduces the study of KT to that of KT . Therefore, the results that are true for C0 -groups can b e directly extended to C0 -cosine op erator functions. Consider the following op erator b ounded on E1 â E: B= and the corresp onding convolution op erator (Kh)(t) = A
0 t

0 0 0B

U (t - s)B h(s) ds,

t R,

(12.20)

where h = [g, f ]T ([0,T ],E 1 â E ). Also, for T > 0, we define the convolution op erator GT : ([0,T ],E ) C ([0,T ],E 1 ) by the relation
t

(GT f )(t) =
0

C (t - s, A)Bf (s) ds,

f ([0,T ],E ),

0 t T.

164


Further, we have (KT h)(t) = [(GT f )(t), (KT f )(t)]T , for h = [g, f ]T ([0,T ],E 1 â E ). Lemma 12.8.1 ([232]). Let p1 , p2 [1, +], and let T > 0. (i) KT is continuous as a mapping from Lp1 ([0,T ],E 1 â E ) into Lp2 ([0,T ],E 1 â E ) iff GT is continuous as a mapping from Lp1 ([0,T ]; E ) into Lp2 ([0,T ],E 1 ) and KT is continuous from Lp1 ([0,T ]; E ) into Lp2 ([0,T ]; E ). (ii) Let T > T . If KT is continuous from Lp1 ([0,T ],E ) into Lp2 ([0,T ],E ) and if p2 = +, then KT is continuous from Lp1 ([0,T ],E 1 â E ) into Lp2 ([0,T ],E 1 â E ). Assertion (ii) of the lemma with T = T and 1 p2 < + also holds. This result is a consequence (see Corollary 12.8.2) of the main lifting theorem (Theorem 12.8.1). Theorem 12.8.1 ([232]). Assume that there exist T0 and p1 ,p
2

0 t T,

[1, +] such that

KT0



B (Lp1 ([0,T ]; E ),Lp2 ([0,T ]; E )). Then there exist constants L > 0 and > 0 such that KT
B (Lp1 ([0,T ];E ),C ([0,T ];E ))

LeT ,

T > 0.

(12.21)

Corollary 12.8.1 ([232]). Assume that the conditions of Theorem 12.8.1 hold. Then there exist L > 0 and > 0 such that SV(KT ,T ) LeT T
1/p
1

,

T R.
B (Lp1 ([0,T ];E ),Lp2 ([0,T ];E ))

Corollary 12.8.2 ([232]). Let p1 , p2 [1, +], and let T > 0. Then KT + iff KT
B (Lp1 ([0,T ];E ),Lp2 ([0,T ];E ))

<

< +.

Corollary 12.8.3 ([232]). Assume that E is a Hilbert space and that the conditions of Theorem 12.8.1 hold. Then there exist L > 0 and > 0 such that SV(KT ,T ) LeT T
1/2

,

T > 0.

In particular, setting p1 = p2 = + in the obtained assertion, we obtain the following interesting consequence. Corollary 12.8.4 ([232]). Assume that E is Hilbert and that SV(KT ,T0 ) < + for a certain T0 > 0. Then there exist L > 0 and > 0 such that SV(KT ,T ) LeT T
1/2

,

T R+ . 165


In thesameway as for C0 -groups, under certain additional conditions, we can strengthen the assertion of Theorem 12.8.1. Definition 12.8.1. We say that the op erator KT in (12.19) satisfies Condition HC if there exist constants C0 > 0and T0 > 0 such that for any x E , there exists a function hx L ([0,T ]; E ) having the prop erties HC1 hx and HC2 Bhx (t) = C (t - T, A)Bx, 0 t T . Note that at least in two cases, Condition HC holds; namely, (i) when B commutes with C (t, A); for t R, we can set hx (t) = C (t - T, A)x, 0 t T , and (ii) when B is invertible; we can set hx (t) = B
-1 L ([0,T ];E )

C x

C (t - T, A)Bx, 0 t T .

Theorem 12.8.2 ([232]). Assume that there exist T0 > 0 and p1 ,p2 [1, ] such that KT0
B (Lp1 ([0,T ];E ),Lp2 ([0,T ];E ))

< .

Also, assume that Condition HC holds. Then there exist L > 0, > 0 such that KT
B (L1 ([0,T ];E ),C ([0,T ];E ))

< LeT .

12.9. Perturbation of C0 -Groups of Operators In this section, we use lifting theorems for studying multiplicative p erturbations. Let exp(·A) b e a C0 -group of linear b ounded op erators on a Banach space E . Let B B (E ) b e a certain linear b ounded op erator on E . Here we study the question whether the multiplicatively p erturb ed op erator Am = A(I +B ) defined on the natural domain D(Am ) = {x E : (I + B )x D(A)} is an infinitesimal op erator of another C0 -group under a condition on the op erator KT . First, assume that Am is the infinitesimal op erator of the C0 -group exp(·Am ). It is easy to verify that for any x E , the following relation holds:
t

exp(tAm )x = exp(tA)x + A
0

exp((t - s)A)B exp(sAm )xds,

t [0,T ].

(12.22)

Consider the convolution op erator KT : ([0,T ],E ) C ([0,T ]; E ) defined in (12.16). Assume that there exists T > 0 such that SV(KT ,T ) = KT : [0,T ] E defined by f f 166
x,T B (L ([0,T ];E ),L([0,T ];E ))

< +. Then KT can b e considered

as a b ounded op erator KT : C ([0,T ]; E ) C ([0,T ]; E ). Relation (12.22) means that for any x E , the mapping f
x,T x,T

(t) = exp(tAm )x for t [0,T ] satisfies the condition
x,T

(t) = exp(tA)x +(KT f

)(t),

t [0,T ].


The converse is also true. For a given x E , we study the solvability in C ([0,T ]; E ) of the convolution equation f (t) = exp(tA)x +(KT f )(t), 0 t T, (12.23)

and then show that f (t) can b e considered as exp(tAm )x. As in Sec. 12.1, this approach is divided into two steps: (Step 1) prove that there exists T > 0 such that KT maps C ([0,T ]; E ) into itself continuously and (Step 2) prove that SV(KT ,T ) = KT
B (L ([0,T ];E ),L ([0,T ];E ))

< 1 for a certain sufficiently small T > 0.

If (Step 1) and (Step 2) hold, then by the contraction mapping principle, Eq. (12.23) is uniquely solvable. Therefore, following Sec. 12.1, we obtain that the p erturb ed op erator Am is the generator of a C0 -semigroup. Theorem 12.9.1 ([232]). Assume that there exist T0 > 0, 1 p1 < +, and 1 p2 + such that KT B (Lp1 ([0,T ]; E ),Lp2 ([0,T ]; E )). Then there exist L > 0 and > 0 such that for any T > 0, SV(KT ,T ) LeT T Moreover, the operator Am generates a C0 -group and exp(tA) - exp(tAm ) = O(t
1/p
1

1/p

1

.

(12.24)

) as t 0.

(12.25)

Proof. Estimate (12.24) for t R+ was obtained in Corollary 12.7.1. This prop erty implies that assertions (Step 1) and (Step 2) hold. Therefore, as in Sec. 12.1, we obtain that Am generates a C0 semigroup exp(tAm ), t R+ . On the other hand, we fix T > 0 and set M = (12.24), yields the following for 0 t T : exp(tAm ) - exp(tA) Kt from which we obtain (12.25) for t R. It remains to show that Am generates a C0 -group and (12.26) also holds for t < 0. Choose 0 < T0 T so that M 2 LeT0 T0
1/p
1

-T tT

su p

exp(tAm ) . Identity (12.22), together with

B (L ([0,T ];E ),L([0,T ];E ))

CLeT |T |1/p1 ,

(12.26)

1/2. Take 0 t T0 . Then exp(tAm ) - exp(tA) exp(-tA)+ I exp(tA).

exp(tAm ) =

Further, by (12.26), for t > 0, and by the choice of T0 , we also have exp(tAm ) - exp(tA) exp(-tA) 1/2. 167


Therefore, the series expansion shows that the op erator exp(·Am ) is invertible and exp(tAm )-1 2M . In fact, this proves that Am generates a C0 -group. Finally, using the series expansion once again, we obtain 0 t T0 ,
+

exp(-tAm ) - exp(-tA)
j =1 +

exp(tAm ) - exp(tA) exp(-tA)
1/p j

j

=
j =1

M 2 Let t

1

2M 2 Let t

1/p

1

,

which proves (12.26) also for t < 0. For Hilb ert spaces the value p1 = + is also admissible. Indeed, using Corollary 12.7.2, we obtain the following theorem. Theorem 12.9.2 ([232]). Assume that E = H is Hilbert and there exist T0 > 0 and p1 , p2 [1, +], such that KT B (Lp1 ([0,T ]; E ),Lp2 ([0,T ]; E )). Then there exist L > 0 and > 0 such that SV(KT ,T ) LeT T Moreover, the operator Am generates a C0 -group and exp(tA) - exp(tAm ) = O(t
1/2 1/2

,

T R.

(12.27)

) as t 0.

Finally, under Condition HG, we can apply Theorem 12.7.2, which leads to the following theorem, which is true for Banach spaces. Theorem 12.9.3 ([232]). Let Condition HG hold. Also, assume that there exist T0 > 0 and p1 , p2 [1, +], such that KT0
B (Lp1 ([0,T0 ],E ),Lp2 ([0,T0 ],E ))

< +. Then Am = A(I + B ) is a generator of a

C0 -group, and we have the estimate exp(tA) - exp(tAm ) = O(t) as t 0. Moreover, if E is reflexive, then B takes its values in the domain of A, AB is a bounded operator, and Am = A + AB is a bounded perturbation of A. 12.10. Perturbations of C0 -Cosine Operator Functions Let C (·,A) b e a C0 -cosine family of linear b ounded op erators on a Banach space E . Taking a linear b ounded op erator B on E , we p ose the problem: whether the multiplicatively p erturb ed op erator Am = A(I + B ), acting on the domain D(Am ) = {x E : (I + B )x D(A)} is a generator of another 168


C0 -cosine family or not. First, assume that Am is a generator of a C0 -cosine family C (·,Am ). We have already known that
t

C (t, Am )x = C (t, A)x + A
0

S (t - s, A)BC (s, Am )xds,

t R,

x E.

(12.28)

Acting in the same way as in Sec. 12.9, for a fixed x E , we arrive at the convolution equation f (t) = C (t, A)x +(KT f )(t), 0 t T, (12.29)

where KT is defined in (12.19) and f is sought in C ([0,T ]; E ). As was shown in Sec. 12.1, if (i) there exists T > 0 such that KT continuously maps C ([0,T ]; E ) onto itself and (ii) SV(KT ,T ) = KT
B (L ([0,T ];E ),L([0,T ];E ))

< 1 for a sufficiently small T > 0,

then by the contraction mapping principle, Eq. (12.29) is uniquely solvable and Am is a generator of a C0 -cosine family. Theorem 12.10.1 ([232]). Assume that there exist T0 > 0, 1 p1 < +, and 1 p2 + such that KT
B (Lp1 ([0,T ];E ),Lp2 ([0,T ];E ))

< . Then there exist L > 0 and > 0 such that for any T > 0, the

fol lowing inequality holds: SV(KT ,T ) LeT T
1/p
1

.

(12.30)

Moreover, the operator Am generates a C0 -cosine operator family, and C (t, A) - C (t, Am ) = O(t
1/p
1

) as t 0.

(12.31)

Using Corollary 12.8.3, we easily obtain the following theorem. Theorem 12.10.2 ([232]). Assume that E = H is a Hilbert space and that there exist T0 > 0 and p1 , p2 [1, +] such that KT that SV(KT ,T ) LeT T
1/2 B (Lp1 ([0,T ];E ),Lp2 ([0,T ];E ))

< +. Then there exist L > 0 and > 0 such

,

T > 0.

(12.32)

Moreover, the operator Am generates a C0 -cosine operator family, and C (t, A) - C (t, Am ) = O(t
1/2

) as t 0.

Theorem 12.10.3 ([232]). Assume that Condition HC holds. Also, assume that there exist T0 > 0 and p1 , p2 [1, +] such that KT0
B (Lp1 ([0,T0 ];E ),Lp2 ([0,T0 ];E ))

< +. Then Am = A(I + B ) is a generator of

a C0 -cosine operator family, and the fol lowing estimate holds: C (t, A) - C (t, Am ) = O(t) as t 0. 169


Chapter 13
INHOMOGENEOUS EQUATIONS In a Banach space E , let us consider the inhomogeneous Cauchy problem u (t) = Au(t)+ f (t), t [0,T ], u(0) = u0 , (13.1)

with the op erator A generating a C0 -semigroup. For f 0, the well-p osed statement of such problems are describ ed in detail in Chapter 1 of [17]. Formally, as in the finite-dimensional analysis, problem (13.1) has a solution of the form u(t) = exp(tA)u0 +
0 t t

exp((t - s)A)f (s) ds,

t [0,T ].

(13.2)

This is the so-called constant variation formula. The prop erties of the expression
0

exp((t - s)A)f (s) ds

and the corresp onding interpretations of solutions related to representation (13.2) are of interest.

13.1. General Results It is known from the theory of partial differential equations that problems that are written in abstract form (13.1) are usually considered in the spaces of typ es C ([0,T ]; E ) or Lp ([0,T ]; E ). In this chapter, we restrict ourselves to these two statements. Thus, for example, if f (·) C ([0,T ]; E ), then for the C0 -semigroup exp(·A), the expression exp((t - s)A)f (s) is continuous in s [0,T ], and hence there exists
t 0

exp((t - s)A)f (s) ds.

On the other hand, if u(·) is a solution of problem (13.1) with f (·) C ([0,T ]; E ), which b elongs to d C ([0,T ]; E ) C 1 ((0,T ]; E ), then exp((t - s)A)u(s) = exp((t - s)A)f (s) and integrating over (0,t), ds we obtain (13.2). The converse is not true in general, since the function u(·) given by expression (13.2) can b e not differentiable. In the theory of abstract differential equation, the following theorem is well known. Theorem 13.1.1 ([20, 31]). Let A G (M, ), u0 D(A), and let the function f (·) be such that either (i) f (·) C 1 ([0,T ]; E ) or (ii) f (·) C ([0,T ]; E ) takes its values in D(A) and, moreover, Af (·) C ([0,T ]; E ). Then problem (13.1) has a unique solution u(·) C 1 ([0,T ]; E ) with the initial condition u0 that is represented in form (13.2). 170


In the case where f (·) satisfies the H¨ der condition ol f (t) - f (s) M |t - s| for 0 s, t T, (13.3)

with certain constants M > 0 and 0 < 1, the following theorem holds. Theorem 13.1.2 ([20, 31]). Let A H(, ), and let f (·) satisfy the H¨ r condition (13.3). Then the olde function u(·) from (13.2) belongs to C ([0,T ]; E ) C 1 ((0,T ]; E ) and is a solution of problem (13.1) for any u0 E . Moreover, u(·) C 1 ([0,T ]; E ) if u0 D(A). The Cauchy­Kowalewski theorem can also b e written in an abstract form. Let {E : 0 1} be Banach spaces having the prop erties E2 E1 if 1 < 2 and x
E
1

x

E

2

for any x E2 .

Let L b e the set of linear op erators Q L(E2 ,E1 ) for 0 1 < 2 < 1 such that Qx
E
1



2 -

x
1

E

2

for any

x E2 .

A function A(t) : [0,T ] L is said to b e continuous if for any > 0, t0 [0,T ], and > , there exists > 0 such that for |t - t0 | < , we have (A(t) - A(t0 ))x
E


x

E



for any x E .

We call attention to the fact that the space L is a Banach space with the norm Q
L

=

0 <1 x

su p

sup( - ) Qx

E



x

-1 E

.

Theorem 13.1.3 ([29]). Let u0 E1 , f (·) C ([-T, T ]; E1 ), and let A(·) C ([-T, T ]; L ). Then 1. For each [0, 1), there exists a function u(·) defined for 0 t < Ts := min(T, (1 - )(e)-1 ) and taking its values in E . The function u(·) is continuously differentiable and satisfies the equation u (t) = A(t)u(t)+ f (t) and the condition u(0) = u0 . 2. If for certain (0, 1] and 0 < T T , on the set [0,T ], we have two functions with values in E that are continuously differentiable, satisfy (13.4), and coincide for t = 0, then these functions coincide on [0,T ]. For a second order equation, in a formula of form (13.2), instead of exp(·A) we have C (·,A) and S (·,A). However, often, there are no technical distinctions in studying these problems, and, as a rule, we restrict ourselves to proofs only for the case of second order equations. 171 for 0 t < Ts (13.4)


13.2. Inhomogeneous Equations in C ([0,T ]; E ) In a Banach space E , let us consider problem (13.1) with the op erator A generating a C0 -semigroup. Definition 13.2.1. A classical solution of problem (13.1) is a function u(·) such that u(·) C 1 ([0,T ]; E ), u(t) D(A) for all t [0,T ], and relations (13.1) hold. Proposition 13.2.1 ([235]). Let f (·) L1 ([0,T ]; E ). Then for any u0 E , problem (13.1) has not more than one classical solution. If it has a classical solution, then this solution has the form (13.2). Definition 13.2.2. A weakened solution of problem (13.1) is a function u(·) C ([0,T ]; E ) such that u (·) C ((0,T ]; E ) and Eq. (13.1) holds on (0,T ]. Theorem 13.2.1 ([47]). Let problem (13.1) with f (·) C ([0,T ]; E ) and u0 D(A) have a weakened solution u(·), and let (A) = . Then u(·) is given in the form (13.2). As was noted, the function u(·) given in the form (13.2) is neither a classical nor a weakened solution in general, since it can b e not differentiable. Definition 13.2.3. A function u(·) C ([0,T ]; E ) given by (13.2) is called a mild solution of problem (13.1). Theorem 13.2.2 ([235]). Let A G (M, - ) with > 0, and let the function f (·) : [0, ) E be bounded and measurable on [0, ). If s- lim f (t) = f , then a mild solution u(·) defined by (13.2) has
t

the fol lowing behavior: s- lim u(t) = -A-1 f .
t

In a Banach space E , let us consider the Cauchy problem u (t) = Au(t)+ f (t), t [0,T ], u(0) = u0 , u (0) = u1 , (13.5)

with the op erator A generating a C0 -cosine op erator function. Definition 13.2.4. A function u(·) is called a classical solution of problem (13.5) if u(·) is twice continuously differentiable, u(t) D(A) for all t [0,T ], and u(·) satisfies relations (13.5). If f (·) C ([0,T ]; E ) and u(·) is a classical solution of (13.5), then, considering the expression d C (t - s, A)u(s)+ S (t - s, A)u (s) = S (t - s, A)f (s) and integrating it in 0 s < t, we obtain ds u(t) = C (t, A)u0 + S (t, A)u1 +
0 t

S (t - s, A)f (s) ds,

t [0,T ].

(13.6)

172


As in the case of C0 -semigroups of op erators, the function u(·) given by (13.6) is not a classical solution in general, since it can b e not twice continuously differentiable. Proposition 13.2.2 ([135]). Let A C (M, ), and let either (i) f (·),Af (·) C ([0,T ); E ) and f (t) D(A) for t [0,T ] or (ii) f (·) C 1 ([0,T ]; E ). Then the function u(·) from (13.6) with u0 D(A) and u1 E 1 is a classical solution of problem (13.5) on [0,T ]. Definition 13.2.5. The function u(·) C ([0,T ); E ) given by expression (13.6) is called a mild solution of problem (13.5). Theorem 13.2.3 ([47]). Let the operator B = (i) f (·) C 1 ([0,T ); E ); (ii) Bf (·) C ([0,T ); E ). Then for any u0 D(A) and u1 D(B ), there exists a unique classical solution of problem (13.5) given by formula (13.6) in the form u(t) = 1 1 exp(tB )+ exp(-tB ) u0 + exp(tB ) - exp(-tB ) B -1u1 2 2 1t + exp((t - s)B )+ exp(-(t - s)B ) B -1 f (s) ds, t [0,T ]. 20 A in problem (13.5) have a bounded inverse B
-1

B (E )

and be a generator of a C0 -group, and let the function f (·) have one of the fol lowing properties:

(13.7)

Theorem 13.2.4 ([106]). Let A C (M, ). The Cauchy problem u (t) = Au(t)+ f (t), u(0) = x, u (0) = y, (13.8)

has a unique 2 -periodic mild solution of class C 1 for any 2 -periodic function f (·) L2 c (R; E ) iff lo 1 C (2, A) . Note that the condition 1 (C (, A)) is equivalent to the conditions {-4 2 k2 /T 2 }k sup k(4 2 k2 /T 2 + A)-1 < .
k Z Z

(A) and

13.3. Coercivity in the Case of Classical Solutions Since the existence of a classical solution of problem (13.1) presupp oses the continuity of the derivative of the function u(·) on [0,T ], then by representation (13.2), it is natural to consider the differentiability of the expression (exp(·A) f )(t) in the variable t R+ . 173


Definition 13.3.1. We say that a C0 -semigroup exp(·A) has the prop erty of maximal regularity (MRprop erty in brief ) if (exp(·A) f )(·) C 1 ([0,T ]; E ) (or, which is equivalent, (exp(·A) f )(·) C ([0,T ]; D(A)) for all f (·) C ([0,T ]; E )). Proposition 13.3.1 ([124]). The convolution (exp(·A) f )(·) C 1 ([0,T ]; E ) iff (exp(·A) f )(t) D(A) for al l t [0,T ] and (exp(·A) f ) C ([0,T ]; D(A)). Proposition 13.3.2 ([124]). The Cauchy problem (13.1) has a classical solution for any f C ([0,T ]; E ) iff A generates a C0 -semigroup with the MR-property. Proposition 13.3.3 ([271]). An operator A generates a C0 -semigroup with the MR-property iff SV(exp(·A),t) is bounded on [0,T ]. Theorem 13.3.1 ([89, 124]). If a C0 -semigroup exp(·A) has the MR-property, then either A is bounded or the space E contains a closed subspace isomorphic to c0 . As was shown in [124], there exist unb ounded op erators on E = c0 that generate C0 -semigroups with the MR-prop erty. Since an op erator A generating a C0 -semigroup is closed, by the closed graph theorem, in the case of a C0 -semigroup with the MR-prop erty, the op erator A(exp(·A) f ) defined on the whole C ([0,T ]; E ) is continuous as an op erator from C ([0,T ]; E ) into C ([0,T ]; E ). This means that the following inequality holds: A(exp(·A) f )
C ([0,T ];E )

C f

C ([0,T ];E )

with a certain constant C indep endent of f (·). Generalizing the clearness of the previous inequality for formulating the MR-prop erty in the space C ([0,T ]; E ) to the description of well-p osedness of the Cauchy problem for an inhomogeneous equation in spaces of typ es C ,h , and so on, we arrive at the following definition. Definition 13.3.2. Let F b e a Banach space b eing a subspace of the initial space E , ([0,T ]; E ) b e the Banach space of functions with values in E . Problem (13.1) is said to b e coercively solvable in the pair of spaces (F, ([0,T ]; E )) (i.e., the solution u(·) has the maximal regularity property) if for any u0 F and any right-hand side f (·) ([0,T ]; E ), there exists a classical solution u(·) of the Cauchy problem (13.1), and for this solution, we have the coercive inequality u (·) 174
([0,T ];E )

+ Au(·)

([0,T ];E )

M ( f (·)

([0,T ];E )

+ u0

F

).

(13.9)


The formulation of this definition is very convenient. So, for example, Definition 13.3.2 for u(t) = t exp(tA)u0 and f (t) = exp(tA)u0 and Prop osition 13.3.3 trivially imply the following assertion. Proposition 13.3.4 ([271]). Let A be a generator of a C0 -semigroup, and let SV(exp(·A),t) < . Then the semigroup exp(·A) is analytic. However, the analyticity of the C0 -semigroup exp(·A) is not sufficient for the coercive solvability of problem (13.1) in C ([0,T ]; E ) (see [72]). Therefore, taking into account Theorem 13.3.1, we see that the study of coercivity in the space C ([0,T ]; E ) is not interesting. However, one can prove the following theorem. Theorem 13.3.2 ([219]). Let - < a < b < , 1 p, q , 0, and let A H(, ). Then for any
f Bp,q ((a, b); E ) with > 1/p or = 1/p and q = 1, we have exp(·A)f C 1 ((a, b); E )C 1 ((a, b); D(A))

and (13.1) holds for any t (a, b) for the function u(·) = exp(·A) f .
In fact, under the condition of the previous theorem, we can prove that for any f Bp,q ((a, b); E ) +1 L1 ((a, b); E ), it follows that exp(·A) f Bp,q ((a, b); E ), where a < a1 < b, and there exists a constant

c > 0 with the prop erty exp(·A) f
+1 Bp,q ((a,b);E )

c( f

Bp,q ((a,b);E )

+f

L1 ((a,b);E )

).

However, such estimates are not coercive. Proposition 13.3.5 ([70]). Let A H(, ), i.e., A generates a certain analytic C0 -semigroup, F =
D(A), and ([0,T ]; E )) is the H¨ r space of functions C0 ([0,T ]; E ) for which the fol lowing norm is olde

finite: f (·)
C0 ([0,T ];E )

= su p

0tT

f (t)

E

+

0t, ,t+ T

su p

f (t + ) - f (t)

E

t.

Then the Cauchy problem (13.1) is coercively solvable in the pair of spaces (F, C0 ([0,T ]; E )). , Theorem 13.3.3 ([3]). Let v = f (0) - Au(0) E- , f C0 ([0, 1]; E- ),A H(, ) for 0 , , 0 < < 1. Then there exists a unique solution of problem (13.1), Au, u C0 ([0, 1]; E- ),

u C ([0, 1]; E- ), and u
, C0 (E -

)

+ Au

, C0 (E

-

)

+u

C ([0,1];E

-

)

c

v0

-

+

1 f (1 - )

, C0 ([0,1];E

-

)

,

, where C0 has the norm

max f
t

E

+

0
max

f (t + ) - f (t)

E

(t + ) .

175


Remark 13.3.1 ([68]). In the case where E = lp , the op erator A(xk ) = (ik xk ) k =1 k =1 (i = -1),

generates a strongly continuous C0 -semigroup (but not an analytic one!). For the right-hand side f (t) = k
-(1-1/p) eik t k =1

satisfying the Holder condition with any exp onent (0, 1), the function ¨
t

(t) =
0

exp((t - s)A)f (s) ds

does not b elong to D(A) for any t. This example also shows that in the case where the op erator A generates a C0 -semigroup only, the H¨lder prop erty of the right-hand side is not sufficient for the existence of a classical solution. o Denote (S (·,A) f )(t) :=
t 0

S (t - s, A)f (s) ds, t [0,T ].

Definition 13.3.3. We say that a C0 -cosine op erator function C (·,A) has the maximal regularity (MRproperty) if S (·,A) f C 2 ([0,T ]; E ) (or, which is equivalent, C (·,A) f C ([0,T ]; D(A)) for all f (·) C ([0,T ]; E ). Proposition 13.3.6 ([100]). Let x, y D(A). Then the fol lowing assertions are equivalent: (i) problem (13.5) has a classical solution for a given f (·); (ii) S (·,A) f C 2([0,T ]; E ); (iii) (S (·,A) f )(t) D(A) for 0 t T , and A(S (·,A) f )(t) is continuous in t [0,T ], i.e., (S (·,A) f ) C ([0,T ], D(A)). Proof. (i) = (ii). We know that if u(·) is a solution of (13.5), then u(·) is twice continuously differentiable and u(t) = C (t, A)x + S (t, A)y +(S (·,A) f )(t). Therefore, we have (S (·,A) f ) (t) = u (t) - C (t, A)x - S (t, A)y = u (t) - C (t, A)Ax - S (t, A)Ay C ([0,T ]; E ), i.e., S (·,A) f C 2 ([0,T ]; E ). (ii) = (iii). Since 2 (C (h, A) - I )(S h2 1 +2 h 1 =2 h 1 +2 h we have
h0+

(·,A) f )(t) = -
t t+h

1 ((S (·,A) f )(t + h) - 2(S (·,A) f )(t)+ (S (·,A) f )(t - h)) h2
t t-h

S (t - s + h, A)f (s) ds +

S (t - s - h, A)f (s)ds

=

(S (·,A) f )(t + h) - 2(S (·,A) f )(t)+ (S (·,A) f )(t - h) -
t t+h

S (t - s + h, A)f (s)ds +

t t-h

S (t - s - h, A)f (s) ds ,

(13.10)

lim

2 (C (t, A) - I )(S (·,A) f )(t) = (S (·,A) f ) (t) - f (t), h2

176


i.e., (S (·,A) f )(t) D(A) and A(S (·,A) f )(t) = (S (·,A) f ) (t) - f (t). Therefore, A(S (·,A) f )(·) C ([0,T ]; E ). (iii) = (i). By (13.10), 1 2 (S (·,A) f )(t + h) - 2(S (·,A) f )(t)+ (S (·,A) f )(t - h) = 2 C (h, A) - I (S (·,A) f )(t) 2 h h t+h t 1 -2 - S (t - s + h, A)f (s)ds + S (t - s - h, A)f (s)ds . h t t-h Then (c) implies (S (·,A) f ) = A(S (·,A) f )+ f C ([0,T ]; E ). Therefore, S (·,A) f is a solution of the Cauchy problem (13.5) for a given f (·) and zero initial data, and u(t) = C (t, A)x + S (t, A)y +(S f )(t), t R, is a solution of the Cauchy problem (13.5) for a given f (·) for each pair x, y D(A). Theorem 13.3.4 ([100]). For a cosine operator function C (·,A), the fol lowing conditions are equivalent: (i) the generator A is bounded; (ii) C (t, A) - I = O(t2 ) (t 0+ ); (iii) Var(C (·,A),t) = O(t2 )(t 0+ ); (iv) Var(C (·,A),t) = o(1) (t 0+ ); (v) SV(C (·,A),t) = O(t2 )(t 0+ ); (vi) SV(C (·,A),t) = o(1) (t 0+ ); (vii) SV(C (·,A),t) < for some t > 0, i.e., C (·,A) is local ly of bounded semivariation; (viii) R(S (t, A)) D(A) for t (-, ), and AS (t, A) is bounded on [a, b] for some 0 < a < b; 2 (ix) R(S (t, A)) D(A) for t (-, ) and lim sup tAS (t, A) < . e + t0 Definition 13.3.4. Let F b e a Banach space b eing a subspace of the initial space E , and let ([0,T ]; E ) b e the Banach space of functions with values in E . Problem (13.5) is said to b e coercively solvable in the pair of spaces (F, ([0,T ]; E )) (in other words, the solution has u(·) the maximal regularity property) if for any right-hand side f (·) ([0,T ]; E ), there exists a classical solution u(·) of the Cauchy problem (13.5), for each t, the value of the solution u(t) b elongs to F , and the following coercive inequality holds for it: u (·)
([0,T ];E )

+ Au(·)

([0,T ];E )

M

f (·)

([0,T ];E )

+ u0

F

+ u1

F

.

Theorem 13.3.5 ([100]). Let problem (13.5) be coercively solvable in the pair (D(A), C ([0,T ]; E )). Then A B (E ). This result can also b e reformulated as follows: 177


Theorem 13.3.6 ([100]). The fol lowing statements are equivalent: (i) for al l x, y D(A) and f C ([0,r ]; E ), problem (13.5) has a classical solution; (ii) the operator A generates a C0 -cosine operator function that satisfies the MR-property; (iii) the operator A generates a C0 -cosine operator function that is of bounded semivariation on [0,r ]; (iv) A is a bounded linear operator on E . 13.4. Coercivity in Lp ([0,T ]; E ) Let 1 p . Then [92] u(·) W Lp ([0,T ]; E ). Definition 13.4.1. A function u(·) is said to b e absolutely continuous if there exists a function v (·) L1 ([0,T ]; E ) such that
t 1,p

([0,T ]; E ) iff u(·) is absolutely continuous and u(·),u (·)

u(t) = u( )+


v ( ) d

for all

, t [0,T ].

An absolutely continuous function u(·) is continuous and differentiable almost everywhere on [0,T ], and, moreover, u (·) = v (·). Definition 13.4.2. A classical solution of problem (13.1) in the space Lp ([0,T ]; E ) is an absolutely continuous function u(·) such that the function u (·),Au(·) Lp ([0,T ]; E ) satisfies Eq. (13.1) almost everywhere on [0,T ] and u(0) = u0 . Definition 13.4.3. Problem (13.1) is said to b e wel l-posed in Lp ([0,T ]; E ) if for any f (·) Lp ([0,T ]; E ), there exists a unique classical solution u(·) in Lp ([0,T ]; E ) continuously dep ending on u0 and f (·). Definition 13.4.4. Problem (13.1) is said to b e coercively solvable in Lp ([0,T ]; E ) if for any f (·) Lp ([0,T ]; E ) and u0 F , there exists a unique classical solution of problem (13.1) in Lp ([0,T ]; E ) and u (·)
Lp ([0,T ];E )

+ Au(·)

Lp ([0,T ];E )

M (p)( f (·)

Lp ([0,T ];E )

+ u0

F

).

(13.11)

Recall that if a certain prop erty holds locally, then we write Lp c instead of Lp . lo
1 Definition 13.4.5. A function u(·) is called a Wlo,p -solution of the Cauchy problem (13.1) if u(·) c 1 Wlo,p ([0,T ]; E ) Lp ([0,T ]; D(A)) and (13.1) is satisfied. c 1 Proposition 13.4.1 ([77]). Let 1 p , A H(, ), and let u(·) is a Wlo,p -solution of the Cauchy c

problem (13.1). Then we have representation (13.2) for this solution. Proposition 13.4.2 ([72]). The coercive solvability of problem (13.1) in Lp ([0,T ]; E ) and the compactness of the resolvent (I - A)-1 imply the analyticity of the C0 -semigroup exp(·A). 178


To understand which space F should b e in Definition 13.4.3, we consider the function A exp(tA)u0 . Obviously, F is the space with the norm u0
T F 1/p

=
0

A exp(tA)u0

p E

dt

+ u0

E

.

(13.12)

Such a space F is a particular case of the spaces E,p , 0 < < 1, 1 p , with the norm u
0 E
,p

=u u0

0

T E

+
0

t

1-

A exp

(tA)u0 p E
E

dt t

1/p

,

E

,

= su p
>0

1-

A exp(A)u0

.

We denote it by E1- 1 = E1-
p

1 p

,p

.

Theorem 13.4.1 ([82]). For any function f (·) Lp ([0,T ]; E ) and any u0 E1- 1 , formula (13.2) defines
p

an E1- 1 -valued function u(·) on [0,T ] and
p

0tT

max u(t)

E

1 1- p

M

u0

E

1- 1 p

+

p2 f p-1

Lp ([0,T ];E )

.

Theorem 13.4.2 ([82]). Let problem (13.1) be coercively solvable in the space Lp0 ([0,T ]; E ) for a certain 1 < p0 < with M (p0 ) = M . Then it is coercively solvable for any 1 < p < , and estimate (13.11) p2 . holds with M (p) = M p-1 Theorem 13.4.3. If the space E = H is Hilbert and A H(, ), then problem (13.1) is coercively solvable in L2 ([0,T ]; H ). Theorem 13.4.4 ([82]). Under the conditions of Theorem 13.4.2, for any f (·) Lp ([0,T ]; E ) and u0 E1- 1 , problem (13.1) has a unique solution u(·) in Lp ([0,T ]; E ) such that the coercive inequality (13.11)
p

holds in the form u
Lp ([0,T ];E )

+ Au

Lp ([0,T ];E )

+ max u(t)
0tT

E

1 1- p

M

p2 p-1

f

Lp ([0,T ];E )

+ u0

E

1 1- p

.

Theorem 13.4.5 ([82]). Let f (·) Lq ([0,T ]; E,q ), 0 < < 1, 1 q , and let A H(, ). Then there exists a unique absolutely continuous solution u(·) of problem (13.1) with u0 = 0 such that Au(·),u (·) Lq ([0,T ]; E,q ), and u
Lq ([0,T ];E
,q

)

+ Au

Lq ([0,T ];E

,q

)



M f (1 - )

Lq ([0,T ];E

,q

)

,

and, moreover, the constant M is independent of f, , and q . Theorem 13.4.6 ([82]). Let A H(, ), and let 1 < p, q < , or p = q = . Then problem (13.1) admits an absolutely continuous solution u(·) such that u ,Au Lp ([0,T ]; E,q ), and u(·) is a continuous 179


E1+-

1 p

,q

-valued function iff f (·) Lp ([0,T ]; E,p ) and u0 E1+-

1 p

,q

. This solution u(·) satisfies the

inequality u (·)
Lp ([0,T ];E )

,q

+ Au(·)
E
1 1+- p ,q

Lp ([0,T ];E

,q

)

+ max u(t)
0tT Lp ([0,T ];E )

E

1+- 1 ,q p

M

u0

+

M (p, q ) f (1 - )

,q

,

where M (p, q ) =

M (q )p2 if p = q and M (p, p) = 1. p-1

Theorem 13.4.7 ([82]). Let p = q = 1 or p = q = . Then problem (13.1) has an absolutely continuous solution u(·) such that u (·),Au(·) Lp ([0,T ]; E,p ) iff f (·) Lp ([0,T ]; E,q )) and u0 E1+- 1 , .
p

This solution satisfies the inequality u(·)
Lp ([0,T ];E )

,p

+ Au(·)

Lp ([0,T ];E

,p

)

M

u0

1+- 1 ,p p

+

1 f (1 - )

Lp ([0,T ];E

,p

)

.

Proposition 13.4.3 ([82]). Let 1 < p < , f (·) Lp ([0,T ]; E,p ), and let u0 E1+- from formula (13.2) satisfies the estimate max u(t)
E

1 p

,

. Then u(·)

0tT

1 1+- p ,

M

u0

E

1 1+- p ,

+

p2 f p-1

Lp ([0,T ];E

,

)

.

~ ~ ~ We set || = µ() for a µ-measurable set . Definition 13.4.6. If (exp(tA)f )(x) =


k(t, x, y )f (y ) dy , t R, then we say that k satisfies the Poisson

estimate of order m N if |k(t, x, y )| P (t, x, y ) for almost all x, y , where P (t, x, y ) := |B (x, t1/m )|-1 p d(x, y )m t

and p(·) is a b ounded, continuous, and strongly p ositive function satisfying the condition
r

lim r

n+

p(r m ) = 0

for a certain > 0 and |B (x, )| := {y : d(x, y ) < }. Theorem 13.4.8 ([163]). Let 1 < p, q < and let (,µ,d) be a topological space satisfying the fol lowing conditions: (i) |B (x, 2)| C |B (x, )|, where B (x, ) is the bal l of radius centered at a point x; (ii) ess sup |B (x, )| C ess inf |B (x, )|; (iii) the operator A generates an analytic C0 -semigroup on L2 () with (A) < 0. 180
x x


Let a semigroup exp(·A) be represented by a kernel satisfying the Poisson estimate of order m N. Then A MR(p, Lq ()), i.e., for each f Lp (R+ ; Lq ()), there exists a unique solution u W
1,p

(R+ ; Lq ()) Lp (R+ ; D(Aq )) of problem (13.1) with u0 = 0 in the sense of Lp (R+ ; Lq ()). Moreover,


u(t)
0

p Lq ()



dt +
0

u (t)

p Lq ()



dt +
0

Au(t)

p Lq ()

dt C
0



f (t)

p Lq ()

dt

for any f (·) Lp (R+ ; Lq ()). Definition 13.4.7. A p ositive op erator A C (E ) is called an operator of bounded imaginary powers if there exist > 0 and M 1 such that Ait B (E ) and Ait M for - t . Proposition 13.4.4 ([77]). Let A be a positive operator of bounded imaginary powers. Then there exist M 1 and 0 such that {Ait }tR is a C0 -group of operators on E with the generator i log A and Ait Me|t| , t R. Note that if E = H is Hilb ert and A = A I > 0, then A is an op erator of b ounded imaginary powers. Definition 13.4.8. Let S (R; E ) b e the Schwartz space of smooth rapidly decreasing E -valued functions. For u(·) S (R; E ), define the Hilbert transform (Hu)(t) := 1 PV 1 t u = 1 p.v.
-

u( ) d, t-

t R.

For an arbitrary Banach space E , the Hilb ert transform can b e not a b ounded op erator on Lp (R; E ), even for a certain p (1, ). Definition 13.4.9. A space E is called an UMD-space if the Hilb ert transform is a b ounded op erator on Lp (R; E ) for a certain p (1, ). Proposition 13.4.5 ([77]). A Hilbert space H , any Banach space isomorphic to an UMD-space, any interpolation space (X, Y ),p and [X, Y ],p constructed via an interpolation pair of UMD-spaces, and any finite-dimensional space are UMD-spaces. Proposition 13.4.6 ([77]). Let E be an UMD-space. Then the Hilbert transform is a bounded operator on Lp (R,E ) for any p (1, ). Proposition 13.4.7 ([77]). Let E be an UMD-space, and let an operator A H(, ) be such that (-A)it Me|t| , t R.
1 p

Then problem (13.1) is coercively solvable in Lp ([0,T ]; E ) with F =

(E, D(A))1-

,p

. 181


In connection with Prop osition 13.4.7, the following assertion is of interest. Proposition 13.4.8 ([193]). Let A H(0, ) on a Hilbert space E = H . The fol lowing conditions are equivalent: (i) there exist C > 0 and such that (-A)it Cet , t R; (ii) there exists an operator Q B (H ) such that Q-1 B (H ) and Q-1 exp(tA)Q 1, t R+ . It is clear that the study of the coercivity of problems (13.1) is in fact that of the convolution op erator A exp((t - s)A)f (s) ds on the space Lp ([0,T ]; E ). For such an op erator, it is natural to apply the Mikhlin theorem on Fourier multipliers in order to prove its continuity on the Lp (R; E ) space. Recently, this approach was realized in [172, 292, 293]. The Poisson semigroup on L1 (R) and on Lp (R; E ) is not coercively well p osed on the Lp (R,E ) space if E is not an UMD-space (see [189]). Hence the assumption on E to b e an UMD space is necessary in some sense. But it was an op en problem whether every generator of an analytic semigroup on Lq (,µ), 1 < q < , yields the coercive well-p osedness in Lp (R; E ). Recently, Kalton and Lancien [171] gave a strong negative answer to this question. If every b ounded analytic semigroup on a Banach space E is such that problem (13.1) is coercively well p osed, then E is isomorphic to a Hilb ert space. If A generates a b ounded analytic semigroup {exp(zA) : | arg(z )| } on a Banach space E , then the following three sets are b ounded in the op erator norm (i) {(I - A)-1 : iR, = 0}; (ii) {exp(tA),tA exp(tA) : t > 0}; (iii) {exp(zA) : | arg z | }. In Hilb ert spaces, this already implies the coercive well-p osedness in Lp (R+ ; E ), but only in Hilb ert spaces E . The additional assumption that we need in more general Banach spaces E is the R-b oundedness. Aset T B (E )is said tob e R-bounded if there exists a constant C < such that for all Q1 ,... ,Qk T and x1 ,... ,xk E, k N,
1 0 k 0 t

rj (u)Qj (xj ) du C
j =0 0

1

k

rj (u)xj du,
j =0

(13.13)

where {rj } is a sequence of indep endent symmetric {-1, 1}-valued random variables, e.g., the Rademacher functions rj (t) = sign(sin(2j t)) on [0, 1]. The smallest C such that (13.13) is fulfilled is called the Rb oundedness constant of T and is denoted by R(T ).

182


Theorem 13.4.9 ([293]). Let A generate a bounded analytic semigroup exp(tA) on an UMD-space E . Then problem (13.1) is coercively wel l posed in the space Lp (R+ ; E ) iff one of sets (i), (ii), and (iii) presented above is R-bounded. A discrete variant of Theorem 13.4.9 was considered in [83]. Definition 13.4.10. Problem (13.5) is said to b e coercively solvable in Lp ([0,T ]; E ), 1 p , if for any f (·) Lp ([0,T ]; E ), there exists a unique solution u(·) satisfying the equation almost everywhere such that u(0) = u0 , u (0) = u1 , u (·),Au(·) Lp ([0,T ]; E ), and the following coercive inequality holds: u (·)
Lp ([0,T ];E )

+ Au(·)

Lp ([0,T ];E )

M (p)( f (·)

Lp ([0,T ];E )

+ u0

D(A)

+ u1

E

1

).

(13.14)

Theorem 13.4.10 ([232]). Let problem (13.5) be coercively solvable in Lp ([0,T ]; E ) with a certain 1 p . Then A is bounded. Proof.
t 0

For simplicity, we set u0 = u1 = 0.

Then (13.14) implies that the op erator (Kf )(t) :=

A S (t - s, A)f (s) ds, i.e., the op erator K from (12.19) with B = I, is a continuous op erator acting from Lp ([0,T ]; E ) into Lp ([0,T ]; E ). Theorem 12.7.1 implies K B (Lp ([0,T ]; E ),C ([0,T ]; E )). We now take f C ([0,T ]; E ). Then we obtain from Corollary 12.8.2 that AS (·,A) f C ([0,T ];E ) 1 1 CT 1/p T 1/q f C ([0,T ];E ), where + = 1, and, therefore, SV(C (·,A),t) Ct as t 0. Therefore, by p q Prop osition 8.1.14, we obtain the b oundedness of the op erator A. 13.5. Coercivity in B ([0,T ]; C 2 ()) In [153], the following result was proved. Theorem 13.5.1. Let be an open bounded subset of Rn lying to one side of its topological boundary , which is a submanifold of Rn of dimension n - 1 and class C A = A(x, Dx ) =
||2 2+

for some (0, 2) \ {1}. Let
||=2

a (x)Dx be a second-order strongly el liptic operator (i.e., Re

a (x) | |2

for some > 0 and for any (x, ) â Rn ) with coefficients of class C (). Then there exist µ 0, , such that for any C with || µ and | Arg | 0 , the problem 0 2 u -Au = f, (13.15) 0 u = 0, has a unique solution u(·) belonging to C ||1+
2

2+

() for any f (·) C (), and for a certain M > 0,
C
2+

u

C ()

+ || u

C ()

+u

()

M

f

C ()

+ ||

2

0 f

C ( )

,

(13.16)

where 0 is the trace operator on . 183


It is clear from (13.16) that op erator A does not generate C0 -semigroup on E = C () space in general, but, following, say, to [201], we can construct a semigroup exp(tA),t 0, which is analytic. Let us consider the following mixed Cauchy­Dirichlet u (t, x) = Au(t, x)+ f (t, x), t u(t, x ) = g(t, x ), u(0,x) = u (x), 0 parab olic problem: t [0,T ],x , t [0,T ],x , x . (13.17)

Definition 13.5.1. We say that problem (13.17) has a strict solution if there exists a continuous function u(t, x) such that it has the first derivative in t and the derivatives of order less than or equal to 2 in the space variables that are continuous up to the b oundary of [0,T ] â , i.e., u(·) C C [0,T ]; C 2 () , and the equations in (13.17) are satisfied. The space B ([0,T ]; C 2 ()) is defined as the space of b ounded functions u(·) : [0,T ] C 2 () endowed with the usual sup-norm. Theorem 13.5.2 ([153]). Let the fol lowing assumptions hold for some (0, 2) \{1}: (I) is an open bounded subset of Rn lying to one side of its topological boundary , which is a submanifold of Rn of dimension n - 1 and class C (I I) operator A = A(x, x ) = Re
||=2 ||2 2+2 1

[0,T ]; C ()

;

a (x)x

is a second-order strongly el liptic operator (i.e.,

a (x) | |2 for some > 0 and for any (x, ) â Rn ) with coefficients of class C 2 ().
2+2

Then problem (13.17) has a unique strict solution u(·) belonging to B [0,T ]; C u B [0,T ]; C 2 () iff the fol lowing conditions are satisfied: t (a) u0 C 2+2 (); (b) f C [0,T ]; C () B [0,T ]; C 2 () ; (c) g B [0,T ]; C
2+2

()

such that

( ) C [0,T ]; C 2 ( ) C

1

[0,T ]; C ( ) ,

g - f C [0,T ]; C ( ) ; t (d) u0 = g(0, ·); g (0, ·) - f (0, ·) = Au0 . (e) t 13.6. Boundary Value Problems Let us consider the following two-p oint problem: u(m) (t) = Au(t)+ f (t), t [0,T ], u(j ) (0) = u0 j , j 1 , u(k) (T ) = u1 k , k 2 ,

g B [0,T ]; C 2 ( ) , t

(13.18)

184


with a continuous function f (·) C ([0,T ]; E ). By a solution of problem (13.18), we mean a function u(·) C m ([0,T ]; E ) taking its values in D(A) and satisfying (13.18) with u0 , u1 D for a certain set D dense in D(A). j k Obviously, the definition of well-p osedness is as follows: if fn (t) 0 uniformly in t from the closed interval [0,T ] and u0 0, u1 n 0 as j 1 , k 2 , and n , then un (t) 0 uniformly in j,n k, t [0,T ]. Theorem 13.6.1 ([136]). Let problem (13.18) be wel l posed for u0 = 0, u1 = 0 for any j 1 ,k 2 . j k Then m0 + m1 m, where m0 = |0 |, m1 = |1 |. Theorem 13.6.2 ([136]). Let problem (13.18) be wel l posed for f (·) 0. Then m0 + m1 m. Theorem 13.6.3 ([136]). Let problem (13.18) be wel l posed, and let either m-2 m-2 or m1 < , (i) m be even and m0 < 2 2 or m-1 m-1 (ii) m be odd and m0 < or m1 < . 2 2 Then A is bounded. Theorem 13.6.4 ([106]). Let A C (M, ) and -N2 (A), and let both limits (5.1) exist for al l x E . 0 Then there exists a unique solution of the Dirichlet problem u (t) = Au(t), and, moreover,
0s

u(0) = x,

u( ) = y,

0 t ,

sup

u(s) c( u(0) + u( ) ).

Let us consider in the Banach space E the b oundary value problem u (t) = Au(t)+ f (t), t (0,T ),u(0) = u0 ,u(T ) = uT , (13.19)

where op erator -A generates a C0 -semigroup and f (·) is some function from [0,T ] to E . Problem (13.19) can b e considered in different functional spaces. A function u(t) is called a solution of the elliptic problem (13.19) if the following conditions are satisfied: (i) u(t) is twice continuously differentiable on the interval [0,T]. The derivatives at the endp oints of the segment are understood as the appropriate unilateral derivatives; (ii) the element u(t) b elongs to D(A) for all t [0,T ] and the function Au(t) is continuous on the segment [0,T]; 185


(iii) u(t) satisfies the equation and b oundary conditions (13.19). A solution of problem (13.19) defined in this manner will from now on b e referred to as a solution of problem (13.19) in the space C ([0,T ],E ). The coercive well-p osedness in C ([0,T ],E ) of the b oundary value problem (13.19) means that the coercive inequality u
C ([0,T ],E )

+ Au

C ([0,T ],E )

M

f

C ([0,T ],E )

+ Au0

E

+ AuT

E

holds for its solution u(·) C ([0,T ],E ) with some M which is indep endent of u0 ,uT , and f (t) C ([0,T ],E ). It turns out that the p ositivity of the op erator A in E is a necessary condition of the coercive wellp osedness of b oundary value problem (13.19) in C ([0,T ],E ). Is the p ositivity of the op erator A in E a sufficient condition for the coercive well-p osedness of b oundary value problem (13.19)? In general case, the answer is negative. The coercive well-p osedness of b oundary value problem (13.19) was established in
, C0T ([0,T ]; E ),

(0 , 0 < < 1),

the space obtained by completion of the space of all smooth E -valued functions (t) on [0,T] in the norm
, C0T

([0,T ];E )

= max (t)
0tT

E

+

0t
su p

(t + ) (T - t) (t + ) - (t)

E

.

, Theorem 13.6.5 ([1]). Let A be the positive operator in a Banach space E and f (·) C0T ([0,T ]; E ) , (0 , 0 < < 1). Then for the solution u(t) in C0T ([0,T ]; E ) of the boundary value problem

(13.19), the coercive inequality u M
, C0T ([0,T ];E )

+ Au

, C0T ([0,T ];E ))

+u
-1

C ([0,T ];E

-

)

f (0) - Au0

E

-

+ f (T ) - AuT

E

-

+

(1 - )-1 f

, C0T ([0,T ];E )

holds, where M is independent of , , u0 ,uT , and f (t). Recall that here the Banach space E , 0 < < 1, consists of those v E for which the norm v is finite. Theorem 13.6.6 ([1]). Let A be the positive operator in a Banach space E and
, f (t) C0T ([0,T ]; E- ), E


= su p z
z>0

1-

A 2 exp(-zA 2 )v

1

1

E

+v

E

(0 , 0 < < 1).

186


, Then for the solution u(·) C0T ([0,T ]; E- ) of boundary value problem (13.19), the coercive inequality

u M

, C0T ([0,T ];E

-

)

+ Au

, C0T ([0,T ];E

-

)

+u

C ([0,T ];E

-

)

f (0) - Au0

E

-

+ f (T ) - AuT

E

-

+ -1 (1 - )-1 f

, C0T ([0,T ];E

-

)

holds, where M is independent of , , , u0 ,uT , and f (t). Theorem 13.6.7 ([71]). Let A be a strongly positive operator in a Banach space E and let problem (13.19) be coercive wel l-posed in Lp0 ([0,T ]; E ) for some 1 < p0 < . Then it is also coercive wel l-posed in Lp ([0,T ]; E ) for any 1 < p < and Au(·)
Lp ([0,T ];E )

+ max u(t)
0tT

E

1-1/p



Mp2 p-1

n

Lp ([0,T ];E )

+ u(0)

E

1-1/p

+ u(T )

E

1-1/p

.

Here the space E coincides with an equivalent norm with the real interp olation space (E, D(A 2 ))1-1/p,p . Consider the problem u (t) = Au(t)+ f (t), t R with b ounded solutions and a sectorial op erator A on E , i.e., -A H (0, ). Theorem 13.6.8 ([292]). Let A be given on an UMD-space E . Then problem (13.20) is coercively solvable: u
Lp (R;E )
1

(13.20)

+ Au(·)

Lp (R;E )

c f

Lp (R;E )

for any f Lp (R; E ),

iff the set {(I - A)-1 : < 0} is R-bounded in B (E ). In the space E , let us consider the problem u (t) = Au(t)+ f (t), t [0,T ], (13.21)

Li u = i1 u(0) + i2 u (0) + i1 u(T )+ i2 u (T ) = fi , i = 1, 2,

with a p ositive op erator A. The b oundary value problem (13.21) is said to b e uniformly well-p osed in X E and [a, b] [0,T ] if for any f1 ,f2 E , its solution u(·) C 2 ([0,T ]; E ) exists, is unique, and is stable with resp ect to fi ,i = 1, 2, uniformly in t [0,T ], i.e., sup u(t) - u(t) C ~ ~ ~ f1 - f1 + f2 - f2 . 187

t[a,b]


In [30, 47], necessary and sufficient conditions under which problem (13.21) with a p ositive op erator A is uniformly well-p osed are given. Also, in these pap ers, the authors present a numb er of theorems on the well-p osedness and ill-p osedness of elliptic problems and also theorems for problems of the form u (t) = Au(t), 0 < t < T , µu(0) + u(T ) = u0 .

Chapter 14
SEMILINEAR PROBLEMS At present, there is a sufficiently large material (see, e.g., [48, 95, 97]) devoted to studying nonlinear problems u (t) = (Au)(t) and u (t) (Au)(t) by using semigroup methods. In this chapter, we consider only problems with a linear principal op erator A and a smooth nonlinear right-hand side f . Namely for these semilinear problems, the numerical analysis is sufficiently well elab orated, which we present in two theorems only. For general approximation theorems and the numerical analysis, see the first article in this volume. 14.1. First Order Equation In a Banach space E , let us consider the following Cauchy problem: u (t) = Au(t)+ f (t, u(t)), t [0,T ], u(0) = u0 ,

(14.1)

with an op erator A generating a C0 -semigroup. Here, the function f : [0,T ] â E E . The existence and uniqueness of solutions of problem (14.1) is discussed in detail, e.g., in [74]. A classical solution of problem (14.1) is defined analogously to Definition 13.2.1. Note that each classical solution of problem (14.1) satisfies the equation u(t) = (Ku)(t) exp(tA)u0 +
0 t

exp((t - s)A)f (s, u(s)) ds.

(14.2)

Definition 14.1.1. A continuous solution u(·) of Eq. (14.2) is called a mild solution of problem (14.1). It is clear that in the semilinear case, as in the case of inhomogeneous equations, a mild solution can b e not a classical solution. Theorem 14.1.1 ([20]). Let A G (M, ), u0 E , a function f : R+ â E E be continuous in t, and let it be Lipschitz in the second argument, i.e., for each > 0, f (t, x) - f (t, y ) L( ) x - y 188
E


for any x, y E , 0 t , and a certain constant L( ). Then problem (14.1) has a unique mild solution on R+ . If the Lipschitz condition holds locally, then a local existence theorem of mild solutions holds. Conditions for the existence and uniqueness for problems of form (14.1)­(14.2) were studied in detail, e.g., in [20, 74, 75, 77, 158]. Proposition 14.1.1 ([46]). Let -A be strongly positive, and let A-1 B0 (E ). Let f (t, u) be continuous in totality of the variables, and let f (t, u(t)) c(R) < for t [0,t0 ] and u R. Then there exists at least one mild solution of problem (14.1) on [0,t ] [0,t0 ], t t0 . Proposition 14.1.2 ([46]). Let -A be strongly positive. Let f (t, A- w) be continuous on [0,t0 ] for any w E , and let f (t, A- w1 ) - f (t, A- w2 ) c(R) w1 - w2 , w1 , w2 R. Final ly, let u0 D(A ). Then there exists a unique mild, i.e., continuous solution w(·) of the equation w(t) = exp(tA)A u0 +
0 t

A exp((t - s)A)f (s, A- w(s))ds

defined on [0,t ] [0,t0 ]. N. Pavel [233] gave a necessary and sufficient condition for the existence of a local solution of (14.1) in case of an analytic semigroup. Theorem 14.1.2 ([233]). Let D E be a local ly closed subset, f (·) : [0, ) E be a continuous functional, and let exp(·A) B0 (E ). A local classical solution u(·) : [0, ) D of (14.1) exists if and only if for al l z, u0 D,
h 0



lim dist(exp(hA)z + hf (t, z ); D) = 0.

Let b e an op en set in a Banach space E , and let B : E b e a compact op erator having no fixed p oints on the b oundary of . Then for the vector field W (x) = x - B x, the rotation (degree) (I -B ; ) b eing integer-valued characteristics of this field is well defined. Let z b e an isolated fixed p oint of the op erator B in the ball B (z ,r0 )of radius r0 centered at a p oint z . Then (I -B ; B (z ,r0 )) = (I -B ; B (z ,r )) for 0 < r < r0 . This common value of the rotation is called the index of the fixed p oint z and is denoted by ind(z ; I -B ). Theorem 14.1.3 ([26]). Let A H(, ), the resolvent (I - A)-1 be compact for a certain (A), and let the operator K be given by formula (14.2). If u (·) is a unique mild solution of problem (14.1), then ind(u (·); I - K ) = 1. 189


This theorem is used, e.g., in approximating the Cauchy problem (14.1) with resp ect to the space, as well as with resp ect to time [60]. Definition 14.1.2. A solution of the Cauchy problem (14.1) is said to b e Lyapunov stable if for any > 0, there exists > 0 such that the inequality u(0) - u(0) implies max ~ u(·) is a solution of problem (14.1) with the initial condition u(0). ~ ~
0t<

u(t) - u(t) where ~

Definition 14.1.3. A solution of the Cauchy problem (14.1) is said to b e uniformly asymptotical ly stable at a p oint u(0) if it is Lyapunov stable, and for any mild solution u(·) of problem (14.1) with u(0)-u(0) ~ ~ , it follows that lim u(t) - u(t) = 0 uniformly in u(0) B (u(0),), i.e., there exists a function ~ ~ such that u(t) - u(t) ~
t u(0), u(0),

(·)

(t) with

u(0),

(t) 0 as t .

We note that conditions for the existence of uniform asymptotic stability of a solution of problem (14.1) are given, e.g., in [74, Theorem 8.1.8]. These conditions are related to the location of the sp ectrum of the op erator A + f (t, u (·)). u In a Banach space E , let us consider the following p eriodic problem: v (t) = Av(t)+ f (t, v(t)), v(0) = v (T ), t R+ , (14.3)

with an op erator A H(, ). In the case of p eriodic solutions, an analog of Eq. (14.2) is the integral equation v (t) = (Kv )(t) exp(tA)(I +exp(TA))-1
0 t T

exp((T - s)A)f (s, v(s)) ds (14.4)

+
0

exp((t - s)A)f (s, v(s)) ds,

t [0,T ].

As was noted in [17, Prop osition 2.1.36], for the solvability of problem (14.3), it suffices to assume the existence of an inverse op erator (I - exp(tA))-1 for t > t0 . Then (I - exp(tA))-1 B (E ) for any t > 0. Denote by u(·,u0 ) a solution of problem (14.1) with u(0) = u0 . Then the function u(·,u0 ) satisfies Eq. (14.2), and we can define the shift op erator Ku0 = u(T, u0 ) along tra jectories, which maps E into E . If u(·,u0 ) is a T -p eriodic solution of problem (14.1), then u0 is a zero of the vector field of the op erator K, i.e., (I -K)(u0 ) = 0. We call attention to that the op erator K maps C ([0,T ]; E ) into C ([0,T ]; E ) and its fixed p oints, if they exist, are solutions of Eq. (14.4). Theorem 14.1.4 ([96]). Let A H(, ), the resolvent (I - A)-1 be compact for a certain (A), and let a function f be sufficiently smooth, so that there exists a periodic solution v (·) of problem (14.3) such that problem (14.1) has an isolated uniformly asymptotical ly stable solution at the point u(0) = v (0). Then ind(v (0); I -K) = ind(v (·); I - K ). 190


Proof. Let S S (x ,). Then the rotation (I -K; S ) of the field I -K on the sphere S is equal to the index ind(x ; I -K): (I -K; S ) = ind(x ; I -K). Let M = sup max v (t; x) ,
xS 0tT

(14.5)

(14.6)

and consider the domain F = C ([0,T ]; E ) defined by = {u(·) C ([0,T ]; E ) : u(0) S, u(·)
F

M +1}.

The function v (·) is a unique zero of the compact vector field I - K on . Hence (I - K ; ) = ind(v (·); I - K ). In view of (14.5) and (14.6), to prove the contiguous theorem, it suffices to show that (I -K; S ) = (I - K ; ). For this purp ose, on , we consider the following family of compact vector fields: (v (·); ) = v (t) - (1 - )exp(tA) I - exp(TA)
t -1 T

(14.7)

exp (T - s)A f s; v (s) ds
0

- exp(tA)K v (0) -
0

exp (t - s)A f s, v(s) ds (0 1).

(14.8)

The fields v (·); are nondegenerate on . Indeed, if for certain v0 (·) and 0 [0, 1], we have v0 (·); 0 = 0, then v0 (0) = (1 - 0 ) I - exp(TA)
-1 T

exp (T - s)A f s, v0 (s) ds + 0 v0 (T ).
0

(14.9)

Since relation (14.9) and the prop erty that v0 (·); 0 = 0 imply that the function v0 (·) is a mild solution of (14.1), we obtain
T

exp (T - s)A f s, v0 (s) ds = v0 (T ) - exp(TA)v0 (0).
0

(14.10)

Without loss of generality, we may set Re (A) < 0. But (14.9) and (14.10) imply exp(TA) (v0 (0) - v0 (T ) = -1 v0 (0) - v0 (T ) . 0 191


If v0 (0) - v0 (T ) = 0, then this element is an eigenvector of the op erator exp(TA) with the eigenvalue -1 > 1. However, this is imp ossible, since Re (A) < 0, and for analytic C0 -semigroups, exp(TA) \ 0 {0} = eT
(A)

. Hence v0 (0) = v0 (T ), which implies that v0 (·) is a T -p eriodic solution of problem (14.3)

and that it is a zero of the field I - K . We arrive at a contradiction. The fields of families (14.8) are nondegenerate on . Therefore, the fields v (·); 0 = I - K and v (·); 1 are homotopic on . We obtain (I - K ; ) = v (·); 1 ; . On , let us consider the following family of vector fields: (0 T )
t

(14.11)

v (·); = v (t) - P exp(tA)K v (0) +
0

exp (t - s)A f s, v(s) ds ,

(14.12)

with the op erator P : F F defined by (P w)(t) = w(t) for 0 t and (P w)(t) = w() for t T. The op erator Q(v (·))(t) = exp(tA)K v (0) +
t 0

exp (t - s)A f s, v(s) ds, which maps F onto F , is

compact. The op erator P : F F is strongly continuous in . Therefore, the op erator P Q is uniformly continuous in , and family (14.12) is a compact deformation (see [45, Sec. 19.1]). Let us show that families (14.12) are nondegenerate on . Assume that for certain 0 [0, 1] and v0 (·) , we have v0 (·) = v (·; x ) and v0 (·); 0 = 0. The b oundary of the domain consists of two parts G0 = {v (·) C ([0,T ]; E ) : v (0) S, v (·) and G1 = {v (·) C ([0,T ]; E ) : v (0) S, v (·) Let v0 (·) G0 . Then v0 (·)
C ([0,T ];E ) C ([0,T ];E ) C ([0,T ];E )

= M +1}

M +1}.

= M +1.

(14.13)

On the other hand, since the function v0 (·) is a solution of (14.1) on the closed interval [0,] and v0 (0) S , it follows from (14.6) that v0 (·)
C ([0,T ];E )

= max v0 (t)
0tT

E

= max v0 (t)
0t

E

M.

(14.14)

Equations (14.13) and (14.14) contradict one another. Therefore, there is only one p ossibility: v0 (·) G1 and v0 (0) S . But v0 (·); 0 = 0 implies v0 (0) = K(v0 (0)), which is imp ossible by the choice of 192


the radius of the ball S . Therefore, fields (14.12) are nondegenerate on and homotopic. Therefore, v (·); 0 ; = v (·); T ; . But v (·); T = v (·); 1 , and hence v (·); 0 ; = v (·); 1 ; . Consider the vector field v (·); 0 = v (t) -K v (0) (v (·) ). (14.15)

Since the op erator K v (0) can also b e considered as a mapping from F into the space of constant functions, which is denoted by E , its rotation (see [45]) coincides with the rotation of its restriction to E . But the field v (·); 0 on E is isomorphic to the field I -K on S . Therefore, v (·); 0 ; = v (·); 0 ; E = (I -K; S ). From (14.11), (14.15), and (14.16), we obtain (14.7). The theorem is proved. (14.16)

To show how one can use, e.g., Theorem 14.1.4 in practice, let us define the following conditions. (A) Consistency. There exists (A) n (An ) such that the resolvents converge: R(; An ) R(; A). (B) Stability. There are some constants M1 1 and such that R(; An ) M/| - | for Re > . To formulate the convergence theorem, we need the following notation. By a semidiscrete approximation of the T -p eriodic problem (14.3), we mean the T -p eriodic problems vn (t) = An vn (t)+ fn t, vn (t) , vn (t) = vn (t + T ),t R+ , (14.17)

where the op erators An generate analytic semigroups in En , condition (A) is satisfied, the functions fn are uniformly b ounded: sup
t[0,T ], x
n

smooth with fn (t, xn ) = fn (t + T, xn ) for any xn En and t R+ . The mild solutions of (14.17) are determined by the equations vn (t) = (Kn vn )(t) exp(tAn ) In - exp(TAn )
t -1 0 T

c

fn (t, xn ) C2 , the functions fn approximate f and are sufficiently
1

exp (T - s)An fn s, vn (s) ds (14.18)

+
0

exp (t - s)An fn s, vn (s) ds.

193


Theorem 14.1.5 ([96]). Assume that Conditions (A) and (B) hold and the compact resolvents R(; A) and R(; An ) converge: R(; An ) R(; A) compactly for some (A). Assume that (i) the functions f and fn are sufficiently smooth, so that there exists an isolated mild solution v (·) of periodic problem (14.3) with v (0) = x such that the Cauchy problem u (t) = Au(t)+ f t, u(t) , u(0) = x , has a uniformly asymptotical ly stable isolated solution at the point x ; (ii) fn (t, xn ) f (t, x) uniformly in t [0,T ] as xn x; (iii) the space E is separable.
Then for almost al l n, problems (14.17) have periodic mild solutions vn (t),t [0,T ], in a neighborhood of pn v (·), where v (·) is a mild periodic solution of (14.3) with v (0) = x . Each sequence {vn (·)} is P compact, and vn (t) v (t) uniformly with respect to t [0,T ].

(14.19)

Proof. We divide the proof into several steps. Step 1. First, let us show that the compact convergence of resolvents R(; An ) R(; A)is equivalent to the compact convergence of C0 -semigroups exp(tAn ) exp(tA) for any t > 0. Let xn = O(1). M t Then from the estimate An exp(tAn ) e , we obtain the b oundedness of the sequence {(An - t )exp(tAn )xn }. Because of the compact convergence of resolvents, we obtain the compactness of the sequence {exp(tAn )xn }. The necessity will b e proved if for the measure of noncompactness µ(·) (for the definition, see [278]), we establish that µ({(I - An )-1 xn }) = 0 for xn = O(1). We have µ({(I - An )-1 xn }) = µ
0 q 0 Q

e-t exp(tAn )xn +µ

µ

e-t exp(tAn )xn dt .

+µ biggl{

Q

e-t exp(tAn )xn dt

exp( An )
q

e-t exp (t - )An xn dt

Two first terms can b e made less than

by the choice of q, Q. The last term is equal to zero b ecause of

the compact convergence exp( An ) exp( A) for any 0 < < q . Step 2. Consider the op erators K and Kn defined by (14.4) and (14.18) on the spaces F = C ([0,T ]; E ) {u(t) : u and Fn = C ([0,T ]; En ) {un (t) : un 194
F
n

F

= max u(t)
t[0,T ]

E

< }

= max un (t)
t[0,T ]

E

n

< }.


The op erator K defined by (14.4) is compact in F. Indeed, we obtain that the op erator F (uk )(t) = exp( A)
0 t-

exp (t - s - )A f s, uk (s) ds
F

maps any b ounded set of functions {uk (·)}, uk (·)

C, into a compact set in E for any t > 0 and

0 < < t. We see that F (uk )(t) -F (uk )(t) C for any t (0,T ], where F (uk )(t) =
0 t

exp (t - s)A f s, uk (s) ds

and 0 < < t. Then it follows that the op erator F (·)(t) : F E is compact for the same t > 0. For t = 0, the op erator F (·)(0) is also compact. Moreover, the set of functions {Fk (·)},Fk (t) = F (uk )(t),t [0,T ], is an equib ounded and equicontinuous family, since for 0 < t1 < t2 , we obtain Fk (t2 ) - Fk (t1 ) C
0 t1

exp (t2 - s)A - exp (t1 - s)A ds + |t2 - t1 | ,

and exp(·A) is uniformly continuous in t > 0. The sequence {yk },yk = I - exp(TA)
-1 T 0

exp (T - s)A f s, uk (s) ds E, is compact, since

{F (uk )(T )} is a compact set. Therefore, {exp(·A)yk } is a compact sequence of functions in F. By the generalized Arzela­Ascoli theorem, it follows that op erator K is compact. Step 3. It is easy to see that Kn K. Indeed, In I stably and exp(TAn ) exp(TA) compactly; hence In - exp(TAn ) I - exp(TA) regularly, the null space N I - exp(TA) = {0} and the op erators In - exp(TAn ) are Fredholm of index zero. Then it follows from [14] that In - exp(TAn ) I - exp(TA) stably, i.e., In - exp(TAn )
-1

I - exp(TA)

-1

and the convergence Kn K is a consequence of the
F
n

dominated convergence theorem. To show that Kn K compactly, we assume that un

= O(1). Now

{Kn un } is P -compact by the generalized Arzela­Ascoli theorem. To show this, we verify the vanishing of the noncompactness measure µ({(Kn un )(t)}) = 0 for all t [0,T ]. Let us consider the relation
(Kn vn )(t) = exp(tAn )yn + n (t)+ (t), n

where yn = In - exp(TAn )
n (t) = exp(An ) -1 0 t- 0 t t- T

exp (T - s)An fn s, vn (s) ds,

exp (t - s - )An fn s, vn (s) ds,

(t) = n

exp (t - s)An fn s, vn (s) ds.
F
n

By virtue of the b oundedness of fn ·,vn (·)

, we can choose the term (·) n

F

n

sufficiently small with

small enough and µ({n }) = 0. The sequence {yn } is P -compact.

195


Step 4. The condition of existence of an isolated uniformly asymptotically stable solution u(t; x ) of problem (14.19) implies that in a small neighb orhood of x , say in S (x ,) E, the op erator K is compact, since the set F (uk )(T ) is compact for any {uk },uk (t) S (x , ),t [0,T ], with uk (0) - x . The p oint x is an isolated zero of the compact vector field I -K and ind(x ; I -K) is defined. In the same way function v (t) = u(t; x ),t [0,T ], that is the solution of problem (14.4), is an isolated zero of the field I - K and ind(v (·); I - K ) is defined. From Theorem 14.1.4, it follows that the relation ind(x ; I -K) = ind(v (·); I - K ) holds. Step 5. The condition of uniform asymptotic stability of the solution u (·) of problem (14.1) at the point x implies that there exists an integer m such that the op erator K itself; more precisely, Km (x ) - Km (x) x


m

maps the ball S (x ,) into

,

(mT ) < for any x S (x ,). Therefore, this means

that ind(x ; I -Km ) = 1, and by [45, Theorem 31.1], we obtain ind(x ; I -K) = 1. Using Step 4, we have ind(v (·); I - K ) = 1. Now, Kn K compactly, ind(v (·); I - K ) = 1, and applying the result from [278],
we obtain that the set of solutions of problems (14.18) is nonempty, any sequence of solutions {vn (·)} is P -compact, and, moreover, vn (t) v (t) uniformly in t [0,T ] as n . The theorem is proved.

14.2. Second Order Equation In a Banach space E , let us consider the following semilinear Cauchy problem: u (t) = Au(t)+ f (t, u(t),u (t)), u(0) = u0 ,u (0) = u1 , t R, (14.20)

with the op erator A being a generator of a C0 -cosine op erator function and a continuous function f : R âD E , where D E 1 â E is a locally closed subset of E 1 â E . Definition 14.2.1. A classical solution of (14.20) on the closed interval [0,T ] is a function u(·) : R E such that u(·) is twice continuously differentiable and satisfies (14.20) for all t [0,T ]. As is known, a classical solution u(t) of problem (14.20) also satisfies the following integral equation (see [274]): u(t) = (Ku)(t) C (t, A)u0 + S (t, A)u1 +
0 t

S (t - s, A)f (s, u(s),u (s))ds

(14.21)

and is a mild solution. Recall that a solution u(·) C 1 ([0,T ]; E ) of Eq. (14.21) is called a mild solution of (14.20). A mild solution of (14.20) is a classical solution if f (·,u(·)) is absolutely continuous. Therefore, in general for a continuous function f , a mild solution is not classical. 196


Existence and uniqueness problems for problem (14.20) were studied, e.g., in [158, 274]. In [158], the case where E is a Banach lattice was also considered. Assume that f : J â E â E E satisfies the following conditions: (C1) (C2) f (·,x,y ) is strongly measurable for all x, y E , and f (t, 0, 0) L1 (J, E ); for all x, y , h, k E and for a.a. t J , g(t, x + h, y + k) - g(t, x, y ) q (t, h , k ), where f : J â R2 R+ is a Carath´odory function, q (t, ·, ·) is nondecreasing for a.a. t J , the problem e + u (t) = M f (t, u(t),u (t)), u(0) = u0 , u (0) = u1 , (14.22)

with some constant M has an upp er solution on J for each (u0 ,u1 ) R2 , and the zero-function is the + only solution of (14.22) when u0 = u1 = 0. Theorem 14.2.1 ([158]). If Conditions (C1)­(C2) hold, then for each (u0 ,u1 ) E 2 â E , problem (14.20) has a unique weak solution u(·) on J . Moreover, u(·) is of the form u(t) = u0 + y (·) is the uniform limit of the sequence {yn } of the successive approximations n=0 y
n+1 t 0

y (s) ds, t J , where

(t) = S (t)Au0 + C (t)u1 +
0

t

C (t - s, A)f s, u0 +
0

s

yn ( ) d, yn (s)) ds,

with t J, n N, and with an arbitrarily chosen y0 C (J, E ). Consider the existence of mild solutions of the problem u (t) = Au(t)+ f (t, u(t)), u(0) = u0 , u (0) = u1 , (14.23)

that lies b etween assumed upp er and lower mild solutions, when E is an ordered Banach space with regular order cone and f : J â E E . Given u0 , u1 E â E , we say that u(·) C (J, E ) is a lower mild solution of problem (14.23) on J if u(t) < C (t, A)u0 + S (t, A)u1 +
0 t

S (t - s, A)f (s, u(s)) ds

(14.24)

for each t J . An upp er mild solution of (14.23) is defined similarly, by reversing the inequality sign in (14.24). If equality holds in (14.24), we say that u(·) is a mild solution of (14.23). Let us introduce the following hyp otheses on the mappings f : J â E E and C : J B (E ): (C3) (14.23) has a lower mild solution u(·) and an upp er mild solution u(·) such that u(·) u(·), ¯ ¯ f (·,u(·)) is strongly measurable whenever u(·) C (J, E ); f (t, ·) is nondecreasing for a.a. t J ; 197 and the functions f (·,u(·)) and f (·, u(·)) are Bochner integrable; ¯ (C4) (C5)


(C6)

C (t, A) 0 for all t J .

If (C6) holds, it follows from (2.8) that S (t, A) 0 for each t J . Theorem 14.2.2 ([158]). If Conditions (C3)­(C6) hold, then problem (14.23) has extremal mild solutions lying between u(·) and u(·). ¯ Theorem 14.2.3 ([96]). Assume that Conditions (A) and (B) hold and the compact resolvents R(; A),R(; An ) converge: R(; An ) R(; A) compactly for some (A) and u0 u0 ,u1 u1 . Assume that n n (i) the functions fn ,f are continuous in both arguments and f is such that there exists a unique mild solution u (t) of problem (14.23) on [0,T ] (in this situation, as we wil l show, ind u = 1); (ii) fn (t, xn ) f (t, x) uniformly in t [0,T ] for xn x; (iii) the space E is separable. Then for almost al l n, the problems un (t) = An un (t)+ fn (t, un (t)), un (0) = u0 ,un (0) = u1 , n n

(14.25)

have mild solutions u (t),t [0,T ] in a neighborhood of pn u (t). Each sequence {u (t)} is P -compact n n and u (t) u (t) uniformly in t [0,T ]. n Proof. First, let us prove that the compact convergence of resolvents R(; An ) R(; A) is equivalent to the compact convergence of sine op erator functions Sn (t, An ) S (t, A) for any t 0. Let xn = O(1). We are going to show that from the compact convergence of resolvents R(; An ) R(; A) it follows µ({Sn (t, An )xn }) = 0 for any t, where µ is the measure of noncompactness of sequences. From the identities 2 (2 I - An )-1 Sn (t, An ) - Sn (t, An ) =
0

e- Cn (An )Sn (t, An )d - Sn (t, An )

1 = 2 we obtain the estimate

0

e- Sn (t + , An )+ Sn (t - , An ) - 2Sn (t, An ) d

2 (2 I - An )-1 Sn (t, An ) - Sn (t, An ) 198 1 2
0

1 e- Sn (t + , An ) - 2Sn (t, An )+ Sn (t - , An ) d + 2



e- Me d,


where the first term on the right-hand side is less than

for small and the second one is less than

for

large enough (we recall that if resolvents converge compactly for some , then they converge compactly for any with sufficiently large Re ). Estimating the measure of noncompactness by µ({Sn (t, An )xn }) µ({2 (2 In - An )-1 Sn (t, An )xn })+ 2 (2 In - An )-1 Sn (t, An ) - Sn (t, An ) we obtain the compact convergence of C0 -sine op erator functions. The necessity will b e proved if we establish that µ({(In - An )-1 xn }) = 0 for xn = O(1) under the condition that Sn (t, An ) S (t, A) compactly. We have µ({(2 In - An )-1 xn }) = µ
0

e-t Sn (t, An )xn
Q

µ
0

q

e-t Sn (t, An )xn dt



+µ
Q

e-t Sn (t, An )xn dt

+µ
q

e-t Sn (t, An )xn dt

.

If q is small enough and Q is large enough, then the first and second terms b ecome less than . The third term is equal to zero by the uniform continuity of Sn (·,An ) on [q, Q]. Now we are going to prove that the compact convergence of C0 -sine op erator functions and condition (ii) imply that Kn K compactly. It is clear that Kn K . Let {un } b e a sequence of functions un C (0,T ; En ) such that un
C (0,T ;En )

= O(1) as n . To prove that {Kn un } is compact, we apply

the theorem from [46]. The sequence of functions {Kn un },Kn un C (0,T ; En ), is uniformly b ounded, equicontinuous, and for any t [0,T ], the op erator Kn maps the b ounded set of functions {un } into a precompact set. Therefore, Kn K compactly. Now, from [278], it follows that (I - K ; Sr ) = (In - Kn ; S
n,r

) as n n0 . If we establish that (I - K ; Sr ) = 0, then by Theorem 3 in [278], it follows

that solutions of (14.25) do exist in a neighb orhood of pn u (t), each sequence {u (t)} is P -compact, and n u (t) u (t) uniformly with resp ect to t [0,T ]; this will prove the theorem. n Therefore, let us show that (I - K ; Sr ) = 1. It follows from the assumption of the theorem that the op erator K has no fixed p oints on the b oundary Sr , where Sr = {u : u - u < r }. We want to show that for the op erator (Ku)(t) = C (t)u0 + S (t, A)u1 + S (t - s, A)f (s, u(s))ds, with a continuous function f , the index of the fixed p oint u is equal to (I - K ; Sr ) = 1. To do this, we define the op erator G u = K (P u)+ u - K (P u ), where K (u ) = u and the op erator P is defined by the formulas (P u)(t) = u(t - ) for < t T, (P u)(t) = u(0)) for t [0,]. 199
0 t


We complete the proof if we prove the following two assertions: (I - G ; Sr ) = (1 ; Sr ) = 1, where 1 (u) = u - u , and (I - G ; Sr ) = (I - G0 ; Sr ) = (I - K ; Sr ) for sufficiently small . The fields 1 (u) = u - u and 2 (u) = u - G u are connected by a linear compact nondegenerate deformation (see [45, Sec. 19.1]): H (µ, )u = u - µG u - (1 - µ)u , 0 µ 1,

i.e., 1 and 2 are linearly homotopic. The op erator H has no singular p oints on Sr . To prove this, we assume the contrary: there exist v = u and H (µ, )v = 0. Then by the formula v = µK (P v ) - µK (P u )+ u , we first obtain v (0) = u (0), and, therefore, b ecause of the relation (P v )(t) = (P u )(t) for 0 t , we have v (t) = u (t) for t [0,]. Rep eating the same arguments, we obtain v (t) = u (t) for t [0, 2], since (P v )(t) = (P u )(t) for 0 t 2 by virtue of (14.26). In this way, we can arrive at 3 and so on; this means that u = v . Since the op erator H has no singular p oints on Sr , H is a linear compact deformation. Clearly, (I - H (0,); Sr ) = (1 ; Sr ) = 1, and, therefore, by [45, Theorem 20.1], 1 = (I - H (1,); Sr ) = (I - G ; Sr ). The op erator G is compact for any (see [284]), and, moreover, {
[0,T ]

(14.26)

G u : u Sr } is precompact

(if this set is not relatively compact, then there exist two sequences {k } and {uk } such that {Gk uk } is not compact, which contradicts the compactness of K ). Clearly, P P0 strongly in C ([0,T ]; E ) as 0 . Let vk v0 , and let k 0 . Since Gk vk - G0 v0 = K (Pk vk )+ u - K (Pk u ) - K (P0 v0 ) - u + K (P0 u ) = K (Pk vk ) - K (P0 v0 )+ K (P0 u ) - K (Pk u ) 0 as k 0 , 200


we have that the op erator G is continuous in b oth arguments, and, as we have seen, the set {y : y = G u, u - u r, 0 T } is relatively compact. Since G G0 compactly as 0, by [278], the op erators G have no fixed p oints on Sr for small , and for the same , we obtain (I - G ; Sr ) = (I - G0 ; Sr ) = (I - K ; Sr ). The theorem is proved. Acknowledgments. S. Piskarev acknowledges the supp ort of Russian Foundation for Basic Research (01-01-00398), DAAD of Germany and Chinese Academy of Sciences and Control System Lab oratory of Academy of Mathematics and System Sciences, Academia Sinica. REFERENCES 1. A. Ashyralyev, Wel l-Posedness of the El liptic Equations in a Space of Smooth Functions. Boundary Value Problems For Nonclassical Equations of Mathematical Physics [in Russian], Sibirsk. Otdel, SSSR-Novosibirsk, 2, No. 2, 82­86 (1989). 2. A. Ashyralyev, "Well-p osed solvability of the b oundary value problem for difference equations of elliptic typ e," Nonlinear Anal. Theory, Meth. Appl., 24, 251­256 (1995). 3. A. Ashiraliev, "On the coercive solvability of parab olic equations in spaces of smooth functions," Izv. Akad. Nauk TurkSSSR, Ser. Fiz.-Mat., Khim., and Geol. Nauk, 3, 3­13 (1989). 4. G. Bab escu, "On a certain connection b etween second-order linear differential and cosine op erator functions on Banach spaces," Lucr. Semin. Mat. Fiz. Inst. Politehn. Trian Vuia, Timisoara, 146, 13­16 (1982). 5. G. Bab escu, "Sp ectral theorems for functions with values of cosine and sine op erators," Bul l. Sci. Tehn. Inst. Politehn. Timisoara. Mat. Fiz., 27, No. 1, 13­20 (1982). 6. A. G. Baskakov, "Harmonic analysis of cosine and exp onential op erator functions," Mat. Sb., 124(166), No. 1(5), 68­95 (1984). 7. A. G. Baskakov, Harmonic Analysis of Linear Operators [in Russian], Voronezh Univ., Voronezh (1987). 8. O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1996). 9. L. Berg and I. L¨ strem, Interpolation Spaces. An Introduction [Russian translation], Mir, Moscow ef (1980). 10. G. M. Vainikko and M. Shlapikiene, "To a certain S. G. Krein theorem on p erturbation of op erators generating analytic semigroups," Uch. Zap. Tartu Univ., 281, 186­189 (1971). 201


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