Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://srcc.msu.ru/nivc/english/about/home_pages/piskarev/review1.ps Äàòà èçìåíåíèÿ: Thu Jun 5 17:53:24 2008 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 23:07:07 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: south pole

APPROXIMATION OF ABSTRACT DIFFERENTIAL
EQUATIONS
Davide Guidetti, B¨ulent Karas¨ozen and Serguei Piskarev
April 25, 2002
Abstract
This review paper is devoted to the numerical analysis of abstract differential equations in Banach
spaces. The presentation is given on general approximation scheme and based on semigroup theory
and functional analysis approach. The paper concerns mainly numerical analysis of differential
equations of first order in time.
1991 Mathematics Subject Classification: 65J, 65M
Keywords and phrases: Abstract differential equations, C 0 ­semigroups, Banach spaces, semidiscretiza­
tion, coerciveness, maximal regularity inequality, Lax's equivalence theorem, difference schemes, semilin­
ear equations, approximation of the spectrum of operators, discrete semigroups, rational approximations,
backward evolution equations, ill­posed problems, stochastic regularization
Contents
1 Introduction 2
2 General approximation scheme 2
2.1 Approximation of spectrum of linear operators : : : : : : : : : : : : : : : : : : : : : : : : 3
2.2 Regions of convergence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4
2.3 Convergence in one space dimension : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5
2.4 Compact convergence of resolvents : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6
3 Discretization of semigroups 7
3.1 The simplest discretization schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7
3.2 Rational approximation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10
3.3 Richardson's Extrapolation method : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12
3.4 Lax­type equivalence theorems with orders : : : : : : : : : : : : : : : : : : : : : : : : : : 12
4 Backward Cauchy problem 15
4.1 C­semigroups and ill­posed problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16
4.2 Semidiscrete approximation theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18
4.3 Approximation by discrete C­semigroups : : : : : : : : : : : : : : : : : : : : : : : : : : : 19
1

5 Coercive inequalities 21
5.1 Coercive inequality in C Ün ([0; T ]; E n ) spaces : : : : : : : : : : : : : : : : : : : : : : : : : 22
5.2 Coercive inequality in C ff;0
Ün ([0; T ]; E n ) spaces : : : : : : : : : : : : : : : : : : : : : : : : : 23
5.3 Coercive inequality in L p
Ün ([0; T ]; E n ) spaces : : : : : : : : : : : : : : : : : : : : : : : : : 23
5.4 Coercive inequality in B Ün
i
[0; T ]; C `
h(\Omega h )
j
`` C h ([0; T ]; C h (
¯\Omega h )) : : : : : : : : : : : : : : : 26
6 Approximations of Semilinear Equations 31
6.1 Approximations of Cauchy problem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31
6.2 Approximation of periodic problem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 32
1 Introduction
This review paper is devoted to the numerical analysis of abstract differential equations in Banach spaces.
Most of the finite difference, finite element and projection methods could be considered from the point
of view of general approximation schemes (see for example [204], [208], [207] for such a representation).
Results obtained for general approximation schemes make the formulation of concrete numerical methods
easier and gives an overview of methods which are suitable for different classes of problems.
The qualitative theory of differential equations in Banach spaces presented in many brilliant papers
and books. We can refer to the bibliography [216] which contains about 2500 references. Unfortunately
there are no books or reviews on approximation theory for differential equations in abstract spaces last
20 years. Any information on the subject can be found only in the original papers. It looks that such a
review is the first step on the way to describe a complete picture of discretisation methods for abstract
differential equations in Banach spaces.
In Section 2 we describe the general approximation scheme, different types of convergence of operators
and the relation of convergence with the approximation of spectra. Also such a convergence analysis
could be used if one considers elliptic problems, i.e. the problems which do not depend on time.
Section 3 contains a complete picture of the theory of discretization of semigroups in Banach spaces.
It summarizes Trotter­Kato and Lax­Richtmyer theorems from the general and common point of view
and related problems.
Approximation of ill­posed problems is considered in Section 4, which are based on the theory of
approximation of local C­semigroups. Since the backward Cauchy problem is very important in applica­
tions and admits stochastic noise, we consider also approximation using stochastic regularization. Such
an approach has never been considered in the literature before to the best our knowledge.
In Section 5 we present discrete coercive inequalities for abstract parabolic equations in C Ün ([0; T ]; E n ),
C ff
Ün ([0; T ]; E n ), L p
Ün ([0; T ]; E n ) and B Ün ([0; T ]; C
ff(\Omega h )) spaces.
The last Section 6 deals with semilinear problems. We consider approximations of Cauchy problems
and also the problems with periodic solutions. The approach we describe is based on the theory of
rotation of vector fields and the principle of compact approximation of operators.
2 General approximation scheme
Let B(E) denote the Banach algebra of all linear bounded operators on a complex Banach space E.
The set of all linear closed densely defined operators in E will be denoted by C(E): We denote by oe(B)
the spectrum of the operator B; by ae(B) the resolvent set of B; by N(B) the null space of B and by
R(B) the range of B. Let us recall that B 2 B(E) is said to be a Fredholm operator if R(B) is closed,
dimN(B) ! 1 and codim R(B) ! 1, the index of B is defined as ind B = dimN(B) \Gamma codim R(B).
2

The general approximation scheme, due to [80]­[82], [184], [207], [204] can be described in the following
way. Let E n and E be Banach spaces and fp n g be a sequence of linear bounded operators p n : E !
E n ; p n 2 B(E;E n ); n 2 IN = f1; 2; \Delta \Delta \Delta g; with the property:
kp n xkEn ! kxkE as n !1 for any x 2 E:
Definition 2.1 The sequence of elements fx n g; x n 2 E n ; n 2 IN ; is said to be P­convergent to x 2 E iff
kx n \Gamma p n xkEn ! 0 as n !1 and we write this x n ! x:
Definition 2.2 The sequence of elements fx n g; x n 2 E n ; n 2 IN ; is said to be P­compact if for any
IN 0 ` IN there exist IN 00 ` IN 0 and x 2 E such that x n ! x; as n !1 in IN 00 :
Definition 2.3 The sequence of bounded linear operators B n 2 B(E n ); n 2 IN ; is said to be PP­
convergent to the bounded linear operator B 2 B(E) if for every x 2 E and for every sequence fx n g; x n 2
E n ; n 2 IN ; such that x n ! x one has B n x n ! Bx: We write then B n ! B:
For general examples of notions of P\Gammaconvergence see [79], [184], [200], [208].
Remark 2.1 If we put E n = E and p n = I for each n 2 IN , where I is the identity operator on E, then
Definition 2.1 leads to the traditional pointwise convergent bounded linear operators which we denote by
B n ! B:
Let us denote by E + the positive cone in a Banach lattice E: The operator B is called positive if for any
x + 2 E + it follows Bx + 2 E + ; we write 0 ¯ B:
Definition 2.4 The system fp n g is called discrete order preserving if for all sequences fx n g; x n 2 E n ;
and element x 2 E the following implication holds: x n ! x implies x +
n ! x + :
It is known [96] that fp n g preserves the order iff kp n x + \Gamma (p n x) + kEn ! 0 as n !1 for any x 2 E:
In the case of unbounded operators, and we know in general infinitesimal generators are unbounded,
we consider the notion of compatibility.
Definition 2.5 The sequence of closed linear operators fA n g; A n 2 C(E n ); n 2 IN ; are said to be com­
patible with a closed linear operator A 2 C(E) iff for each x 2 D(A) there is a sequence fx n g; x n 2
D(A n ) ` E n ; n 2 IN ; such that x n ! x and A n x n ! Ax: We write (A n ; A) are compatible.
Usually in practice Banach spaces E n are finite dimensional, although, in general, say for the case of a
closed operator A; we have dimE n !1 and kA n k B(En) !1 as n !1:
2.1 Approximation of spectrum of linear operators
The most important role in approximations of equation Bx = y and approximations of spectra of operator
B, is played by the notions of stable and regular convergence. These notions are used in different areas
of numerical analysis (cf. [10], [15], [78], [83]­[86], [207], [209], [204], [220]).
Definition 2.6 A sequence of operators fB n g; B n 2 B(E n ); n 2 IN ; is stably convergence to an operator
B 2 B(E) iff B n ! B and kB \Gamma1
n k B(En) = O(1); n !1: We will write B n ! B stably.
Definition 2.7 A sequence of operators fB n g; B n 2 B(E n ); is called regularly convergent to the operator
B 2 B(E) iff B n ! B and the following implication holds:
kx n kEn = O(1) & fB n x n g is P \Gamma compact =) fx n g is P­compact:
We write this as B n ! B regularly.
3

Theorem 2.1 [207] For B n 2 B(E n ) and B 2 B(E) the following conditions are equivalent:
(i) B n ! B regularly, B n are Fredholm operators of index 0 and N(B) = f0g;
(ii) B n ! B stably and R(B) = E;
(iii) B n ! B stably and regularly;
(iv) if one of the conditions (i)­(iii) is fulfilled, then there exist B \Gamma1
n 2 B(E n ); B \Gamma1 2 B(E) and
B \Gamma1
n ! B \Gamma1 regularly and stably.
This theorem admits an extension to the case of closed operators B 2 C(E); B n 2 C(E n ) [210].
Let \Lambda ` C be some open connected set and let B 2 B(E): For an isolated point – 2 oe(B); the
corresponding maximal invariant space (or generalized eigenspace) will be denoted by W(–;B) = P (–)E,
where P (–) = 1
2úi
R
ji \Gamma–j=ffi (iI \Gamma B) \Gamma1 di and ffi is small enough so that there are no points of oe(B) in the disc
fi : ji \Gamma –j Ÿ ffig other than –: The isolated point – 2 oe(B) is a Riesz point of B if –I \Gamma B is a Fredholm
operator with index zero and P (–) is of finite rank. Denote by W(–; ffi; B n ) =
S
j–n \Gamma–j!ffi;– n2oe(Bn ) W(– n ; B n );
where – n 2 oe(B n ) are taken from a ffi­neighborhood of –: It is clear that W(–; ffi; B n ) = P n (–)E n ; where
P n (–) = 1
2úi
R
ji \Gamma–j=ffi (iI \GammaB n ) \Gamma1 di: The following theorems state the complete picture of the approximation
of the spectrum.
Theorem 2.2 [79], [205]­[206] Assume that L n (–) = –I \Gamma B n and L(–) = –I \Gamma B are Fredholm operators
with index zero for any – 2 \Lambda and that L n (–) ! L(–) stably for any – 2 ae(B) `` \Lambda 6= ;: Then
(i) for any – 0 2 oe(B) `` \Lambda there exists a sequence f– n g; – n 2 oe(B n ); n 2 IN , such that – n ! – 0 as
n !1;
(ii) if for some sequence f– n g; – n 2 oe(B n ); n 2 IN , one has – n ! – 0 2 \Lambda as n !1; then – 0 2 oe(B);
(iii) for any x 2 W(– 0 ; B) there exists a sequence fx n g; x n 2 W(– 0 ; ffi; B n ), n 2 IN ; such that x n ! x
as n !1;
(iv) there exists n 0 2 IN such that dimW(– 0 ; ffi; B n ) – dimW(– 0 ; B) for any n – n 0 :
Remark 2.2 The inequality in (iv) can be strict for all n 2 IN as is shown in [204].
Theorem 2.3 [207] Assume that L n (–) and L(–) are Fredholm operators with index zero for all – 2 \Lambda:
Suppose that L n (–) ! L(–) regularly for any – 2 \Lambda and that ae(B) `` \Lambda 6= ;: Then the statements (i)­(iii)
of Theorem 2.2 hold and also
(iv) there exists n 0 2 IN such that dimW(– 0 ; ffi; B n ) = dimW(– 0 ; B) for all n – n 0 ;
(v) any sequence fx n g; x n 2 W(– 0 ; ffi; B n ); n 2 IN ; with kx n kEn = 1 is P­compact and any limit point
of this sequence belongs to W(– 0 ; B):
Remark 2.3 Estimates of j– n \Gamma – 0 j; gap
i
W(– 0 ; ffi; B n ); W(– 0 ; B)
j
and j “ – n \Gamma – 0 j are given in [207],
where “
– n denotes the arithmetic mean (counting algebraic multiplicities) of the spectral values of B n that
contribute to W(– 0 ; ffi; B n ): For the notion of gap and its properties, see [102].
2.2 Regions of convergence
Theorems 2.2 and 2.3 have been generalized in [210] to the case of closed operators using the following
notions introduced by Kato [102].
Definition 2.8 The region of stability \Delta s = \Delta s (fA n g); A n 2 C(B n ); is defined to be the set of all
– 2 C such that – 2 ae(A n ) for almost all n and such that the sequence fk(–I \Gamma A n ) \Gamma1 kg n2IN is bounded.
The region of convergence \Delta c = \Delta c (fA n g); A n 2 C(E n ); is defined as the set of all – 2 C such that
– 2 \Delta s (fA n g) and such that the sequence of operators f(–I \Gamma A n ) \Gamma1 g n2IN is PP­convergent to some
operator S(–) 2 B(E):
4

It is clear that S(\Delta) is pseudo­resolvent and S(\Delta) is a resolvent of some operator iff N(S(–)) = f0g for
some – (cf. [102]).
Definition 2.9 A sequence of operators fK n g; K n 2 C(E n ); is called regularly compatible with the op­
erator K 2 C(E) if (K n ; K) are compatible and, for any bounded sequence kx n kEn = O(1) such that
x n 2 D(K n ) and such that fK n x n g is P­compact, it follows that fx n g is P­compact and the P­convergence
of fx n g to some x and of fK n x n g to some y for n !1 in IN 0 ` IN imply that x 2 D(K) and Kx = y:
Definition 2.10 The region of regularity \Delta r = \Delta r (fA n g; A); is defined as the set of all – 2 C such that
(K n ; K) are regularly compatible, where K n = –I \Gamma A n and K = –I \Gamma A:
The relationships between these regions are given by
Proposition 2.1 [210] Suppose that \Delta c 6= ; and N
i
S(–)
j
= f0g at least for one point – 2 \Delta c so that
S(–) = (I– \Gamma A) \Gamma1 . Then (A n ; A) are compatible and
\Delta c = \Delta s `` ae(A) = \Delta s `` \Delta r = \Delta r `` ae(A):
It is shown in [210] that the following three conditions : (A n ; A) are compatible, –I \Gamma A n and –I \Gamma A
are Fredholm operators with index zero for any – 2 \Lambda and ae(A) `` \Lambda 6= ; imply (i)­(iv) of Theorem 2.2
when ae(A) `` \Lambda ` \Delta s and imply (i)­(iii) of Theorem 2.2 and (iv)­(v) of Theorem 2.3, when \Lambda ` \Delta r :
Definition 2.11 The Riesz point – 0 2 oe(A) is said to be strongly stable in Kato's sense if dimW(– 0 ; ffi; B n ) Ÿ
dimW(– 0 ; B) for all n – n 0 :
Theorem 2.4 [210] The Riesz point – 0 2 oe(A) is strongly stable in Kato's sense iff – 0 2 \Lambda `` \Delta r `` oe(A):
Investigations of approximation of spectra and types of convergence, but not on general approximation
scheme are given in [13], [37], [49], [127], [128], [139], [142].
2.3 Convergence in one space dimension
Throughout this subsection we assume that E n = E and p n = I for all n 2 IN : Hence, the symbol P
will be omitted in all notations of this subsection.
Let us recall that if B n ! B compactly (see Definition 2.12), then for any – 6= 0 we have –I \Gamma B n !
–I \Gamma B regularly [204]. When B n ! B compactly and B is a compact operator, Anselone [10] has proved
that
k(B n \Gamma B)B n k ! 0; k(B n \Gamma B)Bk ! 0 as n !1:
(2.1)
Considering an approximation of a weakly singular compact integral operator Ahues [4] has proved
that these convergence properties (2.1) are sufficient to state that a Riesz point is strongly stable in
Kato's sense.
Theorem 2.5 [7] Assume that B 2 B(E) is compact and that B n ! B: If k(B n \Gamma B)B n k ! 0 as
n ! 1 then for any nonzero – 0 2 oe(B) the assertions (i)­(iii) of Theorem 2.2 and assertions (iv)­(v)
of Theorem 2.3 hold.
Theorem 2.6 [7] Assume that B n ! B and (2.1) holds. Then for any non­zero Riesz point – 0 2 oe(B)
the assertions (i)­(iii) of Theorem 2.2 and assertions (iv)­(v) of Theorem 2.5 hold.
5

Corollary 2.1 [5] Assume that B n ! B; that –I \Gamma B n are Fredholm operators with index zero for
– 2 fz : jz \Gamma – 0 j Ÿ ffig and k(B n \Gamma B)B k
n k ! 0 as n !1 for some k 2 IN : Then for any non zero Riesz
point – 0 2 oe(B) the assertions (i)­(iii) of Theorem 2.2 and assertions (iv)­(v) of Theorem 2.3 hold.
Theorem 2.7 [7] Assume that B is compact, B n ! B and kB n (B n \Gamma B)k ! 0: Then – 0 I \Gamma B n ! – 0 I \Gamma B
regularly for any – 0 6= 0.
Theorem 2.8 [7] Assume that B is compact, B n ! B and kB k
n (B n \Gamma B)k ! 0 for some k 2 IN : Then
– 0 I \Gamma B n ! – 0 I \Gamma B regularly for any – 0 6= 0:
Let r(B) be a spectral radius of operator B 2 B(E):
Theorem 2.9 [18] Let E be a Banach lattice. Let 0 ¯ B n ; B 2 B(E) be such that B n ! B and
k(B n \Gamma B) + k ! 0 as n ! 1: Suppose that r(B) is a Riesz point of oe(B): Then r(B n ) is a Riesz point
of oe(B n ) and r(B n ) ! r(B) as n !1:
The conclusion on order of convergence of eigenvectors in Theorem 2.9 also is given in [17].
The application of Theorems 2.5 ­ 2.8 to the numerical solution of a mathematical model used in the
jet printer industry is considered [6], [115].
2.4 Compact convergence of resolvents
We consider now the important class of operators which have compact resolvents. We will use this
property of generator as assumption in section 6. It is natural to consider in this case approximate
operators which ``keep'' this property.
Definition 2.12 A sequence of operators fB n g; B n 2 B(E n ); n 2 IN ; converges compactly to an
operator B 2 B(E); if B n ! B and the following compactness condition holds:
kx n kEn = O(1) =) fB n x n g is P­compact.
Definition 2.13 The region of compact convergence of resolvents, \Delta cc = \Delta cc (A n ; A); where A n 2 C(E n )
and A 2 C(E) is defined as the set of all – 2 \Delta c `` ae(A) such that (–I \Gamma A n ) \Gamma1 ! (–I \Gamma A) \Gamma1 compactly.
Theorem 2.10 Assume that \Delta cc 6= ;: Then for any i 2 \Delta s the following implication holds:
kx n kEn = O(1) & k(iI \Gamma A n )x n kEn = O(1) =) fx n g is P­compact:
(2.2)
Conversely, if for some i 2 \Delta c `` ae(A) the implication (2.2) holds, then \Delta cc 6= ;:
Proof. Let (¯I \Gamma A n ) \Gamma1 ! (¯I \Gamma A) \Gamma1 compactly for some ¯ 2 \Delta cc : Then for kx n kEn = O(1) and
k(iI \Gamma A n )x n kEn = O(1); we get from Hilbert's identity
(iI \Gamma A n ) \Gamma1 \Gamma (¯I \Gamma A n ) \Gamma1 = (¯ \Gamma i)(iI \Gamma A n ) \Gamma1 (¯I \Gamma A n ) \Gamma1 ;
(2.3)
that x n = (¯I \Gamma A n ) \Gamma1 (iI \Gamma A n )x n \Gamma (i \Gamma ¯)(¯I \Gamma A n ) \Gamma1 x n and it follows that fx n g is P­compact.
Conversely, let the implication (2.2) hold for some i 0 2 \Delta c `` ae(A): We are going to show then that
i 0 2 \Delta cc . Taking a bounded sequence fy n g; n 2 IN ; one gets that the sequence k(i 0 I \GammaA n ) \Gamma1 y n kEn = O(1)
for n 2 IN : Let us apply the implication (2.2) to the sequence x n = (i 0 I \Gamma A n ) \Gamma1 y n : It is easy to see that
fx n g is P­compact. Hence i 0 2 \Delta cc . 2
6

Corollary 2.2 Assume that \Delta cc 6= ;: Then \Delta cc = \Delta c `` ae(A):
Proof. It is clear that \Delta cc ` \Delta c `` ae(A): To prove that \Delta cc ' \Delta c `` ae(A) let us consider the Hilbert
identity (2.3). Now let ¯ 2 \Delta cc : Then ¯ 2 \Delta cc `` \Delta c `` ae(A): Hence for every i 2 \Delta c `` ae(A) and for any
bounded sequence fx n g; n 2 IN ; the sequence f(iI \Gamma A n ) \Gamma1 x n g is P­compact. 2
Comparing Definitions 2.7, 2.8, 2.13 and implication (2.2) we see that \Delta cc ` \Delta r :
Theorem 2.11 Assume that \Delta cc 6= ;: Then \Delta r = C :
Proof. Take any point – 1 2 C : We have to show that (– 1 I \Gamma A n ; – 1 I \Gamma A) are regularly compatible.
Assume that kx n kEn = O(1) and that f(– 1 I \Gamma A n )x n g is P­compact. To show that fx n g is P­compact
take ¯ 2 \Delta cc : Using (2.3) with i = – 1 one obtains x n = (¯I \Gamma A n ) \Gamma1 (– 1 I \Gamma A n )x n +(– 1 \Gamma ¯)(¯I \Gamma A n ) \Gamma1 x n
and therefore fx n g is P­compact. Assume now that x n ! x and (– 1 I \Gamma A n ) \Gamma1 x n ! y; as n ! 1 in
IN 0 ` IN : Then x = (¯I \Gamma A) \Gamma1 y \Gamma (– 1 \Gamma ¯)(¯I \Gamma A) \Gamma1 x and it follows that x 2 D(A) and (– 1 I \Gamma A)x = y.
2
3 Discretization of semigroups
Let us consider the well­posed Cauchy problem in the Banach space E with operator A 2 C(E)
u 0 (t) = Au(t); t 2 [0; 1);
u(0) = u 0 ;
(3.1)
where operator A generates C 0 ­semigroup exp(\DeltaA). It is well­known that the C 0 ­semigroup gives the
solution of (3.1) by the formula u(t) = exp(tA)u 0 for t – 0. The theory of well­posed problems and
numerical analysis of these problems have been developed extensively, see for instance [72], [85], [102],
[158], [160], [197], [213].
Let us consider on the general discretization scheme the semidiscrete approximation of the problem
(3.1) in the Banach spaces E n :
u 0
n (t) = A n u n (t); t 2 [0; 1);
u n (0) = u 0
n ; :
(3.2)
with the operators A n 2 C(E n ) , such that they generate C 0 ­semigroups, which are consistent with the
operator A and u 0
n ! u 0 :
3.1 The simplest discretization schemes
We have the following version of Trotter­Kato's Theorem on general approximation scheme:
Theorem 3.1 [200] ( Theorem ABC ) The following conditions (A) and (B) are equivalent to condition
(C).
(A) Consistency. There exists – 2 ae(A) `` `` n ae(A n ) such that the resolvents converge
(–I \Gamma A n ) \Gamma1 ! (– \Gamma A) \Gamma1 ;
(B) Stability. There are some constants M – 1 and !; which are not depending on n and such that
k exp(tA n )k Ÿ M exp(!t) for t – 0 and any n 2 N ;
(C) Convergence. For any finite T ? 0 one has max t2[0;T ] k exp(tA n )u 0
n \Gamma p n exp(tA)u 0 k ! 0 as
n !1; whenever u 0
n ! u 0 :
7

The analytic C 0 ­semigroup case is slightly different from the general case, but with the same property
(A).
Theorem 3.2 [158] Let operators A and A n generate analytic C 0 ­semigroups. The following conditions
(A) and (B 1 ) are equivalent to condition (C 1 ).
(A) Consistency. There exists – 2 ae(A) `` `` n ae(A n ) such that the resolvents converge
(–I \Gamma A n ) \Gamma1 ! (–I \Gamma A) \Gamma1 ;
(B 1 ) Stability. There are some constants M 2 – 1 and ! 2 such that
k(–I \Gamma A n ) \Gamma1 k Ÿ M 2
j– \Gamma ! 2 j ; Re– ? ! 2 ; n 2 N ;
(C 1 ) Convergence. For any finite ¯ ? 0 and some 0 ! ` ! ú
2
we have
max
j2\Sigma(`;¯)
k exp(jA n )u 0
n \Gamma p n exp(jA)u 0 k ! 0
as n !1 whenever u 0
n ! u 0 : Here \Sigma(`; ¯) = fz 2 \Sigma(`) : jzj Ÿ ¯g; and \Sigma(`) = fz 2 C : jarg zj Ÿ `g:
Definition 3.1 The linear operator A : D(A) ` E ! E is said to have the positive off­diagonal (POD)
property if hAu; OEi – 0 whenever 0 ¯ u 2 D(A) and 0 ¯ OE 2 E \Lambda with hu; OEi = 0:
Definition 3.2 An element e 2 E + is said to be an order unit in E if for every x 2 E there exists
0 Ÿ – 2 R such that \Gamma–e ¯ x ¯ –e: For e 2 intE + we can define an order unit norm by
kxk e = inff– – 0 : \Gamma–e ¯ x ¯ –eg:
The ordered Banach space E is called an order unit space if there exists e 2 intE + such that k \Delta kE = k \Delta k e :
We can state now Trotter­Kato's version for positive semigroups.
Theorem 3.3 [166] Let the operators A n and A from (3.1) and (3.2) be consistent and let E; E n be
order unit spaces and e n 2 D(A n ) `` intE +
n : Assume that the operators A n have the POD property and
A n e n ¯ 0 for sufficiently large n: Then exp(tA n ) ! exp(tA) uniformly in t 2 [0; T ]:
We can assume that conditions (A) and (B) for the corresponding semigroup case are satisfied without
any restriction of generality if any discretization processes in time are considered. If we denote by T n (\Delta) a
family of discrete semigroups as in [102], i.e. –
A n = 1
Ün (T n (Ü n ) \Gamma I) 2 B(E n ) and T n (t) = T n (Ü n ) kn ; where
k n = [ t
Ün ]; as Ü n ! 0; n !1; then one gets
Theorem 3.4 [200] ( Theorem ABC­discr ) The following conditions (A) and (B') are equivalent to
condition (C').
(A) Consistency. There exists – 2 ae(A) `` `` n ae( –
A n ) such that the resolvents converge
(–I \Gamma –
A n ) \Gamma1 ! (–I \Gamma A) \Gamma1 ;
(B') Stability. There are some constants M 1 – 1 and ! 1 such that
kT n (t)k Ÿ M 1 exp(! 1 t) for t 2 R+ = [0; 1);n 2 N ;
(C') Convergence. For any finite T ? 0 one has max t2 [0;T ] kT n (t)u 0
n \Gamma p n exp(tA)u 0 k ! 0 as n !1;
whenever u 0
n ! u 0 :
8

Theorem 3.5 [200] Assume that conditions (A) and (B) of Theorem 3.1 hold. Then the implicit dif­
ference scheme
U n (t + Ü n ) \Gamma U n (t)
Ü n
= A n U n (t + Ü ); U n (0) = u 0
n ;
(3.3)
is stable, i.e. k(I n \Gamma Ü n A n ) \Gammak n k Ÿ M 1 e ! 1 t ; t = k n Ü n 2 IR+ ; and gives an approximation to the solution
of the problem (3.1), i.e. U n (t) j (I n \Gamma Ü n A n ) \Gammak n u 0
n ! exp(tA)u 0
n P­converges uniformly with respect to
t = k n Ü n 2 [0; T ] as u 0
n ! u 0 ; n !1; k n !1; Ü n ! 0:
Here in Theorem 3.5 –
A n = A n (I n \Gamma Ü n A n ) \Gamma1 and therefore (I n \Gamma Ü n A n ) \Gammak n = (I n + Ü n –
A n ) kn .
Theorem 3.6 [200] Assume that conditions (A) and (B) of the Theorem 3.1 hold and condition
Ü n kA 2
n k = O(1)
(3.4)
is fulfilled. Then the difference scheme
U n (t + Ü n ) \Gamma U n (t)
Ü n
= A n U n (t); U n (0) = u 0
n ;
(3.5)
is stable, i.e. k(I n + Ü n A n ) kn k Ÿ Me !t ; t = k n Ü n 2 IR+ ; and gives an approximation to the solution of the
problem (3.1), i.e. U n (t) j (I n +Ü n A n ) kn u 0
n ! u(t) P­converges uniformly with respect to t = k n Ü n 2 [0; T ]
as n !1; k n !1; Ü n ! 0:
Theorem 3.7 [158] Assume that conditions (A) and (B 1 ) of Theorem 3.2 hold and condition
Ü n kA n k Ÿ 1=(M + 2); n 2 IN
(3.6)
is fulfilled. Then the difference scheme (3.5) is stable and gives an approximation to the solution of
the problem (3.1), i.e. U n (t) j (I n + Ü n A n ) kn u 0
n ! u(t) discretely P­converge uniformly with respect to
t = k n Ü n 2 [0; T ] as u 0
n ! u 0 ; n !1; k n !1; Ü n ! 0:
Let us introduce the following conditions:
(B 0
1 ) Stability. There are constants M 0 ; ! 0 such that
k exp(tA n )k Ÿ M 0 e ! 0 t ; kA n exp(tA n )k Ÿ M 0
t
e ! 0 t ; t 2 IR+ :
(B 00
1 ) Stability. There are constants M 00 ; ! 00 and Ü \Lambda ? 0 such that
k(I \Gamma Ü n A n ) \Gammak k Ÿ M 00 e ! 00 kÜn ; kkÜ n A n (I \Gamma Ü n A n ) \Gamma1 k Ÿ M 00 e ! 00 kÜn ; 0 ! Ü n ! Ü \Lambda ; n; k 2 IN :
Proposition 3.1 [180] Conditions (B 1 ), (B 0
1 ), (B 00
1 ) are equivalent.
Theorem 3.8 The conditions (A) and (B 00
1 ) are equivalent to the condition (C 1 ).
Theorem 3.9 [161] Let the assumptions of Theorem 3.7 and (3.4) be satisfied. Then
tA n (I + Ü n A n ) kn ! tA exp(tA) uniformly in t = k n Ü n 2 [0; T ]:
(3.7)
Conversely, if (I +Ü n A n ) kn ! exp(tA) uniformly in t = k n Ü n 2 [0; T ] and (3.7) is satisfied, then condition
(C 1 ) holds.
9

Theorem 3.10 [161] Let condition (B 1 ) hold. Then
k exp(tA n ) \Gamma (I \Gamma Ü n A n ) \Gammak n k Ÿ c
Ü n
t e !t :
If, moreover, stability condition (3.6) holds, then
k exp(tA n ) \Gamma (I + Ü n A n ) kn k Ÿ c
Ü n
t
e !t ;
k(exp(tA n ) \Gamma (I + Ü n A n ) kn )x n k Ÿ cÜ n e !t kA n x n k;
kA n (exp(tA n ) \Gamma (I + Ü n A n ) kn )x n k Ÿ c
Ü n
t
e !t kA n x n k; t = k n Ü n :
In the case of analytic C 0 ­semigroups for the forward scheme, as we saw, the following stability
condition: Ü n kA n k ! 1=(M+2); which can't be improved even in Hilbert spaces for self­adjoint operators.
In the case of almost periodic C 0 ­semigroups and the forward scheme for differential equations of first
order in time (3.1), one obtains necessary and sufficient stability condition Ü n kA n k ! 1 [160]. It was
discovered that the stability condition of the forward scheme like (3.5) for the positive C 0 ­semigroups
also can be written in the form Ü n kA n k ! 1; see [165].
Stability of difference schemes for differential equations in Hilbert spaces in the energy norm are in­
vestigated in [176], [177], where also schemes with weights were considered. Semidiscrete approximations
are studied in [177].
3.2 Rational approximation
Let us denote by P p (z) the element of the set of all real polynomials of degree not greater than p and by
ú p;q the set of all rational functions r p;q (z) = Pp (z)
Pq (z)
and P q (0) = 1: Then a Pad'e (p,q)­approximation for
e \Gammaz is defined as an element R p;q (z) 2 ú p;q such that
je \Gammaz \Gamma R p;q (z)j = O(jzj p+q+1 ) as jzj ! 0:
It is well known that a Pad'e approximation for e \Gammaz exists, is unique and is representable by the
formula R p;q (z) = P p;q (z)=Q p;q (z); where
P p;q (z) = \Sigma p
j=0
(p + q \Gamma j)!p!(\Gammaz) j
(p + q)!j!(p \Gamma j)!
; Q p;q (z) = \Sigma q
j=0
(p + q \Gamma j)!q!z j
(p + q)!j!(q \Gamma j)!
:
In the papers [171]­[172] details of the location of poles and order of convergence of rational approxima­
tions in different regions are given.
Definition 3.3 A rational approximation r p;q (\Delta) 2 ú p;q for e \Gammaz is said to be
a) A­acceptable if jr p;q (z)j ! 1 for Re(z) ? 0;
b) A(`)­acceptable if jr p;q (z)j ! 1 for z 2 \Sigma(`) = fz : \Gamma` ! arg(z) ! `; z 6= 0g:
It is well known that R q;q (z); R q\Gamma1;q (z) and R q\Gamma2;q (z) are A­acceptable. But for q – 3 and p = q \Gamma 3 the
Pad'e functions are not A­acceptable.
Theorem 3.11 [172] For any q – 2 and p – 0 the Pad'e approximation to e \Gammaz has no poles in the sector
S p;q = fz : jarg(z)j ! cos \Gamma1
i q\Gammap\Gamma2
p+q
j
g; in particular for p Ÿ q Ÿ p + 4 all poles lies in the left­half plane.
Since r(\Delta) 2 ú p;q is an approximation of e \Gammaz it is natural to construct an operator­function r(Ü n A n ) k
which could be considered as an approximation of exp(tA n ) for t = kÜ n : We assume in this section for
simplicity that k exp(tA n )k Ÿ M; t 2 IR+ :
10

Theorem 3.12 [41] Let condition (B) be satisfied. There is a constant C depending on r such that if
r is A­acceptable, then
kr(Ü n A n ) k k Ÿ CM
p
k for Ü n ? 0; k 2 IN :
Remark 3.1 The term
p
k in Theorem 3.12 can not be removed in general, moreover, there are examples
[52], [94] which shows that the inequality kr(Ü n A n ) k k – c
p
k; k 2 IN ; hold.
We say that r(\Delta) 2 ú p;q is accurate of order 1 Ÿ d Ÿ p + q if je \Gammaz \Gamma r(z)j = O(jzj d+1 ) as jzj ! 0:
Theorem 3.13 [41] Let condition (B) be satisfied. Then there is a constant C depending on r such
that, for r which is A­acceptable and accurate of order d, one gets
kr(Ü n A n ) k u 0
n \Gamma exp(tA n )u 0
n k Ÿ CMÜ d
n kA d+1
n u 0
n k for Ü n ? 0; k 2 IN ; u 0
n 2 D(A d+1
n ):
Theorem 3.14 [41] Let condition (B 1 ) is satisfied. Then there is a constant C depending on r such
that, for r which is A­acceptable and accurate of order d, one gets
kr(Ü n A n ) k u 0
n \Gamma exp(tA n )u 0
n k Ÿ CMÜ d
n kA d
n u 0
n k for Ü n ? 0; k 2 IN ; u 0
n 2 D(A d
n ):
Theorem 3.15 [182], [159] Let condition (B 1 ) be satisfied. Then there is a constant C depending on
r such that for r which is A­acceptable and accurate of order d with jr(1)j ! 1 or condition (3.6) is
satisfied, then one gets
kr(Ü n A n ) k u 0
n \Gamma exp(tA n )u 0
n k Ÿ CM
Ü fl
n
t d\Gammafl kA fl
n u 0
n k for Ü n ? 0; 0 Ÿ fl Ÿ d; t = kÜ n ; k 2 IN :
In [51] , [149], [151] the analogy of Theorems 3.13­3.15 was proved for multistep methods.
Let us recall that constant M 2 in condition (B 1 ) defines 0 ! ff ! ú
2
by M 2 sin ff ! 1 [107] such that
k(–I \Gamma A n ) \Gamma1 k Ÿ M
j– \Gamma !j
for any – 2 \Sigma(ú=2 + ff):
(3.8)
Theorem 3.16 [52], [147] Let condition (B 1 ) be satisfied. Then there is a constant C depending on r
such that, for r which is A(`)­acceptable, accurate of order d and ` 2 (ú=2 \Gamma ff; ú=2] for ff from condition
(3.8), one gets
kr(Ü n A n ) k k Ÿ CM for Ü n ? 0; k 2 IN ;
and
kr(Ü n A n ) k \Gamma exp(tA n ) \Gamma fl k exp(\GammaÜ \Gammab
n ak n (\GammaA n ) \Gammab )k Ÿ CM(k \Gammad
n + k \Gamma1=b
n ); t = k n Ü n ;
where fl = r(1) and a; b are some positive constants.
It is possible to show [148] that \Pi k
j=1 r(Ü n;j A n ) is a stable approximation for exp(\Sigma k
j=1 Ü n;j A) with variable
stepsize, but under condition 0 ! c Ÿ Ü i =Ü j Ÿ C ! 1; i; j 2 IN .
11

3.3 Richardson's Extrapolation method
Let us consider the schemes (3.3) and (3.5) which have an order of convergence O(Ü n ) and denote
U Ün
n (k n ) = U n (t)u 0
n and U Ün
n (k n ) = U n (t)u 0
n ; t = k n Ü n : The following deferred approach to the limit is
valid:
Theorem 3.17 [164] Assume that condition (B) is satisfied. Then for V n (t) = 2U Ün
n (k n ) \Gamma U Ün=2
n (2k n )
one has
kV n (t) \Gamma u n (t)k Ÿ Ü 2
n Me !t t 2 kA 3
n u 0
n k; t = k n Ü n :
If in addition the scheme (3.5) is stable, then for V n (t) = 2U Ün
n (k n ) \Gamma U Ün=2
n (2k n ); t = k n Ü n ;
kV n (t) \Gamma u n (t)k Ÿ Ü 2
n Me !t t 2 kA 3
n u 0
n k; t = k n Ü n :
Let us consider the Crank­Nicolson scheme
~
U n (kÜ n + Ü n ) \Gamma ~
U n (kÜ n )
Ü n
= A n
~
U n (kÜ n + Ü n ) + ~
U n (kÜ n )
2 ; ~
U n (0) = I n ; k 2 IN 0 ;
(3.9)
Theorem 3.18 [164] Assume that condition (B) is satisfied and that scheme (3.9) is stable. Then
/ n (t) = 4
3
~
U Ün=2
n (2k n ) \Gamma 1
3
~ U Ün
n (k n ) satisfies
k/ n (t) \Gamma u n (t)k Ÿ cÜ 4
n e !t t 2 kA 6
n u 0
n k; t = k n Ü n :
In general, let us put V Ün
n (t) = R p;q (Ü n A n ) kn u 0
n ; t = k n Ü n :
Theorem 3.19 [164] Assume that condition (B) is satisfied, p = q and the scheme which corresponds
to V Ün
n is stable. Then for / n (t) = \Gamma1
2 2q \Gamma1 V Ün
n (t) + 2 2q
2 2q \Gamma1 V Ün=2
n (t) one gets
k/ n (t) \Gamma u n (t)k Ÿ cÜ 2q+2
n e !t
i t 3=2
p Ü n
kA 2q+3
n u 0
n k + t 3 Ü 2q\Gamma3
n kA 4q+2
n u 0
n k
j
; t = k n Ü n :
Theorem 3.20 [164] Assume that condition (B 1 ) is satisfied, p = q and Ü n kA n k Ÿ const. Then for
0 Ÿ fl Ÿ 2q and / n (t) = \Gamma1
2 2q \Gamma1 V Ün
n (t) + 2 2q
2 2q \Gamma1 V Ün=2
n (t) one gets
k/ n (t) \Gamma u n (t)k Ÿ c Ü 2q+2
n
t 2q+2\Gammafl
e !t kA fl
n u 0
n k; t = k n Ü n :
3.4 Lax­type equivalence theorems with orders
The Lax equivalence Theorem on convergence of solution of approximation problem to the solution of
the given well­posed Cauchy problem states that stability of the method is necessary and sufficient for
convergence, provided it is consistent. More recently they furnish Lax's Theorem with orders, which
make possible consider ''not stable'' approximations.
Definition 3.4 The C 0 ­semigroups exp(tA n ) and exp(tA) are said to be consistent of order O('(Ü n ))
on a linear manifold U ae E with respect to the semigroup exp(\DeltaA) if exp(tA)U ` D(A) and there is a
constant C such that
k(A n p n \Gamma p n A) exp(tA)xk Ÿ CÜ n '(Ü n )e !t jxj U for any x 2 U;
(3.10)
where j \Delta j denotes the seminorm on U.
12

Definition 3.5 The C 0 ­semigroups exp(tA n ) is said to be stable of order O(M n e !n t ) if there are constants
M n and ! n such that
k exp(tA n )k Ÿ M n e !n t for any t 2 IR+ :
(3.11)
The following is a slight modification of [44]­[47] and [63]­[65], which was proved in [161].
Theorem 3.21 Let the C 0 ­semigroup exp(\DeltaA n ) be consistent of order O('(Ü n )) on a linear manifold
U ae E with respect to the semigroup exp(\DeltaA); exp(tA)U ae U and j exp(tA)xj U Ÿ M jxj U : The following
assertions are equivalent:
(i) k
i
exp(tA n )p n \Gamma p n exp(tA)
j
xk Ÿ 2M n e !n t K( Cn
2
t'(Ü n ); x; E; U);
(ii) k
i
exp(tA n )p n \Gamma p n exp(tA)
j
xk Ÿ M n e !n t
(
M x ; x 2 E;
Cn
2
t'(Ü n )jxj U ; t = k n Ü n 2 [0; T ]; x 2 U;
(iii) k exp(tA n )k Ÿ M n e !n t ; k(A n p n \Gamma p n A) exp(tA)xk Ÿ C n Ü n '(Ü n )e !t jxj U for any x 2 U; t 2 IR+ ;
where M x is a constant depending only on x and K(t; x; E; U) = inf y2U
n
kx \Gamma ykE + tjyj U
o
is Peetre
functional.
Definition 3.6 The family of discrete semigroups fU n (k n Ü n )g is said to be consistent of order O('(Ü n ))
on a linear manifold U ae E with respect to the semigroup exp(\DeltaA) provided U = E and
k(U n (Ü n )p n \Gamma p n exp(Ü n A)) exp(tA)xk Ÿ CÜ n '(Ü n )jxj U for any x 2 U:
(3.12)
Theorem 3.22 Let exp(tA n )U n ae U n ; condition (B) holds and j exp(tA n )xj Un Ÿ Ce !t jx n j Un for any
x n 2 U n and t ? 0: Then the following conditions are equivalent:
(a) k(U n (k n Ü n ) \Gamma exp(k n Ü n A n ))x n k Ÿ M n K( Cnkn Ün
2
'(Ü n ); x n ; E n ; U n ); n; k n 2 IN ;
(b) k(U n (k n Ü n ) \Gamma exp(k n Ü n A n ))x n k Ÿ M n
(
M xn ; x n 2 E n ;
Cn
2 te t! '(Ü n )jx n j Un ; x n 2 U n ;
(c) kU n (k n Ü n )k B(En) Ÿ M n ; k(U n (Ü n )\Gammaexp(Ü n A)) exp(tA n )x n k Ÿ CnMn
2
Ü n e !t '(Ü n )jx n j Un ; where k n Ü n =
t 2 [0; T ]:
Definition 3.7 The family of discrete semigroups fU(k n Ü n )g is said to be stable of order O(1=/(n \Gamma1 ))
if
kU n (k n Ü n )k B(En) Ÿ C=/(n \Gamma1 ); for n; k n 2 IN ; 0 ! Ü n Ÿ Ü \Lambda ; Ü n k n 2 [0; T ]:
(3.13)
Theorem 3.23 Let the discrete semigroup fU(k n Ü n )g be consistent of order O('(Ü n )) on a linear man­
ifold U ae E with respect to the semigroup exp(\DeltaA): The following assertions are equivalent:
(i) kU n (k n Ü n )k B(En) Ÿ C=/(n \Gamma1 );
(ii) k(U n (k n Ü n )p n \Gamma p n exp(k n Ü n A))xk Ÿ C
/(n \Gamma1 ) K(k n Ü n '(Ü n ); x; E; U); n; k n 2 IN ;
(iii) k(U n (k n Ü n )p n \Gamma p n exp(k n Ü n A))xk Ÿ C
/(n \Gamma1 )
(
M x ; x 2 E;
k n Ü n '(Ü n )jxj U ; k n Ü n 2 [0; T ]; x 2 U;
where M x is a constant depending only on x:
Theorem 3.24 Let j exp(tA)xj U Ÿ Cjxj U for any x 2 U and t 2 [0; T ]: Then the following conditions
are equivalent:
(i) The family of operators fU(k n Ü n )g is consistent of order O('(Ü n )) on a linear manifold U ae E
with respect to the semigroup exp(\DeltaA) and stable of order O(1=OE(n \Gamma1 ));
(ii) k(U n (k n Ü n )p n \Gamma p n exp(k n Ü n A))xk Ÿ C
/(n \Gamma1 )
K(k n Ü n '(Ü n ); x; E; U); n; k n 2 IN ;
(iii) k(U n (k n Ü n )p n \Gamma p n exp(tA))xk Ÿ C
/(n \Gamma1 )
(
M x ; x 2 E;
k n Ü n '(Ü n )jxj U ; t = k n Ü n 2 [0; T ]; x 2 U:
13

On the extension of Lax­Richtmyer theory see [154], [178].
For the particular case, when E = L p (IR d ) and operator A j P (D) = \Sigma jffjŸr p ff D ff in E, one the can
consider Cauchy problem
@u(x; t)
@t = P (D)u(x; t); u(x; 0) = u 0 (x); x 2 IR+ ;
(3.14)
with P (D) such that (3.14) is well­posed in the sense ku(\Delta; t)k L p (IR) Ÿ c ku 0 (\Delta)k L p (IR) ; t 2 IR+ :
Let us denote “
P (¸) = \Sigma jffjŸr p ff (i¸) ff : It is well­known that (3.14) well­posed iff k exp(t “
P )k Mp Ÿ C; t 2
IR+ ; where M p is a space of Fourier multipliers.
Semidiscrete approximation of (3.14) is given by
@u n (x; t)
@t = P h (D)u n (x; t); u n (x; 0) = u 0
n (x); x 2 IR+ ;
(3.15)
where P h (D h ) = h \Gammar \Sigma jffjŸr p ff (h)\Sigma fi2I ff b fi u n (x+fih; t) and “
P h (¸) = h \Gammar \Sigma jffjŸr p ff (h)\Sigma fi2I ff b fi e ih¸;hfii : Operator
P h (D h ) is said to be consistent with operator P (D) with order ¯ if “
P h (¸) \Gamma “
P (¸) = h ¯ j¸j r+¯ Q(h¸); r =
deg “
P h (¸); Q is an infinitely differentiable function and jQ(j)j – Q 0 ? 0 for 0 ! jjj Ÿ ffl 0 :
Theorem 3.25 [40] Let P (D) and P h (D h ) be consistent of order ¯ and (3.14) and (3.15) be well­posed.
Then for every T ? 0 there exists C ? 0 such that k
i
e tP h (D h ) \Gamma exp(tP (D)
j
)u 0 k L 2 (IR d ) Ÿ ch ¯ ku 0 k W 2;r+¯ (IR d )
and for 0 ! s ! r + ¯
k
i
e tP h (D h ) \Gamma exp(tP (D)
j
)u 0 k L 2 (IR d ) Ÿ ch s¯
¯+r ku 0 kB s
2
;
k
i
e tP h (D h ) \Gamma exp(tP (D)
j
)u 0
n k L 1 (IR d ) Ÿ ch s¯
¯+r ku 0 k B d=2+s
2
;
where B `
p = B `
p;1 is a Besov space.
It is remarked in [29], that, for the quite general case B `
p;q = (L p (IR); D(A)) `;q :
If we consider full discretization scheme for (3.14) in the form L h U k+1
n = B h U k
n ; k = 0; 1; 2; : : : ; where
L h v = \Sigma fi a fi (h)v(x + fih); B h v = \Sigma fi b fi (h)v(x + fih); then a discrete semigroup can be constructed as
U n (kÜ n )u 0
n = F \Gamma1
i “
U k
n (¸)“u 0
n
j
; “
U n (¸) = “
B n (¸)= “
L n (¸); “
B n (¸) = \Sigma fi a fi (h)e h¸;fihi (the time step Ü n is tied to h
by Ü n =h r = constant). Such a finite difference operator, U n (kÜ n ), approximates (3.14) with order ¯ if
“
U n (¸) = e Ün “
P (¸) +O(h r+¯ + j¸j r+¯ ) as ¸; h ! 0:
Theorem 3.26 [40] Let (3.14) be well­posed and U n (kÜ n ) be stable in E = L 2 (IR d ) and approximate
(3.14) with order ¯ ? 0: Then for any T ? 0 there is a constant c ? 0 such that
k
i
U n (t) \Gamma exp(tP (D)
j
)u 0
n k L 2 (IR d ) Ÿ ch ¯ ku 0
n k W 2;r+¯ (IR d )
and for 0 ! s ! r + ¯
k
i
U n (t) \Gamma exp(tP (D)
j
)u 0 k L 2 (IR d ) Ÿ ch s¯
¯+r ku 0
kB s
2
;
k
i
U n (t) \Gamma exp(tP (D)
j
)u 0 k L 1 (IR d ) Ÿ ch ¯ ku 0 k B d=2+¯+r
2;1
;
k
i
U n (t) \Gamma exp(tP (D)
j
)u 0 k L 1 (IR d ) Ÿ ch
s¯
¯+r ku 0 k B d=2+s
2
; t = kÜ n 2 [0; T ]:
Inverse results: order of convergence implies smoothness of u 0
n , see [29], [40].
Time discretization of parabolic problems with memory by backward Euler method was considered
in [24]. Stability and error estimates take place in a Banach space framework, and the results are used to
derive error estimates in the L 2 and maximum norms for piecewise linear finite­element discretizations
in two space dimensions.
14

4 Backward Cauchy problem
Let us consider in the Banach space E the backward Cauchy problem:
v 0 (t) = Av(t); t 2 [0; T ];
v(T ) = v T ;
(4.1)
where we are going to find element v(0). At least in two important cases it is not a well­posed problem,
namely, if A is unbounded and generates an analytic C 0 ­semigroup or if the C 0 ­semigroup exp(\DeltaA) is
compact. Indeed, in these situations the problem exp(TA)x = v T is ill­posed [48], [97], [193] in the sense
that the operator exp(\GammaT A) is not bounded on E and, moreover, in general D(exp(\GammaT A)) 6= E. This
means that the Cauchy problem (4.1) has a solution in general only for some (but not every) initial data
v T and the solution v(0); if it is exists, does not depend continuously on the initial data. After changes
of variables and putting v(j) = u(T \Gamma j) one can rewrite the problem (4.1) in the form
u 0 (t) = \GammaAu(t); t 2 [0; T ];
u(0) = u 0 ;
(4.2)
where u 0 = v T is given and u(T ) is the element which we are going to find. In this section we are going
consider the approximation of (4.2) with operator A; which generates analytic C 0 ­semigroup.
Definition 4.1 A bounded linear operator R ffl;T on the space E is called a regularizator for the Cauchy
problem ( 4.2 ) if for any ffi ? 0 and any u 0 2 E; for which a solution of (4.2) exists, there exists
ffl = ffl(ffi) ? 0 such that ffl(ffi) ! 0 as ffi ! 0 and sup ku ffi \Gammau 0 kŸffi kR ffl(ffi);T u ffi \Gamma exp(\GammaT A)u 0 k ! 0 as ffi ! 0:
In [137] it is proved that for the existence of a linear regularizator of the problem (4.2), which commute
with operator A; it is necessary and sufficient that \GammaA generates C ffl ­semigroups S ffl (t), 0 Ÿ t Ÿ T , such
that C ffl strongly converges to the identity operator I as ffl ! 0.
There are a lot of regularizators, which could be considered for problem (4.2). For example, in [162]
it was shown that if \GammaA 2
n generates a cosine operator function, then the method of quasireversibility,
which is given by the Cauchy problems
u 0
n;ff (t) = \GammaA n u n;ff (t) \Gamma ffA 2
n u n;ff (t); u n;ff (0) = u 0
n ;
is a regularization method for (4.2), and ku n;ff (T ) \Gamma p n u(T )k Ÿ Cff
i
ku 0
n \Gamma p n u 0 k=ffi + ae
j
; where ff = ff(ffi) =
1=
i
ln(1=ffi) \Gamma ln ln(1=ffi) \Gamma o(ln \Gamma1 (1=ffi))
j
: In this case S ff (t) j exp(\GammatA) exp(\GammaffT A 2 ) is a C ff ­semigroup
with C ff = exp(\GammaffT A 2 ) and C ff ! I as ff ! 0: Moreover, the generator of this C ff ­semigroup is \GammaA:
It has been shown in [57] that the stochastic differential equation
du ff (t) = \GammaAu ff (t)dt \Gamma ffAu ff (t)dw(t);
u(0) = u 0 ;
(4.3)
where w(\Delta) is a standard one­dimensional Wiener process, yields a stochastic regularization of (4.2).
Explicitly, the operator­function
U ff (t)u 0 = 1
2úi
Z
\Gamma
e \Gammat–\Gammaff
i
w(t)\Gammaw(0)
j
–\Gamma 1
2 ff 2 – 2 jtj
(– \Gamma A) \Gamma1 u 0 d–; t ? 0;
which represents a solution of (4.3) for any u 0 2 A c (A), possesses the following properties:
lim
ff!0
kU ff (T )u 0 \Gamma exp(\GammaT A)u 0 k = 0;
(4.4)
15

kU ff (t)k Ÿ c 1
ff
q
jtj
exp
0
@ c 2
q
jtj
ff
+ c 3 jtj \Gamma¯ ) 2
1
A + b(ff; jtj) for any ff ? 0:
(4.5)
Here the function b(ff; t) is bounded in the parameters ff and t and A c (A) is the set of entire vectors of
operator A: This means that by virtue of the inequality
kU ff (T )u ffi \Gamma exp(\GammaT A)u 0 k Ÿ kU ff (T )k ku ffi \Gamma u 0 k + kU ff (T )u 0 \Gamma exp(\GammaT A)u 0 k;
(4.6)
there is a dependence on ff = ff(ffi) in such a way that U ff (T ) becomes a regularizator. The operator­
function t 7! exp
i
(T \Gamma t)A
j
U ff (T ), 0 Ÿ t Ÿ T , is a C ff ­semigroup with C ff = exp(TA)U ff (T ): One can
see that C ff ! I as ff ! 0 and that the generator of this C ff ­semigroup is \GammaA:
4.1 C­semigroups and ill­posed problems
Let C be a bounded linear operator on the Banach space E; i.e. C 2 B(E); and let T ? 0 be some finite
number.
Definition 4.2 [188]. A family of bounded operators fS(t) : 0 Ÿ t ! Tg is called a local C­semigroup
on E if
(i) S(t + s)C = S(t)S(s) for t; s; t + s 2 [0; T );
(ii) S(0) = C;
(iii) S(\Delta) is strongly continuous on [0; T ):
Clearly, S(\Delta) is a commutative family. A local C­semigroup is called nondegenerate if condition S(t)x = 0
for all t 2 (0; T ) implies x = 0: It is seen from Definition 4.2 that a local C­semigroup is nondegenerate
[60] if and only if C is injective, i.e., N(C) = f0g. Concerning construction with N(C) 6= f0g see
[109], [110]. It is very interesting question how to apply the case of noninjective C; i.e. degenerate
C­semigroups, for ill­posed problems. Unfortunately this approach still is not realized.
Starting from now on we will consider only the case that C 2 B(E) is an injective operator.
Definition 4.3 The generator of fS(t) : 0 Ÿ t ! Tg is defined as the limit \GammaGx := C \Gamma1 lim h!0+
1
h (S(h)x\Gamma
Cx); x 2 D(G); with a natural domain D(G) :=
n
x 2 E : 9 lim h!0+
1
h (S(h)x \Gamma Cx) 2 R(C)
o
:
Proposition 4.1 [179] The operator G is closed, R(C) ` D(G) and C \Gamma1 GC = G:
We denote the C­semigroup S(\Delta) with generator \GammaG by S(\DeltaG): Next, let Ü 2 (0; T ): Put
L Ü (–)x :=
Z Ü
0
e \Gamma–t S(tG)xdt; x 2 E; – ? 0:
(4.7)
This is the so­called local Laplace transform of S(\DeltaG):
Proposition 4.2 Let S(\DeltaG) be a local C­semigroup and L Ü (\Delta) be the local Laplace transform of S(\DeltaG):
Then for any x 2 E one has L Ü (–)x 2 D(G) and
(– +G)L Ü (–)x = Cx \Gamma e \Gamma–Ü S(ÜG)x for all Ü 2 [0; T ) and – ? 0:
(4.8)
In the case of local C­semigroups spectrum oe(\GammaG) could be located on the line [0; 1): So in this case in
general the Laplace transform of the local C­semigroup does not exist in principle and we follow the ideas
from [27], [179], [188]. The function L Ü (–) with the property (4.8) is said to be asymptotic resolvent.
16

Theorem 4.1 [179] Let A be a closed linear operator in E and let C 2 B(E) be injective.
(i) If operator A is the generator of a local C­semigroup fS(t); 0 Ÿ t ! Tg on E, then there exists an
asymptotic C­resolvent L Ü (–) of \GammaA such that
k d m
d– m L Ü (–)xk Ÿ M Ü
m!
– m+1 kxk; x 2 E;
(4.9)
with 0 Ÿ m=– Ÿ Ü; – ? a; m 2 IN [ f0g; and operator A satisfies C \Gamma1 AC = A:
(ii) If \GammaA has an asymptotic resolvent which satisfies (4.9), and CD(A) is dense in D(A), D(C \Gamma1 AC) ae
D(A), i.e. Cx 2 D(A) and ACx 2 R(C) imply x 2 D(A), then the part A 0 of A in E 0 := D(A) generates
a local C­semigroup on E 0 , with C equal to C 0 := CjE 0
.
In particular, under the assumption that CD(A) = E, the operator \GammaA generates a local C­semigroup
on E if and only if C \Gamma1 AC = A and there exists an asymptotic C­resolvent satisfying (4.9). In this case,
A has dense domain.
Remark 4.1 An asymptotic C­resolvent L Ü (–) of operator \GammaA is compact for some – 2 C (and then
for any – large enough) if and only if S(\DeltaA) is compact and uniformly continuous in t. Indeed, if S(\DeltaA) is
compact then by (4.7) and [217] it follows that L Ü (–) is compact. Conversely, taking derivative of L Ü (–)
in Ü and using the fact that S(\DeltaA) is uniformly continuous in t we have that S(\DeltaA) is compact as the
uniform limit of compact operators. This fact could be used in the approximation of semilinear equations
in case of the C­semigroups approach (cf. section 6 ).
Let us consider the abstract Cauchy problem, which is given by (4.2).
Definition 4.4 A function u(\Delta) is called a solution to (ACP; T,y) if u(\Delta) is continuously differentiable
in t 2 [0; T ); u(t) 2 D(A) for all 0 Ÿ t ! T , and u(\Delta) satisfies (4.2). We denote (ACP;T; y) with
y 2 CD(A), also by (ACP;T; CD(A)).
Definition 4.5 The Cauchy problem (ACP ; T; CD(A)) is said to be generalized well­posed if for every
y 2 CD(A); there is a unique solution u(\Delta; y) to (ACP ; T; y) such that ku(t; y)k Ÿ M(t)kC \Gamma1 yk for
0 Ÿ t ! T and y 2 CD(A); where the function M(t) is bounded on every compact subinterval of [0; T ):
We have to stress here that generalized well­posedness in the sense of Definition 4.5 is more general than
in the case of the problem in (3.1). Moreover, we can state that this generalized well­posedness is a
solvability condition of (4.2) for which a regularizator exists.
Theorem 4.2 [179] Let C be a bounded linear injection on E and let A be a closed linear operator.
Then the following assertions are equivalent:
(I) The operator \GammaA is the generator of a local C­semigroup;
(II) C \Gamma1 AC = A, and the problem: v 0 (t) = \GammaAv(t) + Cx; t 2 [0; T ); v(0) = 0; has a unique solution
for every x 2 X:
If either ae(A) 6= ; or A has dense domain, (I) and (II) are also equivalent to
(III) C \Gamma1 AC = A, and the problem
i
ACP ; T; CD(A)
j
is generalized well­posed. Moreover, u(t; y) =
C \Gamma1 S(tA)y; t 2 [0; T ); is a unique solution for every initial value y 2 CD(A):
Since the local C­semigroups are regularizators of the ill­posed problem (4.2) it is very important to
present approximation theory of local C­semigroups.
17

4.2 Semidiscrete approximation theorem
Let us consider, within the general discretization scheme, the semidiscrete approximation of the problem
(4.2) in the Banach spaces E n :
u 0
n (t) = \GammaA n u n (t); t 2 [0; T );
u n (0) = u 0
n ;
(4.10)
where the operators \GammaA n are generators of local C n ­semigroups, which are compatible with the operator
\GammaA and u 0
n ! u 0 : We understand compatibility in the sense of the general approximation scheme as the
PP­convergence of C n ! C and the PP­convergence of resolvents ( ~
– \Gamma A n ) \Gamma1 ! ( ~
– \Gamma A) \Gamma1 for some
~
– 2 ae(A) `` ae(A n ): Recall that in our general case (4.1) such ~
– exists, since the conditions (A) and (B)
from Theorem 3.1 are naturally assumed to be satisfied.
Theorem 4.3 [211] (Theorem ABC­C) Under the assumption CD(A 2 ) = E, the following conditions
(A c ) together with (B c ) are equivalent to condition (C c ).
(A c ) Consistency. C n ! C and operators A n , A are compatible;
(B c ) Stability. For any 0 ! Ü ! T there is some constant M Ü , not depending on n, such that
kS(tA n )k Ÿ M Ü for 0 Ÿ t Ÿ Ü and n 2 IN ;
(C c ) Convergence. For any 0 ! Ü ! T we have max t2[0;Ü ] kS(tA n )x 0
n \Gamma p n S(tA)x 0 k = 0; as n !
1,whenever x 0
n ! x 0 .
Remark 4.2 In the case of exponentially bounded C­semigroups [61], [62] we can trivially change the
condition (A c ) to condition
(A') C n ! C and ( ~
– \Gamma A n ) \Gamma1 C n ! ( ~ – \Gamma A) \Gamma1 C for some ~ – 2 C ; see [222] for details. Since,
the construction can be done just as with condition (A'), in this case we do not need to assume that
( ~ – \Gamma A n ) \Gamma1 ! ( ~ – \Gamma A) \Gamma1 for some ~ –:
Remark 4.3 We have put the condition CD(A 2 ) = E for simplicity. For the general case one gets
convergence on the set CD(A 2 ): In the case of integrated semigroups such situations have been well in­
vestigated, see for instance [30], [32]. Actually the paper [30] is devoted to the following effect observed
in the study of convergence of semigroups. Suppose we are given a sequence of uniformly bounded semi­
groups fexp(tA n ); t – 0g; n – 1 (k exp(tA n )k Ÿ M; t 2 IR+ ) acting in the Banach space E: Assume
furthermore that the limit lim n!1 (–I \Gamma A n ) \Gamma1 x = S(–)x exists for any x 2 E: If
R(S(–)) = E
(4.11)
(R(S(–)) is common for all – ? 0), the semigroups in question converge strongly, by the Trotter ­ Kato
theorem. One may also show that if condition (4.11) is relaxed, the limit
lim
n!1
exp(tA n )x
(4.12)
exists for all x 2 R(S(–)) (see e.g. [112], [67] p. 34, or [30], [32]). As observed by T.G. Kurtz [112] for
any x 2 E there exists the limit
lim
n!1
Z t
0
exp(sA n )xds:
(4.13)
In general, however, one cannot expect that (4.12) holds for x 62 R(S(–)): This effect is of course related
to Arendt's theorem, or rather to generation theorem for ''absolutely continuous integrated semigroups''
presented in [34].
18

Let us consider a semi­discretization of problem (4.3) in Banach spaces E n
du n;ff (t) = \GammaA n u n;ff (t)dt \Gamma ffA n u n;ff (t)dw(t);
u n;ff (0) = u 0
n ;
(4.14)
where u 0
n ! u 0 ; the operators A n generate analytic semigroups and
f(\Omega ; F ; IP ) ; w(t)g is the standard
one­dimensional Wiener process (Brownian motion). As usual the symbol IE[\Delta] will denote mathematical
expectation.
We emphasize that we consider the situation where oe(A n ), oe(A) ae C n \Sigma( 3
4
ú):
Theorem 4.4 [211] Let the conditions (A) and (B 1 ) of Theorem 3.2 be satisfied and let ffi n ? 0 be a
sequence which converges to 0 if n ! 1. Then there exists a sequence ff n such that u n;ffn (t) ! u(t)
for every t 2 [0; T ] as n ! 1: Here u n;ffn (\Delta) is a solution of (4.14) and u(\Delta) is a solution of (4.2) with
u 0 2 A c (A): Convergence is to be understood in the following sense:
sup
ku 0 n \Gammap nu 0 kŸffi n
ku n;ffn (t) \Gamma p n u(t)k ! 0; IP ­almost surely as ffi n ! 0:
4.3 Approximation by discrete C­semigroups
Following Section 3 we denote by fT n (\Delta)g a family of discrete semigroups, on E n , respectively, i.e., T n (t) =
T n (Ü n ) kn ; where k n = [t=Ü n ]: We define the generator of T n (\Delta) by the formula \GammaA n = 1
Ü n
(T n (Ü n ) \Gamma I) and
we are interested in the process Ü n ! 0; k n ; n !1: We assume that C n 2 B(E n ) is an injective operator
such that T n C n = C n T n : The discrete C n ­semigroup U n (\Delta) is defined as U n (t) = T n (t)C n : In this section
we also assume that the dimension of each space E n is finite, but dim (E n ) !1 as n !1:
Theorem 4.5 [211] ( Theorem ABC­C­discr ) Under condition (A) in Theorem 3.1 and the assumption
CD(A 2 ) = E, the following conditions (A cd ) and (B cd ) together are equivalent to condition (C cd ).
(A cd ) Consistency. C n ! C; operators A n ; A are compatible and A n 2 B(E n ), n 2 IN ;
(B cd ) Stability. For any 0 ! Ü ! T there is some constant M Ü ; not depending on n, such that
kU n (t)k Ÿ M Ü for all 0 Ÿ t Ÿ Ü ! T and n 2 IN :
is satisfied uniformly for any choice of fÜ n g and fk n g as long as Ü n ! 0, and k n = [t=Ü n ];
(C cd ) Convergence. For any 0 ! Ü ! T one gets max t2[0;Ü ] kU n (t)x 0
n \Gamma p n S(tA)x 0 k ! 0 as Ü n !
0; n !1; whenever x 0
n ! x 0 :
Theorem 4.6 [211] Let the conditions (A c ) and (B c ) be satisfied. Assume that condition (A) in The­
orem 3.1 and the assumption CD(A 2 ) = E are satisfied and also that Ü n kA 2
n C \Gamma1
n k Ÿ q
MÜ T with q ! 1:
Then
kU n (t)k Ÿ M Ü (1 \Gamma q) \Gamma1 for 0 Ÿ t Ÿ Ü ! T and any n 2 IN :
is satisfied uniformly for any choice of fÜ n g and fk n g with Ü n ! 0, as long as k n =
h t
Ün
i
. Moreover, for
any 0 ! Ü ! T one gets max t2[0;Ü ] kU n (t)x 0
n \Gamma p n S(tA)x 0 k ! 0 as Ü n ! 0, n !1; whenever x 0
n ! x 0 :
Remark 4.4 As a matter of fact the scheme U n (t) j (I+Ü n A n ) \Gammak n C n with t = k n Ü n could be constructed
even under condition (3.4). Indeed, Ü n A n = Ü n –A n (– \Gamma A n ) \Gamma1 \Gamma Ü n A 2
n (– \Gamma A n ) \Gamma1 , and by the choice of –
we can make the second term less than ffl and then by choosing Ü n appropriately for a fixed –, we obtain
kÜ n A n k Ÿ 2ffl, so that the scheme U n (\Delta) is well defined.
19

Remark 4.5 In contrast to the well­posed case, for ill­posed problems it looks that the implicit and
explicit methods of discretization in time are not so different in the sense of stability advantages ( compare
with Theorems 3.5, 3.6). Moreover, under condition (3.4) from the identity
(I \Gamma Ü n A n ) kn C n = (I \Gamma Ü 2
n A 2
n ) kn (I + Ü n A n ) \Gammak n C n
and inequality k(I \Sigma Ü 2
n A 2
n ) kn k Ÿ Ce tÜnkA 2
n k ; t = k n Ü n ; it follows that the stability properties of the implicit
and explicit methods are the same.
There are a lot of stochastic finite­difference schemes which could be written for problem (4.14). For
example, some of the simplest are
U n;ff (t + Ü n ) \Gamma U n;ff (t) = \GammaÜ n A n U n;ff (t) \Gamma ff \Deltaw(t)A n U n;ff (t);
(4.15)
¯
U n;ff (t + Ü n ) \Gamma ¯
U n;ff (t) = \GammaÜ n A n ¯
U n;ff (t + Ü n ) \Gamma ff \Deltaw(t)A n ¯
U n;ff (t);
(4.16)
where \Deltaw(t) =
i
w(t) \Gamma w(t \Gamma Ü n )
j
, t = k n Ü n ; and U n;ff (0) = ¯
U n;ff (0) = I n :
Theorem 4.7 [211] Let the conditions (A) and (B 1 ) of Theorem 3.2 be satisfied. Assume that the
stability conditions (3.4) and Ü n kA 2
n ke ckAnk = O(1) are fulfilled for some constant c ? 0: Then for
ff n = p Ü n the scheme (4.15) behaves stably in the following sense:
j n := sup
n2IN
sup
ae
IE
Ÿfl fl fl fl U n;ffn (t)u 0
n \Gamma exp
`
\GammatA n + ff n (w(t) \Gamma w(0))A n \Gamma t
2 ff 2
n A 2
n
'
u 0
n
fl fl fl fl
–
: ku 0
n k Ÿ 1
oe
! 0;
and converges in the following sense:
IE
h
kU n;ffn (t)u 0
n \Gamma p n u(t)k
i
Ÿ C
p
Ü n kA exp(\GammaT A)u 0 k + ku n;ffn (t) \Gamma p n u ff n (t)k + Cj n ku 0
n k; 0 ! t Ÿ T:
For the scheme U n;ffn (\Delta) similar notions are employed.
We can also study the convergence behaviour of more sophisticated numerical methods. For example,
in [42] in order to approximate (4.14) the following Runge­Kutta scheme was considered
Y 1 = U n;ff (t) + p
Ü n ffA n U n;ff (t);
U n;ff (t + Ü n ) \Gamma U n;ff (t) = \GammaÜ n A n U n;ff (t) + ff\Deltaw(t)A n U n;ff (t)+
+
p Ü n
2
` (\Deltaw(t)) 2
p Ü n
\Gamma 1
' /
ff n A n Y 1 \Gamma ffA n U n;ff (t)
!
:
(4.17)
Thus the solution can be written in the form
U n;ff (t + Ü n ) =
= (I n \Gamma Ü n A n \Gamma ff 2 Ü n
2 A 2
n ) kn \Pi kn
k=1
`
I n + Z \Gamma1
n ff\Deltaw(t)A n + (Z \Gamma1
n =2)ff 2 \Deltaw(t) 2 A 2
n
'
U n;ff (0);
(4.18)
where Z n = (I n \Gamma Ü n A n \Gamma ff 2 Ün
2
A 2
n ):
Theorem 4.8 [42] Let the conditions (A) and (B) of Theorem 3.2 be satisfied. Assume that the stability
conditions (3.4) and Ü n kA 2
n ke ckAnk = O(1) are fulfilled for some constant c ? 0: Then for ff n = p
Ü n the
scheme (4.18) behaves stably in the following sense:
j n := sup
ae
IE
Ÿfl fl fl fl U n;ffn (t)u 0
n \Gamma e ( \GammatA n+ffn (w(t)\Gammaw(0))A n \Gamma t
2 ff 2
n A 2 n) u 0
n
fl fl fl fl
–
: ku 0
n k Ÿ 1
oe
! 0;
and converges in the following sense:
IE
h
kU n;ffn (t)u 0
n \Gamma p n u(t)k
i
Ÿ
Ÿ C
p
Ü n kA exp(\GammaT A)u 0 k + ku n;ffn (t) \Gamma p n u ff n (t)k + Cj n ku 0
n k; 0 ! t Ÿ T:
20

In the case of the well­posed problem
(
du ff (t) = Au ff (t)dt + ffAu ff (t)dw(t);
u(0) = u 0 ;
(4.19)
where the operator A generates an analytic C 0 ­semigroup, the semidiscrete and full­discretisation schemes
do not need additional stability assumptions and the order of convergence will be defined just by the
consistency property of the scheme. More exactly the term e –t under the integral leads to the absolute
convergence of the integral independently of the behaviour of ff on any compact set. For example we
have the following
Theorem 4.9 [42] Let the conditions (A) and (B'') of Theorem 3.2 be satisfied. Assume that the stability
conditions (3.4) is fulfilled for some constant C ? 0: Then for any ff n 2 [0; ff 0 ] the scheme like (4.18)
behaves stably in the following sense:
sup
ae
IE
Ÿfl fl fl fl U n;ffn (t)u 0
n \Gamma e (tAn+ffn (w(t)\Gammaw(0))A n \Gamma t
2 ff 2
n A 2 n) u 0
n
fl fl fl fl
–
: ku 0
n k Ÿ 1
oe
Ÿ Ü n ;
and converges in the following sense:
IE
h
kU n;ffn (t)u 0
n \Gamma p n exp(tA)u 0 k
i
Ÿ Cff n kA exp(tA)u 0 k + ku n;ffn (t) \Gamma p n u ff n (t)k + CÜ n ku 0
n k; 0 ! t Ÿ T:
5 Coercive inequalities
Let us consider in the Banach space E the nonhomogeneous Cauchy problem:
u 0 (t) = Au(t) + f(t); t 2 [0; T ];
u(0) = u 0 ;
(5.1)
with operator A which generates C 0 ­semigroup and f(\Delta) be some function from [0; T ] to E. The problem
(5.1) can be considered in different functional spaces. The most popular situations are the following
settings: well­posedness in C([0; T ]; E); C ff;0 ([0; T ]; E) and L p ([0; T ]; E) spaces (see [9], [22], [136], [214]).
We say that problem (5.1) is well­posed, say in C([0; T ]; E); if for any f(\Delta) 2 C([0; T ]; E) and any
u 0 2 D(A) the problem (5.1) is
i). uniquely solvable, i.e. u(\Delta) satisfies the equation and boundary condition (5.1) , u(\Delta) is continuously
differentiable on [0; T ]; u(t) 2 D(A) for any t 2 [0; T ] and Au(\Delta) is continuous on [0; T ];
ii). operator (f(\Delta); u 0 ) ! u(\Delta), regarded as an operator from C([0; T ]; E) \Theta D(A) to C([0; T ]; E) is
continuous.
In case u 0 j 0; the coercive well­posedness in C([0; T ]; E) means that kAu(\Delta)k C([0;T ];E) Ÿ ckf(\Delta)k C([0;T ];E) :
In general coercive well­posedness in the space \Upsilon([0; T ]; E) means for problem (5.1) that it is well­posed
in the space \Upsilon([0; T ]; E) and one has
ku 0 (\Delta)k \Upsilon([0;T ];E) + kAu(\Delta)k \Upsilon([0;T ];E) Ÿ C (kf(\Delta)k \Upsilon([0;T ];E) + ku 0 k F );
where F is some subspace of E: For results of coercive well­posedness see [9], [22], [136].
The semidiscrete approximation of (5.1) are the Cauchy problems in Banach spaces E n :
u 0
n (t) = A n u n (t) + f n (t); t 2 [0; T ];
u n (0) = u 0
n ;
(5.2)
21

with operators A n which generate C 0 ­semigroups, A n and A are compatible, u 0
n ! u 0 and f n ! f in
appropriate sense. Following section 3 it is natural to assume that conditions (A) and (B 1 ) are satisfied.
Here we are going to describe the discretization of (5.2) in time. The simplest difference scheme
(Rothe scheme) is
U k
n \GammaU k\Gamma1
n
Ün = A n U k
n + ' k
n ; k 2 f1; :::; [ T
Ün ]g;
U 0
n = u 0
n ;
(5.3)
where, for example, in the case of f n (\Delta) 2 C([0; T ]; E n ) one can put ' k
n = f n (kÜ n ); k 2 f1; :::; Kg;K = [ T
Ün ];
and in the case f n 2 L 1 ([0; T ]; E n ) one can set ' k
n = 1
Ün
R t k
t k\Gamma1
f n (s)ds; t k = kÜ n ; k 2 f1; :::; Kg:
5.1 Coercive inequality in C Ü n
([0; T ]; E n ) spaces
Denote by C Ün ([0; T ]; E n ) the space of the elements ' n = f' k
n g K
k=0 such that ' k
n 2 E n ; k 2 f0; :::; Kg;
with the norm k' n kCÜn ([0;T ];En) = max 0ŸkŸK k' k
n kEn :
We recall that coercive well­posedness in C([0; T ]; E) implies [22] that A generates an analytic C 0 ­
semigroup.
Theorem 5.1 [22] Let condition (B 1 ) be satisfied. The problem (5.3 ) is stable in the space C Ün ([0; T ]; E n );
i.e.
kU n k CÜn ([0;T ];En) Ÿ C
i
k' n k CÜn ([0;T ];En) + ku 0
n k
j
:
Theorem 5.2 [22] Let condition (B 1 ) be satisfied. The problem (5.3 ) is almost coercive stable in the
space C Ün ([0; T ]; E n ); i.e.
kA n U n k CÜn ([0;T ];En) Ÿ M
i
kA n u 0
n kEn + min
i
ln(1=Ü n ); 1 +
fi fi fi ln kA n k
fi fi fi
j
k' n k CÜn ([0;T ];En)
j
:
We have note that if (5.1) is coercive well­posed in the space C([0; T ]; E); then [66] operator A should
be bounded or the space E contains a subspace isomorphic to c 0 : It means that in general problem (5.3)
is not coercive well­posed in C Ün ([0; T ]; E n ) space.
For the explicit scheme
U k
n \GammaU k\Gamma1
n
Ü = A n U k\Gamma1
n + ' k
n ; k 2 f1; :::; Kg;
U 0
n = u 0
n ;
(5.4)
Theorem 5.2 could be reconstructed, but under a stability condition.
Theorem 5.3 [22] Let condition (B 1 ) is satisfied and Ü n ln( 1
Ün ) kA n k Ÿ ffl for sufficiently small ffl ? 0:
Then the problem (5.4) is almost coercive stable in the space C Ün ([0; T ]; E n ); i.e.
kA n U n k CÜn ([0;T ];En) + kU n k CÜn ([0;T ];E n;1\Gamma 1
ln 1
Ün
Ÿ
Ÿ M
`
kA n u 0
n kE n;1\Gamma 1
ln 1
Ün
+ min
i
ln(1=Ü n ); 1 +
fi fi fi ln kA n k
fi fi fi
j
k' n kC Ün ([0;T ];En)
'
;
where ku n kE n;ff =
i R 1
0 kA n exp(tA n )u n k
1
1\Gammaff
En dt
j 1\Gammaff
.
Remark 5.1 The space E n;ff coincides with equivalent norm with the real interpolation space
(E n ; D(A n )) 1\Gamma1=p;p see [136].
22

5.2 Coercive inequality in C ff;0
Ü n
([0; T ]; E n ) spaces
Denote by C ff;0
Ün ([0; T ]; E n ) for 0 ! ff ! 1 the space of the elements ' n with the norm k' n k C ff;0
Ün ([0;T ];En)
=
max 0ŸkŸK k' k
n kEn + max 1Ÿk!k+lŸK k' k+l
n \Gamma ' k
n kEn (Ü n k) ff (lÜ n ) \Gammaff :
Theorem 5.4 [180] Let condition (B 1 ) hold. Then the scheme (5.3) is coercive well­posed in C ff;0
Ün ([0; T ]; E n )
with 0 ! ff ! 1; i.e.
kA n U n k C ff;0
Ün ([0;T ];En) Ÿ M
ff(1 \Gamma ff)
i
kA n u 0
n kEn + k' n k C ff;0
Ün ([0;T ];En)
j
:
Roughly speaking, assumption (B 1 ) is necessary and sufficient for coercive well­posedness in C ff;0
Ün ([0; T ]; E n )
space.
5.3 Coercive inequality in L p
Ü n
([0; T ]; E n ) spaces
Denote by L p
Ün ([0; T ]; E n ) for 1 Ÿ p ! 1 the space of the elements ' n with the norm k' n k L p
Ün ([0;T ];En) =
i
\Sigma K
j=0 k' k
n k p
En Ü n
j 1=p
:
Theorem 5.5 [180] Let condition (B 1 ) hold. Let the difference scheme (5.3) be coercive well­posed in
L p 0
Ün ([0; T ]; E n ) for some 1 ! p 0 ! 1: Then it is coercive well­posed in L p
Ün ([0; T ]; E n ) for any 1 ! p ! 1
and
kA n U n k L p
Ün ([0;T ];En) + max
0ŸkŸK
kU k
n kE n;1\Gamma1=p Ÿ Mp 2
p \Gamma 1
i
k' n k L p
Ün ([0;T ];En) + kU 0
n k 1\Gamma1=p
j
:
We have to note that to the contrary to C ff;0 space case the analyticity of semigroup exp(\DeltaA) is not
enough to get coersive well­posedness in L p space [124], so to state coercive well­posedness in L p we need
some additional assumptions.
Theorem 5.6 [180] Let 1 ! p; q ! 1; 0 ! ff ! 1 and condition (B 1 ) hold. Then the difference scheme
(5.3) is coercive well­posed in L p
Ün ([0; T ]; E n;ff;q ); i.e.
kA n U n k L p
Ün ([0;T ];E n;ff;q ) + max
0ŸkŸK
kU k
n kE n;1\Gamma1=p Ÿ Mp 2
(p \Gamma 1)ff(1 \Gamma ff)
i
k' n k L p
Ün ([0;T ];E n;ff;q ) + kU 0
n k 1\Gamma1=p
j
;
where E n;ff;q is the interpolation space (E n ; D(A n )) ff;q with the norm ku n kE n;ff;q =
i R 1
0 k– ff A n (– \Gamma
A n ) \Gamma1 k q
En
d–
–
j 1=q
:
For general Banach space E we have the following results. Assume that A is the generator of the analytic
semigroup exp(tA); t 2 IR+ ; of the linear bounded operators with exponentially decreasing norm when
t !1: That means that stability condition (B 00
1 ) holds with ! 00 Ÿ 0:
Theorem 5.7 [21] Let condition (B 1 ) hold. Then the solution of difference scheme (5.3) is almost
coercive stable, i.e.
kA n U n k L p
Ün ([0;T ];En) Ÿ M
i
kA n U 0
n kEn + minfln 1
Ü n
; 1 + j ln kA n k B(En) jg k' n k L p
Ün ([0;T ];En)
j
holds for any p – 1; where M does not depend on Ü n ; u 0
n ; and ' n :
23

Of course for schemes like
U k
n \GammaU k\Gamma1
n
Ün = A n ( U k
n +U k\Gamma1
n
2
) + ' k
n ; n 2 f1; :::; Kg;
U 0
n = u 0
n :
(5.5)
coercive well­posedness could be considered.
Theorem 5.8 [21] Let the condition (B 1 ) holds. Then the solution of difference scheme (5.5) is almost
coercive stable, i.e. the estimate
kfA n
U j
n + U j \Gamma1
n
2 gk L p
Ün ([0;T ];En) Ÿ M
i
kA n u 0
n kEn + minfln
1
Ü n
; 1 + j ln kA n kEn7!En jg k' n k L p
Ün ([0;T ];En)
j
holds for any p – 1; where M does not depend on Ü n ; u 0
n ; and ' n :
Theorem 5.9 [21] Let the condition (B 1 ) holds and condition (3.6) is satisfied. Then the solution of
difference scheme (5.5) is almost coercive stable, i.e. the estimate
kA n U n k L p
Ün ([0;T ];En) Ÿ M
i
kA n u 0
n kEn + minfln 1
Ü n
; 1 + j ln kA n kEn7!En jg k' n k L p
Ün ([0;T ];En)
j
holds for any p – 1; where M does not depend on Ü n ; u 0
n ; and ' n :
The necessary and sufficient conditions for coercive well­posedness of the problem (5.1) in L p ([0; T ]; E)
was obtained in [219], [218], [98]. More precisely, a Banach space E has the UMD--property, whenever
the Hilbert transform Hf(t) = 1
ú PV \Gamma
R 1
\Gamma1
1
t\Gammas f(s)ds extends to a bounded operator on L p (IR; E) for
some (all) p 2 (1; 1): It is well known, that all subspaces and quotient spaces of L
q(\Omega ; ¯) with 1 ! q ! 1
have this property.
The Poisson semigroup on L 1 (IR) and on L p (IR; E) does not coercive well posed on L p (IR; E) space
if E is not an UMD--space (cf. [124]). Hence the assumptions on E to be an UMD space is necessary in
some sense.
But it was an open problem whether every generator of an analytic semigroup on L
q(\Omega ; ¯), 1 ! q ! 1,
provide coercive well­posedness in L p (IR; E). Recently Kalton and Lancien [100] gave a strong negative
answer to this question. If every bounded analytic semigroup on a Banach space E is such that problem
(5.1) is coercive well­posed, then E is isomorphic to a Hilbert space.
If A generates a bounded analytic semigroup fexp(zA) : j arg (z)j Ÿ ffig, on a Banach space E, then
the following three sets are bounded in the operator norm
i) f–(– \Gamma A) \Gamma1 : – 2 iIR; – 6= 0g;
ii) fexp(tA); tA exp(tA) : t ? 0g;
iii) fexp(zA) : j arg zj Ÿ ffig.
In Hilbert spaces this already implies coercive well­posedness in L p (IR + ; E), but only in Hilbert spaces
E. The additional assumption that we need in more general Banach spaces E will be R--boundedness.
A set T ae B(E) is called R--bounded, if there is a constant C ! 1, such that for all Z 1 ; : : : ; Z k 2 T
and x 1 ; : : : ; x k 2 E; k 2 IN ;
Z 1
0
k
k
X
j=0
r j (u)Z j (x j )k du Ÿ C
Z 1
0
k
k
X
j=0
r j (u)x j k du;
(5.6)
where fr j g is a sequence of independent symmetric f\Gamma1; 1g-- valued random variables, e. g. the Rademacher
functions r j (t) = sign (sin(2 j út)) on [0; 1]. The smallest C, such that (5:6) is fulfilled, is called the R--
boundedness constant of T and is denoted by R(T).
24

Theorem 5.10 [219] Let A generates a bounded analytic semigroup exp(tA) on a UMD--space E. Then
the problem (5.1) is coercive well­posed in the space L p (IR + ; E) if and only if one of the sets i), ii) or
iii) above is R--bounded.
The interpretation of discrete coercive inequality and discrete semigroup define the convolution operator
in the form –
A n \Sigma k
j=0 T k\Gammaj
n Q n ' n Ü n with some bounded operator Q n 2 B(E n ); which usually has smoothness
property as it is clear from the Proofs of the Theorems 5.7 and 5.8. Here T n (Ü n ) k is discrete semigroup
say as in Subsection 3.1. Boundedeness of convolution operator in L p
Ün (ZZ + ; E n ) space implies discrete
coercive well­posedness in L p
Ün (ZZ + ; E n ):
We assume also that Banach spaces E n satisfy in this section the collectively UMD--property, i.e. we
assume that the Hilbert transforms H n f n (t) = 1
ú PV \Gamma
R 1
\Gamma1
1
t\Gammas f n (s)ds extend to a bounded operators
on L p (IR; E n ) for some (all) p 2 (1; 1); such that all of them are bounded by constant which does not
depend on n: This assumption holds for example if all E n can be embedded into a fixed space L
p(\Omega\Gamma
with 1 ! p ! 1:
Definition 5.1 A discrete semigroup T n (\Delta) with generator –
A n is said generates coercive well­posedness on
L p
Ün (ZZ + ; E n ) space if the corresponding convolution operator ' n 7! f –
A n \Sigma k
j=0 T k\Gammaj
n Q n ' j
n Ü n g is continuous
on L p
Ün (ZZ + ; E n ) space.
Theorem 5.11 [21] Assume that for convolution operator ' n 7! f –
A n \Sigma k
j=0 T k\Gammaj
n Q n ' j
n Ü n g the following
conditions hold
1 0 . the set f –
A n (– \Gamma T n ) \Gamma1 Q n Ü n : j–j = 1; – 6= 1; – 6= \Gamma1g is R­bounded;
2 0 . the set f(– \Gamma 1)(– + 1) –
A n (– \Gamma T n ) \Gamma2 Q n Ü n : j–j = 1; – 6= 1; – 6= \Gamma1g is R­bounded.
Then a discrete semigroup T n (\Delta) generates coercive well­posedness on L p
Ün (ZZ + ; E n ) space.
Theorem 5.12 [21] Let E n be UMD Banach spaces. Assume also that the set f–(– \Gamma A n ) \Gamma1 : – 2
iIR; – 6= 0g is R­bounded with the R--boundedness constant which does not depend on n: Then the solution
of difference scheme (5.3) is coercive stable, i.e.
k –
A n U n k L p
Ün (ZZ + ;En ) Ÿ Mk' n k L p
Ün (ZZ + ;En )
(5.7)
holds for any p – 1; where M does not depend on Ü n ; u 0
n ; and ' n :
Remark 5.2 We have to note that Theorem 3.2 could be reformulates in the notions of R­boundedness
with changes of condition (B 1 ) for condition: there is a 0 ! ` ! ú=2 such that the set f–(– \Gamma A n ) \Gamma1 : – 2
\Sigma(` + ú=2)g is R­bounded with the R--boundedness constant which does not depend on n: The condition
(C 1 ) could be written, due to [219] Theorem 4.2, in the form exp(tA n ) ! exp(tA) converges for any
t 2 IR and there is a 0 ! ` ! ú=2 such that the set fexp(zA n ) : z 2 \Sigma(`)g is R­bounded with the
R--boundedness constant which does not depend on n: So one of our assumption in Theorems 5.12 and
5.13 is in some sense condition (B 1 ) changed for R­boundedness condition.
Theorem 5.13 [21] Let E n be UMD Banach spaces. Assume also that the set f–(– \Gamma A) \Gamma1 : – 2
iIR; – 6= 0g is R­bounded with the R--boundedness constant which does not depend on n: Then the solution
of difference scheme (5.5) is coercive stable, i.e.
kf –
A n
U k
n + U k\Gamma1
n
2
gk L p
Ün ([0;T ];En) Ÿ Mk' n k L p
Ün ([0;T ];En)
(5.8)
holds for any p – 1; where M does not depend on Ü n ; u 0
n ; and ' n :
25

Remark 5.3 Analyzing the Proofs of Theorems 5.12 and 5.13 it is easy to see that one can put –
A n = A n
in the statements (5.7) and (5.8). Moreover, the statement (5.8) could be written in the form
k –
A n U n k L p
Ün ([0;T ];En) Ÿ Mk' n k L p
Ün ([0;T ];En) :
The proof of that fact based on equality –
A n = A n (I n \Gamma Ün
2
A n ) \Gamma1 :
It is possible to consider a more general Pad'e difference scheme [22] for p = q \Gamma 1 or p = q \Gamma 2: In
this case the difference scheme will be written in the form
U k
n \Gamma U k\Gamma1
n
Ü n
= ( –
A n U n ) k\Gamma1 + ' p;q;k
n ; U 0
n = u 0
n ; 1 Ÿ k Ÿ K:
(5.9)
where ( –
A n U n ) k = ( R p;q (ÜnAn )\GammaI
Ün U n ) k\Gamma1 and k' k
n \Gamma ' p;q;k
n kEn Ÿ MÜ p+q
n : To formulate coercive statements of
subsections 5.1­5.3 we just need to change operator A n for –
A n : We know from Theorem 3.16 that Pad'e
approximation under condition (B 1 ) with p = q is stable, but in general it will be not coercive stable.
To get coercive inequality we need condition (3.6). Spaces where the problem considered also could be
very different [22].
5.4 Coercive inequality in B Ü n
([0; T ]; C `
h(\Omega h )) `` C h ([0; T ]; C h (
¯\Omega h ))
From the numerical analysis point of view it would be very interesting to consider the Problem (5.1)
in the space \Upsilon([0; T ]; E) that E would be smoother than
C(\Omega\Gamma (elements of such a space could easily
be well approximated) and \Upsilon([0; T ]; E) be like C([0; T ]; E) or bounded functions space. The interesting
fact is that such a situation actually possible at least for the strongly elliptic operator of order 2 with
coefficients of class C `
(\Omega ): Since in such space E operator (p n v) i = v(ih) very concrete, i.e. taking values
in the grid points, we consider the case omitting in this section the notion of p n :
Theorem 5.14 [36]
Let\Omega be an open bounded subset of IR d , lying on one side of its topological boundary
@\Omega , which is a submanifold of IR d of dimension d \Gamma 1 and class C 2+` , for some ` 2 (0; 2) n f1g: Let A =
A(x; D x ) =
P
jffjŸ2 a ff (x)D ff
x be a strongly elliptic operator of order two (thus, Re
P
jffj=2 a ff (x)¸ ff – šj¸j 2
for some š ? 0 and for any (x; ¸)
2\Omega \Theta IR d ) with coefficients of class C `
(\Omega ). Then, there exist ¯ – 0,
OE 0 2 ( ú
2
; ú) such that for any – 2 C , with j–j – ¯ and jArg–j Ÿ OE 0 the problem
–v \Gamma Av = y;
fl 0 v = 0;
has for any y 2 C ` (\Omega\Gamma a unique solution v belonging to C 2+` (\Omega\Gamma and for a certain M ? 0,
j–j 1+ `
2 kvk C(\Omega\Gamma + j–j kvk C ` (\Omega\Gamma + kvk C 2+` (\Omega\Gamma Ÿ M
i
kyk C ` (\Omega\Gamma + j–j `
2 kfl 0 ykC(@
j
;
(5.10)
where fl 0 is the trace operator on
@\Omega .
It is clear from (5.10) that operator A in general does not generate C 0 ­semigroup in E = C ` (\Omega\Gamma space,
but, following say [136], one can construct semigroup exp(tA); t – 0; which will be analytic.
Let I = ZZ and E be a Banach space with norm k \Delta k: For a grid function U : I ! E, writing U j or
(U) j instead of U(j) for any j 2 I; we set
B(I; E) := fU : I ! E : sup
j2I
kU j k ! +1g; kUk B(I;E) := sup
j2I
kU j k:
26

It is easily seen that B(I; E) is a Banach space with the norm k \Delta k B(I;E) . If the set I is some interval,
say I = (a; 1); we denote by B(I; E) the set of all bounded functions from I to E:
For the grid function U : I ! E and h ? 0, we define operator @ h by formula
(@ h U) j := h \Gamma1 (U j+1 \Gamma U j ):
For any m 2 ZZ we set (@ m
h U) j := h \Gammam P m
i=0
i m
i
j
(\Gamma1) m\Gammai U j+i : If U 2 B(I; E), we set
kUk C m
h (I;E)
:= max
n
k@ r
h Uk B ( I;E) : 0 Ÿ r Ÿ m
o
:
Finally, let ` 2 (0; 1): We define
[U ] C `
h (I;E)
:= sup
ni
(k \Gamma j)h
j \Gamma`
kU k \Gamma U j k : j; k 2 I; j ! k
o
and, if m 2 IN 0 ,
kUk C m+`
h (I;E)
:= max
n
kUk C m
h (I;E)
; [@ m
h U ] C `
h (I;E)
o
:
In the same context we shall indicate B(I; E) with C 0
h (I; E). If E = C we write simply B(I) or C 0
h (I).
Let f 2 B(IN ; E): We indicate with ~
f ; the extension of f to IN 0 such that ~
f 0 = 0: For a nonnegative
real number !, we define
kfk C !
h;0 (IN ;E) := k ~
fk C !
h (IN 0 ;E) :
(5.11)
Let now L ? 0; n 2 IN ; n – 3, and h = L
n : For j 2 I := f1; :::; n \Gamma 1g we are given complex numbers
a j ; b j ; b 0
j ; c j satisfying the following conditions ('):
('1): there exists š ? 0 such that Re(a j ) – š, for every j 2 I;
('2): max
n
ja j j; jb j j; jb 0
j j; jc j j
o
Ÿ Q for every j 2 I with Q ? š;
('3); there exists ! : [0; L] ! [0; +1) such that !(0) = 0 and ! is continuous in 0 such that for
j; k 2 I with j Ÿ k
ja k \Gamma a j j Ÿ !
i
(k \Gamma j)h
j
:
For – 2 C we are going to study the following problem:
–U j \Gamma a j (@ 2
h U) j \Gamma1 \Gamma b j (@ h U) j \Gamma b 0
j (@ h U) j \Gamma1 \Gamma c j U j = f j for j = 1; :::; n \Gamma 1;
U 0 = U n = 0:
(5.12)
To this aim, we set I := f0; 1; :::; n \Gamma 1; ng and for U 2 B(I; E); we define
~
U j =
(
U j if j 2 I;
0 if j 2 f0; ng:
We introduce operator A h in B(I; E) defined as follows:
(A h U) j := a j (@ 2
h
~
U ) j \Gamma1 + b j (@ h ~
U) j + b 0
j (@ h ~
U) j \Gamma1 + c j ~
U j for j 2 I:
(5.13)
Further, we assume that
(' ` 1): there exists š ? 0 such that Re(a j ) – š for every j 2 I;
(' ` 2): max
n
kak C `
h (I)
; kbk C `
h (I)
; kb 0 k C `
h (I)
; kbk C `
h (I)
o
Ÿ Q with Q ? š:
27

Proposition 5.1 [92] Assume that assumptions (' ` ) are satisfied for some ` 2 (0; 2) n f1g. Fix OE 0 2
[0; ú \Gamma arccos( š
Q )). Then there exists ¯ 0 ? 0 such that f– 2 C : j–j – ¯ 0 ; jArg(–)j Ÿ OE 0 g ` ae(A h ) where
A h is the operator defined in (5.13). Moreover, for every r 2 [0; 2] there exists c ? 0 depending only on
L; š; Q; r such that for every f 2 B(I; E); and any F 2 B(I; E) with F j I = f , one has
k(– \Gamma A h ) \Gamma1 fk C `+r
h;0 (I;E)
Ÿ cj–j r
2 \Gamma1
i
kFk C `
h (I;E)
+ j–j `
2 maxfkF 0 k; kF n kg
j
:
Let us consider the following mixed Cauchy­Dirichlet parabolic problem:
@u
@t (t; x) = Au(t; x) + f(t; x); t 2 [0; T ]; x 2 [0; L];
u(t; x 0 ) = 0; t 2 [0; T ]; x 0 2 f0; Lg;
u(0; x) = 0; x 2 [0; L];
(5.14)
where A is a second order differential operator and L ? 0. We say that problem (5.14) has a strict
solution if there exists a continuous function u(t; x) with the first derivative with respect to t and the
derivatives of order less than or equal to 2 with respect to x which are continuous up to boundary of
[0; T ] \Theta [0; L]; i.e. u 2 C 1
i
[0; T ];
C(\Omega )
j
`` C
i
[0; T ]; C 2 (\Omega\Gamma
j
, and the equations in (5.14) are satisfied.
Theorem 5.15 [88] Consider problem (5.14) under the following assumptions:
(I) T and L are positive real numbers;
(II) ` 2 (0; 1) n f 1
2
g;
(III) Au(x) = a(x) @ 2 u
@x 2 (t; x) + b(x) @u
@x (t; x) + c(x)u(t; x), with a; b; c 2 C 2` ([0; L]);
(IV) a is real valued and mina = š ? 0;
(V) f 2 C([0; T ] \Theta [0; L]), t ! f(t; :) 2 B
i
[0; T ]; C 2` ([0; L])
j
, t ! f(t; 0) and t ! f(t; L) belong to
C ` ([0; T ]), f(0; 0) = f(0; L) = 0.
Then there exists a unique strict solution u(\Delta) of problem (5.14). Such solution belongs to
B
i
[0; T ]; C 2+2` ([0; L])
j
and @u
@t 2 B
i
[0; T ]; C 2` ([0; L]
j
.
Now let I be a set which can depend on a positive parameter h and the Banach space X h = B(I).
Next we introduce a linear operator A h in X h depending on h. In every case ae(A h ) contains f– 2
C n f0g : j–j – R and jArg(–)j Ÿ OE 0 g; where R ? 0 and OE 0 2 ( ú
2
; ú); and there exists M ? 0 such that
for – in the specified subset of C ; k(– \Gamma A h ) \Gamma1 k L(X h ) Ÿ M j–j \Gamma1 : Here R; OE 0 and M are independent of h:
Then we consider another set ~
I such that I ` ~
I; we put ~
X h := B( ~ I). We define an extension operator
E h from X h to ~
X h : in all our concrete cases this was the extension with zero. Next, for ` 2 (0; 1), we
introduce the norms k \Delta k 2`;h and k \Delta k 2+2`;h in ~
X h . The first of these norms is connected to k \Delta kX and
the operator A h by the following property: there exist two positive constants c 1 and c 2 independent of h
such that for every U 2 X h
c 1 kE h Uk 2`;h Ÿ kUk (X h ;D(A h )) ` Ÿ c 2 kE h Uk 2`;h :
Then for every h we consider a restriction operator R h 2 L( ~
X h ; X h ), such that R h E h = I X h : Let us also
introduce a seminorm p h in ~
X h : in the concrete cases we have p h (U) = kU j ~
InI k B( ~
InI) . We assume that,
if j–j – R and jArg(–)j Ÿ OE 0 , for every G 2 ~
X h ; then
j–j kE h (– \Gamma A h ) \Gamma1 R h Gk 2`;h + kE h (– \Gamma A h ) \Gamma1 R h Gk 2+2`;h Ÿ M
i
kGk 2`;h + j–j ` p h (G)
j
:
Another inequality we impose is the following. If j–j – R, jArg(–)j Ÿ OE 0 and G 2 ~
X h , then
kA h (– \Gamma A h ) \Gamma1 R h GkX h Ÿ M j–j \Gamma`
i
kGk 2`;h + j–j ` p h (G)
j
:
Such an inequality can be easily deduced in each of our examples. In the formulation below we remove
the parameter h. In the case of the backward Euler scheme (5.3) we have
28

Theorem 5.16 [92] Let X and ~
X be Banach spaces with norms k \Delta kX and k \Delta k ~
X respectively, A 2
B(X);E 2 B(X; ~
X); R 2 B( ~
X;X) such that RE = I X . Assume, moreover, that ` 2 (0; 1) and k \Delta k 2`
and k \Delta k 2+2` are norms in ~
X, while p is a seminorm in the same space. Finally, assume that there exist
R ? 0; OE 0 2 ( ú
2
; ú); M ? 0; such that the following conditions are satisfied:
(a) f– 2 C : j–j – R; jArg(–)j Ÿ OE 0 g ` ae(A) and, for – in this set,
k(– \Gamma A) \Gamma1 k B(X) Ÿ M(1 + j–j) \Gamma1 ;
(b) for every F 2 X
kEFk 2`
Ÿ MkFk (X;D(A)) ` ;
(c) for every V 2 ~
X ; – 2 C with j–j – R and jArg(–)j Ÿ OE 0
(1 + j–j) \Gamma1 kE(– \Gamma A) \Gamma1 RV k 2` + kE(– \Gamma A) \Gamma1 RV k 2+2`
+(1 + j–j) ` kA(– \Gamma A) \Gamma1 RV kX Ÿ M
i
kV k 2` + (1 + j–j) ` p(V )
j
;
(d) p(V ) Ÿ kV k 2` for every V 2 ~
X and p(EF ) = 0 for every F 2 X;
(e) kRV kX Ÿ kV k 2` for every V 2 ~
X :
Let T ? 0, K 2 IN ; K – 2, Ü = T
K . Assume that ÜR ! 1.
Let G 2 B(f0; 1; :::; Kg; ~
X) be such that G 0 = 0 and consider problem (5.3) with ' k = RG k for k =
1; :::; K and U 0 = 0. Then for U 2 B(f0; 1; :::; Kg;X) which solves (5.3) one has
kEU k
k 2+2` Ÿ c
i
max
0ŸkŸK
kG k k 2` + max
0Ÿk 1 !k 2 ŸK
((k 2 \Gamma k 1 )Ü ) \Gamma` p(G k 2 \Gamma G k 1 )
j
;
(5.15)
for k = 0; 1; :::; K; where c is a positive constant depending only on `, R, OE 0 , M , T and independent of
Ü n and G.
Now we consider the Crank­Nicolson scheme: we replace (5.3) with (5.5). Theorem 5.16 has the
following correspondent:
Theorem 5.17 [91] Assume that the assumptions of Theorem 5.16 are all satisfied and, moreover,
(f) kÜAkB(X) Ÿ S with some S ? 0;
(g) if j–j – 2S, then
kE(– \Gamma ÜA) \Gamma1 RV k 2` Ÿ M
i
kV k 2` + Ü \Gamma` p(V )
j
for every V 2 ~
X;
(h) kERV k 2` Ÿ M
i
kV k 2` + Ü \Gamma` p(V )
j
for every V 2 ~
X;
(i) 2ÜR ! 1.
Let G 2 B(f0; 1; :::; Kg; ~
X) be such that G 0 = 0 and consider problem (5.5) with ' k = RG k for k =
1; :::; K and U 0 = 0. If U 2 B
i
f0; 1; :::; Kg;X
j
solves (5.5) for k = 0; 1; :::; K, then
kEU k k 2+2` Ÿ c
i
max
0ŸkŸK
kG k k 2` + max
0Ÿk 1 !k 2 ŸK
((k 2 \Gamma k 1 )Ü ) \Gamma` p(G k 2 \Gamma G k 1 )
j
;
(5.16)
where c is a positive constant depending only on `; R; OE 0 ; M; S; T and is independent of Ü n and G.
29

An application of Theorems 5.16 and 5.17 to the discretization of problem (5.14) is the following. Let
K;n 2 IN : We put Ü := T
K ; h := L
n . We assume that K – 2; n – 3. For j = 0; 1; :::; n we set a j := a(jh),
b j = 1
2
b(jh); c j := c(jh); N n\Gamma1 := f1; :::; n \Gamma 1g, N n := f0; 1; :::; n \Gamma 1; ng,
X := B(N n\Gamma1 ); ~
X := B(N n ):
(5.17)
If V 2 X, we put, as before for i 2 N n\Gamma1 ;
(A h V ) i := a i
~
V i+1 \Gamma 2 ~
V i + ~
V i+1
h 2
+ b i
~
V i+1 \Gamma ~
V i\Gamma1
2h + c i ~
V i ;
(5.18)
where
~
V i = (EV ) i =
(
V i if 1 Ÿ i Ÿ n \Gamma 1;
0 if i 2 f0; ng:
Next, we define
R 2 B( ~
X;X);RV := V j Nn\Gamma1
(5.19)
for every V 2 ~
X. Then, again for V 2 ~
X, we set, for ` 2 (0; 1
2
),
kV k 2` := maxfkV k ~
X ; max
0Ÿi 1 !i 2 Ÿn
((i 2 \Gamma i 1 )h) \Gamma2` jV i2 \Gamma V i 1
jg;
(5.20)
kV k 2+2` := maxfkV k ~
X ; max
0ŸiŸn\Gamma1
j(@ h V ) i j; max
0ŸiŸn\Gamma2
j(@ 2
h V ) i j;
max
0Ÿi 1 !i 2 Ÿn\Gamma2
((i 2 \Gamma i 1 )h) \Gamma2` j(@ 2
h V ) i2 \Gamma (@ 2
h V ) i 1
jg;
(5.21)
with
(@ h V ) i := V i+1 \Gamma V i
h
for 0 Ÿ i Ÿ n \Gamma 1; (@ 2
h V ) i := V i+2 \Gamma 2V i+1 + V i
h
for 0 Ÿ i Ÿ n \Gamma 2;
p(V ) := maxfjV 0 j; jV n jg:
(5.22)
One has the following result:
Theorem 5.18 [91] With the positions (5.17),(5.18), (5.19) and (5.20) the assumptions (a) \Gamma (e) of
Theorem 5.16 are satisfied, with R; OE 0 ; M independent of h. If we impose the further condition
Ü n Ÿ ffh 2 ;
(5.23)
the same holds true also for assumptions (f) \Gamma (h) of Theorem 5.17 (with even S independent of n).
As a consequence, we have:
Theorem 5.19 [92] Consider the problem (5.14) under the assumptions of Theorem 5.15. With the
conventions (5.17),(5.18), (5.19), (5.20), set G k
j := f(kÜ n ; jh) for k 2 f1; :::; Kg; j = 0; :::; n: Put
' k := RG k and indicate with G 0 the zero in B(N n ).
Then if Ü n is sufficiently small,the problem
~
U k
j \Gamma ~
U k\Gamma1
j
Ün = a i
~
U k
i+1 \Gamma2 ~
U k
i + ~
U k
i\Gamma1
h 2 + b i
~
U k
i+1 \Gamma ~
U k
i\Gamma1
2h
+ c i ~
U k
i + ' k
j ; ;
~
U 0
j = 0;
(5.24)
for j 2 1; : : : ; n \Gamma 1; k 2 f1; :::; Kg has a unique solution such that
k ~
U k k C 2+2`
h (Nn )
Ÿ c
i
kfk B([0;T ];C 2` ([0;L])) + maxfkf(\Delta; 0)k C ` ([0;T ]) ; kf(\Delta; L)k C ` ([0;T ]) g
j
;
(5.25)
with c independent of h and Ü n .
30

An analogous result holds true for the Crank­Nicholson scheme (5.5). Then we set G k
j := f((k \Gamma 1
2
)Ü n ; jh);
under the further condition (5.23).
Remark 5.4 From Theorem 5.18 it follows that, for the scheme (5.5) with u 0
n = 0 and under condition
(5.23)
kA h Uk C 2`
h (Nn\Gamma1 ) Ÿ c
i
kfk B([0;T ];C 2` ([0;L])) + maxfkf n (\Delta; 0)k C ` ([0;T ]) ; kf(\Delta; L)k C ` ([0;T ]) g
j
;
(5.26)
with c independent of h and Ü n . In the quoted papers Theorems 5.16 and 5.17 are applied also to the
discretization of the heat equation in a square.
A counter example in [91] and [90] shows that condition (5.23) cannot be removed in general. Finally
estimates of order of convergence are given in [90].
6 Approximations of Semilinear Equations
Let us consider in a Banach space E the semilinear Cauchy problem
u 0 (t) = Au(t) + f(t; u(t)); u(0) = u 0 ;
(6.1)
with operator A, which generates analytic C 0 ­semigroup of type !(A) ! 0 and function f is smooth
enough. Existence and uniqueness of solution of problem (6.1) have been discussed in [9], [28], [93], [95],
[136] for example.
6.1 Approximations of Cauchy problem
By a semidiscrete approximation of the problem (6.1) we mean the Cauchy problems in the Banach
spaces E n
u 0
n (t) = A n u n (t) + f n (t; u n (t)); u n (0) = u 0
n ;
(6.2)
where the operators A n generate analytic semigroups in E n , A n and A are compatible, functions f n
approximate f and u 0
n ! u 0 :
Let
\Omega be an open set in a Banach space F and let B :
¯\Omega ! F be a compact operator having no
fixed points on the boundary
of\Omega : Then for the vector field F(x) = x \Gamma Bx there is defined the rotation
fl(I \Gamma B; @
\Omega\Gamma ; being an integer­valued characteristic of this field. Let x \Lambda be a unique isolated fixed point
of the operator B in the ball S r 0
of radius r 0 with the centre at x \Lambda . Then fl(I \Gamma B; @S r ) = fl(I \Gamma B; @S r 0
)
for 0 ! r Ÿ r 0 ; and this common value of the rotations is called the index of the fixed point x \Lambda and it is
denoted by ind x \Lambda :
Theorem 6.1 [163] Assume that conditions (A) and (B 1 ) hold and compact resolvents (–I \Gamma A) \Gamma1 ; (–I \Gamma
A n ) \Gamma1 converge (–I \Gamma A n ) \Gamma1 ! (–I \Gamma A) \Gamma1 compactly for some – 2 ae(A) and u 0
n ! u 0 : Assume that
(i) the functions f n ; f are bounded and sufficiently smooth so that there exists a unique mild solution
u \Lambda (\Delta) of the problem (6.1) on [0; T ] ( in this situation ind u \Lambda (\Delta) = 1);
(ii) f n (t; x n ) ! f(t; x) uniformly with respect to t 2 [0; T ] for x n ! x;
(iii) the space E is separable .
Then for almost all n the problems (6.2) have in the neighbourhood of p n u \Lambda (\Delta) mild solutions u \Lambda
n (t); t 2
[0; T ]: Each sequence fu \Lambda
n (t)g is P­compact and u \Lambda
n (t) ! u \Lambda (t) uniformly with respect to t 2 [0; T ]:
31

Let us consider time discretization with respect to the explicit difference scheme
U n (t + Ü n ) \Gamma U n (t)
Ü n
= A n U n (t) + f n (t; U n (t)); U n (0) = u 0
n ; t = kÜ n ; k = f0; \Delta \Delta \Delta ; Kg:
(6.3)
Theorem 6.2 [163] Assume that the conditions of Theorem 6.1 and condition (3.6) are satisfied. Then
the functions U n (t) from (6.3) give an approximate mild solution u \Lambda (\Delta) of the problem (6.1) and, moreover,
U n (t) ! u \Lambda (t) uniformly with respect to t 2 [0; T ]:
Let us define the operator !(u n )(t) = u n (t) \Gamma
R t
0 exp((t \Gamma s)A n )f(s; u n (s))ds:
Remark 6.1 If we assume that the conditions of Theorem 6.1 hold and the functions f(\Delta); f n (\Delta) have
Fr'echet derivatives in some balls containing the solutions u \Lambda and u \Lambda
n and moreover, assume that f 0
nu (t; p n u \Lambda (t))
are uniformly continuous with respect to the first and second arguments and f 0
nun (t; u n (t)) ! f 0
u (t; u \Lambda (t))
uniformly with respect to t 2 [0; T ] for u n ! u \Lambda ; then [163] for almost all n the problems (6.2) have
in the neighbourhood of p n u \Lambda (\Delta) mild solutions u \Lambda
n (t); t 2 [0; T ]: Each sequence fu \Lambda
n (\Delta)g is P­compact and
u \Lambda
n (t) ! u \Lambda (t) uniformly with respect to t 2 [0; T ] and, moreover, for sufficiently large n – n 0 and some
T \Lambda Ÿ T we have
c 1 ffl n (u \Lambda ; u 0
n ) Ÿ ku \Lambda
n \Gamma p n u \Lambda k Fn Ÿ c 2 ffl n (u \Lambda ; u 0
n );
where the constants c 1 ; c 2 are not dependent on n, F n = C([0; T ]; E n ) , while
ffl n (u \Lambda ; u 0
n ) = max
t2[0;T \Lambda ]
k!(p n u \Lambda )(t) \Gamma exp(tA n )u 0
n kEn :
Let U n (t) = (I n + Ü n A n ) k ; = n (u n )(t) = u n (t) \Gamma \Sigma k\Gamma1
l=1 U n ((k \Gamma l)Ü n ) f n (lÜ n ; u n (lÜ n )) Ü n :
Remark 6.2 If we assume that the conditions of Theorem 6.2 hold and the functions f(\Delta); f n (\Delta) have
Fr'echet derivatives in some balls containing the solutions u \Lambda (\Delta) and u \Lambda
n (\Delta) and moreover, assume that
f 0
nun (t; p n u \Lambda (t)) are uniformly continuous with respect to the first and second arguments and f 0
nun (t; u n (t)) !
f 0
u (t; u \Lambda (t)) uniformly with respect to t 2 [0; T ] for u n ! u \Lambda , and condition (3.6) holds, then [163] the
functions U n (t) from (6.3) give an approximate mild solution of the problem (6.1) and U \Lambda
n (t) ! u \Lambda (t)
uniformly with respect to t 2 [0; T ] and , moreover, for sufficiently large n – n 0 and some T \Lambda Ÿ T we
have
c 1 ffl n (u \Lambda ; u 0
n ) Ÿ kU \Lambda
n \Gamma p n u \Lambda k F Ün
n Ÿ c 2 ffl n (u \Lambda ; u 0
n );
where the constants c 1 ; c 2 do not depend on n; F Ün
n = fu n (kÜ n ) : max 0ŸkÜnŸT ku n (kÜ n )k En ! 1 g; while
ffl n (u \Lambda ; u 0
n ) = max t2[0;T \Lambda ] k= n (p n u \Lambda )(t) \Gamma U n (t)u 0
n kEn :
Schemes which have higher order of convergence than (6.3) are considered in [143], [163]. The Runge­
Kutta methods for semilinear equations were considered in [76], [134], [133], [132], [143], [146].
6.2 Approximation of periodic problem
Let us consider in a Banach space E the semilinear T­periodic problem
v 0 (t) = Av(t) + f(t; v(t)); v(t) = v(T + t); t 2 IR+ ;
(6.4)
with operator A, which generates an analytic compact C 0 ­semigroup and with the function f which is
smooth enough and f(t; x) = f(t + T; x) for any x 2 E and t 2 IR+ : Let u(\Delta; u 0 ) be the solution of
32

Cauchy problem (6.1) with initial data u(0; u 0 ) = u 0 : This function u(\Delta; u 0 ) is also a mild solution, i.e. it
satisfies the integral equation
u(t) = exp(tA)u 0 +
Z t
0
exp ((t \Gamma s)A)f(s; u(s))ds; t 2 IR+ :
(6.5)
Then the shift operator K(u 0 ) = u(T ; u 0 ) can be defined and it maps E to E: If u(\Delta; x \Lambda ) is a periodic
solution of (6.1), then x \Lambda is a zero of the compact vector field defined by I \Gamma K; i.e. K(x \Lambda ) = x \Lambda :
Remark 6.3 We assume here that operator (I \Gamma exp(TA)) \Gamma1 exists and is bounded. In the mean time
it is just enough to assume that (I \Gamma exp(tA)) \Gamma1 2 B(E) holds for t – t 0 with some t 0 ? 0: This
assumption is not restrictive as without loss of generality we are able to change A to A \Gamma !I and get
k exp (t(A \Gamma !))k Ÿ Me \Gammaffit for ffi ? 0; t – 0: It follows then [26] that (I \Gamma exp(tA)) \Gamma1 2 B(E) for any
t ? 0:
Remark 6.4 We said that function f is smooth enough in the sense that it has to be at least continuous
in both arguments, sup t2[0;T ];kxkŸc 1
kf(t; x)k Ÿ C 2 and such that there exists the global mild solution of
the problem u 0 (t) = Au(t) + f(t; u(t)); u(0) = u 0 ; t 2 IR+ :
Definition 6.1 The solution u(\Delta) of Cauchy problem (6.1) is said to be stable in the Lyapunov sense if
for any ffl ? 0 there is ffi ? 0 such that inequality ku(0) \Gamma ~ u(0)k Ÿ ffi implies max 0Ÿt!1 ku(t) \Gamma ~
u(t)k Ÿ ffl;
where ~
u(\Delta) is a mild solution of (6.1) with the initial value ~ u(0):
Definition 6.2 The solution u(\Delta) of Cauchy problem (6.1) is said to be uniformly asymptotically stable
at the point u(0) if it is stable in the Lyapunov sense and for any mild solution ~
u(\Delta) of (6.1) with
ku(0) \Gamma ~
u(0)k Ÿ ffi it follows that lim t!1 ku(t) \Gamma ~ u(t)k = 0 uniformly in ~
u(\Delta) 2 B(u(0); ffi); i.e. there
is a function OE u(0);ffi (\Delta) such that ku(t; u(0)) \Gamma u(t; ~
u(0))k Ÿ OE u(0);ffi (t) with OE u(0);ffi (t) ! 0 as t ! 1 and
ku(0) \Gamma ~ u(0)k Ÿ ffi:
Constructive conditions on operator A and f to ensure that equation u 0 (t) = Au(t) + f(u(t)); u(u) = u 0
is asymptotically k \Gammadimensional are given in [170], [169]. They concern the location of eigenvalues of A,
i.e. – k+1 \Gamma – k ? 2L; – k+1 ? L:
Theorem 6.3 [35] Assume that conditions (A) and (B'') hold and compact resolvents (–I \Gamma A) \Gamma1 ; (–I \Gamma
A n ) \Gamma1 converge (–I \Gamma A n ) \Gamma1 ! (–I \Gamma A) \Gamma1 compactly for some – 2 ae(A): Assume that
(i) the functions f; f n are sufficiently smooth, so that there exists an isolated mild solution v \Lambda (\Delta)
of periodic problem (6.4) with v \Lambda (0) = x \Lambda such that the Cauchy problem (6.1) with u(0) = x \Lambda has an
uniformly asymptotically stable isolated solution at the point x \Lambda (in this case ind v \Lambda (\Delta) = 1);
(ii) f n (t; x n ) ! f(t; x) for any t 2 [0; T ] as x n ! x;
(iii) the space E is separable .
Then for almost all n the problems
v 0
n (t) = A n v n (t) + f n (t; v n (t)); v n (t) = v n (t + T ); t 2 IR+ ;
(6.6)
have periodic mild solutions v \Lambda
n (t); t 2 [0; T ]; in the neighbourhood of p n v \Lambda (\Delta); where v \Lambda (\Delta) is a mild periodic
solution of (6.4) with v \Lambda (0) = x \Lambda : Each sequence fv \Lambda
n (\Delta)g is P­compact and v \Lambda
n (t) ! v \Lambda (t) uniformly with
respect to t 2 [0; T ]:
We say that a fixed point x \Lambda to operator K in Banach lattice E is stable from above [95] if given ffl ? 0;
there is a ffi ? 0 such that kK k x \Gamma x \Lambda k Ÿ ffl for all k 2 IN if x \Lambda ¯ x and kx \Gamma x \Lambda k Ÿ ffi: Using this notion
Theorem 6.3 can be reformulated for positive semigroups due to result from [59].
33

Theorem 6.4 Let the operators A n and A from the problems (6.4) and (6.6) be consistent and let
E; E n be order unit spaces and e n 2 D(A n ) `` intE +
n : Assume that the operators A n have the POD
property and A n e n ¯ 0 for sufficiently large n and compact resolvents (–I \Gamma A) \Gamma1 ; (–I \Gamma A n ) \Gamma1 converge
(–I \Gamma A n ) \Gamma1 ! (–I \Gamma A) \Gamma1 compactly for some – 2 ae(A): Assume that
(i) the functions f; f n are sufficiently smooth, bounded and positive, so that there exists a mild solution
u \Lambda (\Delta) of the Cauchy problem (6.3) such that elements u \Lambda (0) = x \Lambda is a stable from above and fixed points
of operator K; with x \Lambda OE y; Ky ¯ y ( in this situation ind x \Lambda = 1);
(ii) f n (t; x n ) ! f(t; x) uniformly with respect to t 2 [0; T ] for x n ! x;
(iii) the space E is separable .
Then for almost all n the problems (6.6) have periodic mild solutions v \Lambda
n (t); t 2 [0; T ] in the neigh­
bourhood of p n v \Lambda (\Delta); where v \Lambda (\Delta) is any stable from above mild periodic solution of (6.4). Each sequence
fv \Lambda
n (\Delta)g is P­compact and v \Lambda
n (t) ! v \Lambda (t) uniformly with respect to t 2 [0; T ]:
Remark 6.5 The technique which is used here could be applied to the case of condensing operators [3].
For example, the resolvent of \Delta in L 2 (IR d ) is condensing, but it is not compact.
In [117] they study the qualitative behavior of spatially semidiscrete finite element solutions of a semi­
linear parabolic problem near an unstable hyperbolic equilibrium.
The shadowing approach to study the long­time behavior of numerical approximations of semilinear
parabolic equations was studied in [116].
ACKNOWLEDGEMENTS
The authors acknowledge the support of NATO­CP Advanced Fellowship Programme of T ¨
UBITAK
(Turkish Scientific Research Council), University of Antwerpen, Russian Foundation for Basic Research
(01­01­00398) and University of Bologna. We would also like to thank Professor E.H. Twizell for his
valuable comments.
References
[1] N. H. Abdelaziz. On approximation by discrete semigroups. J. Approx. Theory, 73(3):253--269,
1993.
[2] N. H. Abdelaziz and Paul R. Chernoff. Continuous and discrete semigroup approximations with
applications to the Cauchy problem. J. Operator Theory, 32(2):331--352, 1994.
[3] S. B. Agase and V. Raghavendra. Existence of mild solutions of semilinear differential equations
in Banach spaces. Indian J. Pure Appl. Math., 21(9):813--821, 1990.
[4] M. Ahu'es. A class of strongly stable operator approximations. J. Austral. Math. Soc. Ser. B,
28(4):435--442, 1987.
[5] M. Ahu'es and F. Hocine. A note on spectral approximation of linear operations. Appl. Math. Lett.,
7(2):63--66, 1994.
[6] M. Ahu'es and A. Largillier. Two numerical approximations for a class of weakly singular integral
operators. Appl. Numer. Math., 17(4):347--362, 1995.
34

[7] M. Ahues and S. Piskarev. Spectral approximation of weakly singular integral operators. i: Conver­
gence theory. In et. al. Constanda. C., editor, Integral methods in science and engineering. Vol. II:
Approximation methods. Proceedings of the 4th international conference, IMSE '96, Oulu, Finland,
June 17--20, Harlow, 1996. Longman, Pitman Res.
[8] H. A. Alibekov and P. E. Sobolevskii. Stability and convergence of difference schemes of a high
order of approximation for parabolic partial differential equations. Ukrain. Mat. Zh., 32(3):291--300,
1980.
[9] H. Amann. Linear and quasilinear parabolic problems. Vol. I. Birkh¨auser Verlag, Basel, 1995.
Abstract linear theory.
[10] P. M. Anselone. Collectively compact operator approximation theory and applications to integral
equations. Prentice­Hall Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis,
Prentice­Hall Series in Automatic Computation.
[11] P. M. Anselone. Nonlinear operator approximation. In Moderne Methoden der numerischen Math­
ematik (Tagung 200­Jahrfeier, Tech. Univ. Clausthal, Clausthal, 1975), volume 32 of Internat. Ser.
Numer. Math., pages 17--24. Birkh¨auser, Basel, 1976.
[12] P. M. Anselone. Recent advances in operator approximation theory. In Methods of functional
analysis in approximation theory (Bombay, 1985), volume 76 of Internat. Schriftenreihe Numer.
Math., pages 309--324. Birkh¨auser, Basel, 1986.
[13] P. M. Anselone and T. W. Palmer. Spectral properties of collectively compact sets of linear
operators. J. Math. Mech., 17:853--860, 1967/1968.
[14] P. M. Anselone and T. W. Palmer. Collectively compact sets of linear operators. Pacific J. Math.,
25:417--422, 1968.
[15] P. M. Anselone and M. L. Treuden. Regular operator approximation theory. Pacific J. Math.,
120(2):257--268, 1985.
[16] P. M. Anselone and M. L. Treuden. Spectral analysis of asymptotically compact operator sequences.
Numer. Funct. Anal. Optim., 17(7­8):679--690, 1996.
[17] F. Ar`andiga and V. Caselles. Approximations of positive operators and continuity of the spectral
radius. III. J. Austral. Math. Soc. Ser. A, 57(3):330--340, 1994.
[18] F. Ar`andiga and V. Caselles. On strongly stable approximations. Rev. Mat. Univ. Complut. Madrid,
7(2):207--217, 1994.
[19] W. Arendt. Vector­valued Laplace transforms and Cauchy problems. Israel J. Math., 59(3):327--
352, 1987.
[20] A. Ashyralyev. Difference schemes of higher order accuracy for elliptic equations. In Application
of the methods of functional analysis to nonclassical equations in mathematical physics (Russian),
pages 3--14. Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1989.
[21] A. Ashyralyev, S. Piskarev, and L. Weis. On well­posedness of difference schemes for abstract
parabolic equations in L p ([0; T ]; E) spaces. Submitted.
35

[22] A. Ashyralyev and P. E. Sobolevski–i. Well­posedness of parabolic difference equations, volume 69
of Operator Theory: Advances and Applications. Birkh¨auser Verlag, Basel, 1994. Translated from
the Russian by A. Iacob.
[23] A. O. Ashyralyev. Some difference schemes for parabolic equations with nonsmooth initial data.
Izv. Akad. Nauk Turkmen. SSR Ser. Fiz.­Tekhn. Khim. Geol. Nauk, 6:70--72, 1983.
[24] N. Yu. Bakaev, S. Larsson, and V. Thom'ee. Backward Euler type methods for parabolic integro­
differential equations in Banach space. RAIRO Mod'el. Math. Anal. Num'er., 32(1):85--99, 1998.
[25] M. E. Ballotti, J. A. Goldstein, and M. E. Parrott. Almost periodic solutions of evolution equations.
J. Math. Anal. Appl., 138(2):522--536, 1989.
[26] V. Barbu and N. H. Pavel. On the invertibility of I \Sigma exp(\GammatA), t ? 0, with A maximal monotone.
In World Congress of Nonlinear Analysts '92, Vol. I--IV (Tampa, FL, 1992), pages 2231--2237. de
Gruyter, Berlin, 1996.
[27] B. Baumer and F. Neubrander. Existence and uniqueness of solutions of ordinary linear differential
equations in Banach spaces. Preprint, Departmeent of Mathematics, 1996. Louisiana state Univ.,
Baton Rouge.
[28] Ph. Benilan, N. Crandall, and A. Pazy. Nonlinear evolution equations in Banach spaces. book in
print, 1998.
[29] J. Bergh and J. L¨ofstr¨om. Interpolation spaces. An introduction. Springer­Verlag, Berlin, 1976.
Grundlehren der Mathematischen Wissenschaften, No. 223.
[30] A. Bobrowski. Degenerate convergence of semigroups. Semigroup Forum, 49(3):303--327, 1994.
[31] A. Bobrowski. Integrated semigroups and the trotter­kato theorem. Bull. Polish Acad. Sci.,
41(4):297--303, 1994.
[32] A. Bobrowski. Examples of a pointwise convergence of semigroups. Ann. Univ. Mariae Curie­
Sk/lodowska Sect. A, 49:15--33, 1995.
[33] A. Bobrowski. Generalized telegraph equation and the Sova­Kurtz version of the Trotter­Kato
theorem. Ann. Polon. Math., 64(1):37--45, 1996.
[34] A. Bobrowski. On the generation of non­continuous semigroups. Semigroup Forum, 54(2):237--252,
1997.
[35] N. A. Bobylev, J. K. Kim, S. K. Korovin, and S. Piskarev. Semidiscrete approximations of semi­
linear periodic problems in Banach spaces. Nonlinear Anal., 33(5):473--482, 1998.
[36] P. Bolley, J. Camus, and Pha . m T. L. Estimation de la r'esolvante du probl`eme de Dirichlet dans
les espaces de H¨older. C. R. Acad. Sci. Paris S'er. I Math., 305(6):253--256, 1987.
[37] R. Bouldin. Operator approximations with stable eigenvalues. J. Austral. Math. Soc. Ser. A,
49(2):250--257, 1990.
[38] J. H. Bramble, J. E. Pasciak, P. H. Sammon, and V. Thom'ee. Incomplete iterations in multistep
backward difference methods for parabolic problems with smooth and nonsmooth data. Math.
Comp., 52(186):339--367, 1989.
36

[39] P. Brenner, M. Crouzeix, and V. Thom'ee. Single­step methods for inhomogeneous linear differential
equations in Banach space. RAIRO Anal. Numer., 16:5--26, 1982.
[40] P. Brenner, V. Thom'ee, and L. B. Wahlbin. Besov spaces and applications to difference methods
for initial value problems. Springer­Verlag, Berlin, 1975. Lecture Notes in Mathematics, Vol. 434.
[41] P. Brenner and Thom'ee V. On rational approximations of semigroups. SIAM J. Numer. Anal.,
16:683--694, 1979.
[42] K. Burrage and S. Piskarev. Stochastic methods for ill­posed problems. BIT, 40(2):226--240, 2000.
[43] S. Busenberg and B. Wu. Convergence theorems for integrated semigroups. Differential Integral
Equations, 5(3):509--520, 1992.
[44] P. L. Butzer, W. Dickmeis, L. Hahn, and R. J. Nessel. Lax­type theorems and a unified approach
to some limit theorems in probability theory with rates. Resultate Math., 2:30--53, 1979.
[45] P. L. Butzer, W. Dickmeis, Hu. Jansen, and R. J. Nessel. Alternative forms with orders of the Lax
equivalence theorem in Banach spaces. Computing, 17(4):335--342, 1976/77.
[46] P. L. Butzer, W. Dickmeis, and R. J. Nessel. Lax­type theorems with orders in connection with
inhomogeneous evolution equations in Banach spaces. In Linear spaces and approximation (Proc.
Conf., Math. Res. Inst., Oberwolfach, 1977), volume 21 of Lecture Notes in Biomath., pages 531--
546. Springer, Berlin, 1978.
[47] P. L. Butzer, W. Dickmeis, and R. J. Nessel. Numerical analysis of evolution equations. In Sym­
posium on Numerical Analysis, Kyoto, July 11--12, 1978, volume 1 of Lecture Notes in Numerical
and Applied Analysis, pages 1--163. Kinokuniya Book Store Co., Ltd., Tokyo, 1979.
[48] A. S. Carasso, J. G. Sanderson, and J. M. Hyman. Digital removal of random media image degrada­
tions by solving the diffusion equation backwards in time. SIAM J. Numer. Anal., 15(2):344--367,
1978.
[49] F. Chatelin. Spectral approximation of linear operators. Computer Science and Applied Math­
ematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1983. With a
foreword by P. Henrici, With solutions to exercises by Mario Ahu'es.
[50] A.J. Chorin, M.F. McCracken, T. J. R. Hughes, and J. E. Marsden. Product formulas and numerical
algorithms. Comm. Pure Appl. Math., 31(2):205--256, 1978.
[51] M. Crouzeix. On multistep approximation of semigroups in Banach spaces. In Proceedings of the
2nd international conference on computational and applied mathematics (Leuven, 1986), volume 20,
pages 25--35, 1987.
[52] M. Crouzeix, S. Larsson, S. Piskarev, and V. Thom'ee. The stability of rational approximations of
analytic semigroups. BIT, 33(1):74--84, 1993.
[53] M. Crouzeix, S. Larsson, and V. Thom'ee. Resolvent estimates for elliptic finite element operators
in one dimension. Math. Comp., 63(207):121--140, 1994.
[54] M. Crouzeix and V. Thom'ee. On the discretization in time of semilinear parabolic equations with
nonsmooth initial data. Math. Comp., 49(180):359--377, 1987.
37

[55] G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions, volume 44 of Encyclopedia
of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.
[56] Yu. L. Dalecky and S. V. Fomin. Measures and differential equations in infinite­dimensional space,
volume 76 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group,
Dordrecht, 1991. Translated from the Russian, With additional material by V. R. Steblovskaya,
Yu. V. Bogdansky [Yu. V. Bogdanski–i] and N. Yu. Goncharuk.
[57] Yu. L. Dalecky and N. Yu. Goncharuk. Random operator functions defined by stochstic differential
equations. In Measures and differential equations in infinite­dimensional space. 1990. Translated
from the Russian. With additional material by V. R. Steblovskaya, Yu. V. Bogdansky.
[58] Yu­L. Dalecky and N­Yu. Goncharuk. On a quasilinear stochastic differential equation of parabolic
type. Stochastic Anal. Appl., 12(1):103--129, 1994.
[59] E. N. Dancer. Upper and lower stability and index theory for positive mappings and applications.
Nonlinear Anal., 17:205--217, 1991.
[60] E. B. Davies and M. M. H. Pang. The Cauchy problem and a generalization of the Hille­ Yosida
theorem. Proc. London Math. Soc. (3), 55(1):181--208, 1987.
[61] R. deLaubenfels. Integrated semigroups, C­semigroups and the abstract Cauchy problem. Semi­
group Forum, 41(1):83--95, 1990.
[62] R. deLaubenfels. C­semigroups and the Cauchy problem. J. Funct. Anal., 111(1):44--61, 1993.
[63] R. J. Dickmeis, W.and Nessel. On uniform boundedness principles and Banach­Steinhaus theorems
with rates. Numer. Funct. Anal. Optim., 3:19--52, 1981.
[64] W. Dickmeis and R. J. Nessel. Classical approximation processes in connection with Lax equivalence
theorems with orders. Acta Sci. Math. (Szeged), 40(1­2):33--48, 1978.
[65] W. Dickmeis and R. J. Nessel. A unified approach to certain counterexamples in approximation
theory in connection with a uniform boundedness principle with rates. J. Approx. Theory, 31:161--
174, 1981.
[66] B. Eberhardt and G. Greiner. Baillon's theorem on maximal regularity. Acta Appl. Math., 27:47--54,
1992. Positive operators and semigroups on Banach lattices (Cura¸cao, 1990).
[67] S. N. Ethier and T. G. Kurtz. Markov processes. Wiley Series in Probability and Mathematical
Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986.
Characterization and convergence.
[68] H. Fujita. On the finite element approximation for parabolic equations: an operator theoretical
approach. In Computing methods in applied sciences and engineering (Second Internat. Sympos.,
Versailles, 1975), Part 1, volume 134 of Lecture Notes in Econom. and Math. Systems, pages
171--192. Springer, Berlin, 1976.
[69] H. Fujita and A. Mizutani. On the finite element method for parabolic equations. I. Approximation
of holomorphic semi­groups. J. Math. Soc. Japan, 28(4):749--771, 1976.
38

[70] H. Fujita and T. Suzuki. On the finite element approximation for evolution equations of parabolic
type. In Computing methods in applied sciences and engineering (Proc. Third Internat. Sympos.,
Versailles, 1977), I, volume 704 of Lecture Notes in Math., pages 207--221. Springer, Berlin, 1979.
[71] H. Fujita and T. Suzuki. Evolution problems. In Handbook of numerical analysis, Vol. II, pages
789--928. North­Holland, Amsterdam, 1991.
[72] J. A. Goldstein. Semigroups of linear operators and applications. Oxford Mathematical Mono­
graphs. The Clarendon Press Oxford University Press, New York, 1985.
[73] J. A. Goldstein. A survey of semigroups of linear operators and applications. In Semigroups of
linear and nonlinear operations and applications (Cura¸cao, 1992), pages 9--57. Kluwer Acad. Publ.,
Dordrecht, 1993.
[74] J. A. Goldstein and Jr. Sandefur, J. T. An abstract d'Alembert formula. SIAM J. Math. Anal.,
18(3):842--856, 1987.
[75] J.A. Goldstein, R. deLaubenfels, and Jr. Sandefur, J. T. Regularized semigroups, iterated Cauchy
problems and equipartition of energy. Monatsh. Math., 115(1­2):47--66, 1993.
[76] C. Gonz'alez and C. Palencia. Stability of Runge­Kutta methods for abstract time­dependent
parabolic problems: the H¨older case. Math. Comp., 68:73--89, 1999.
[77] E. G¨orlich and D. Pontzen. An approximation theorem for semigroups of growth order ff and an
extension of the trotter­lie formula. J. Funct. Anal., 50(3):414--425, 1983.
[78] R. D. Grigorieff. Zur Theorie linearer approximationsregul¨arer Operatoren. I, II. Math. Nachr.,
55:233--249; ibid. 55 (1973), 251--263, 1973.
[79] R. D. Grigorieff. Diskrete Approximation von Eigenwertproblemen. I. Qualitative Konvergenz.
Numer. Math., 24(4):355--374, 1975.
[80] R. D. Grigorieff. Diskrete Approximation von Eigenwertproblemen. II. Konvergenzordnung. Numer.
Math., 24(5):415--433, 1975.
[81] R. D. Grigorieff. ¨
Uber diskrete Approximationen nichtlinearer Gleichungen 1. Art. Math. Nachr.,
69:253--272, 1975.
[82] R. D. Grigorieff. Diskrete Approximation von Eigenwertproblemen. III. Asymptotische Entwick­
lungen. Numer. Math., 25(1):79--97, 1975/76.
[83] R. D. Grigorieff. Zur Charakterisierung linearer approximationsregul¨arer Operatoren. In Math­
ematical papers given on the occasion of Ernst Mohr's 75th birthday, pages 63--77. Tech. Univ.
Berlin, Berlin, 1985.
[84] R. D. Grigorieff. Some stability inequalities for compact finite difference schemes. Math. Nachr.,
135:93--101, 1988.
[85] R. D. Grigorieff. Time discretization of semigroups by the variable two­step BDF method. In
Numerical treatment of differential equations (Halle, 1989), volume 121 of Teubner­Texte Math.,
pages 204--216. Teubner, Stuttgart, 1991.
39

[86] R. D. Grigorieff and H. Jeggle. Approximation von Eigenwertproblemen bei nichlinearer Parame­
terabh¨angigkeit. Manuscripta Math., 10:245--271, 1973.
[87] D. Guidetti. Optimal regularity for mixed parabolic problems in spaces of functions which are
H¨older continuous with respect to space variables. Technical report, Department of Mathematics,
University of Bologna, 1995.
[88] D. Guidetti. The parabolic mixed Cauchy­Dirichlet problem in spaces of functions which are H¨older
continuous with respect to space variables. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend.
Lincei (9) Mat. Appl., 7(3):161--168, 1996.
[89] D. Guidetti. The parabolic mixed Cauchy­Dirichlet problem in spaces of functions which are H¨older
continuous with respect to space variables. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend.
Lincei (9) Mat. Appl., 7(3):161--168, 1996.
[90] D. Guidetti and S. Piskarev. On maximal regularity of difference schemes for parabolic equations in
c ` (!) spaces. N 8, Dipartimento di Matematica, Universita'a degli Studi di Bologna, 1997. Preprint.
[91] D. Guidetti and S. Piskarev. On maximal regularity of Crank­Nicholson schemes for parabolic
equations in C ` (\Omega\Gamma space. In Proccedings of International Conference ''Function spaces. Differential
operators. Problems in mathematical education, volume 2, pages 2231--2237. RUDN, Moscow, 1998.
[92] D. Guidetti and S. Piskarev. Stability of Rothe's scheme and maximal regularity for parabolic
equations in C `
(\Omega ). In Progress in partial differential equations, Vol. 1 (Pont­`a­Mousson, 1997),
pages 167--180. Longman, Harlow, 1998.
[93] D. Henry. Geometric theory of semilinear parabolic equations. Springer­Verlag, Berlin, 1981.
[94] R. Hersh and T. Kato. High­accuracy stable difference schemes for well­posed initial value problems.
SIAM J. Numer. Anal., 16(4):670--682, 1979.
[95] P. Hess. Periodic­parabolic boundary value problems and positivity. Longman Scientific & Technical,
Harlow, 1991.
[96] R. H. W. Hoppe. A constructive approach to the Bellman semigroup. Nonlinear Anal., 9(11):1165--
1181, 1985.
[97] V. K. Ivanov, V. V. Vasin, and V. P. Tanana. Theory of linear ill­posed problems and its applica­
tions. Nauka, Moscow, 1978.
[98] N. Kalton and L. Weis. The H 1 calculus and sums of closed operators. submitted.
[99] N. Kalton and L. Weis. Perturbation and interpolation theorems for the H 1 calculus. in prepara­
tion.
[100] N.J. Kalton and G. Lancien. A solution to the problem of L p ­maximal regularity. Math. Z.,
235(3):559--568, 2000.
[101] S. Kanda. Convergence of difference approximations and nonlinear semigroups. Proc. Amer. Math.
Soc., 108(3):741--748, 1990.
[102] T. Kato. Perturbation theory for linear operators. Classics in Mathematics. Springer­Verlag, Berlin,
1995. Reprint of the 1980 edition.
40

[103] N. Kenmochi and S. Oharu. Difference approximation of nonlinear evolution equations and semi­
groups of nonlinear operators. Publ. Res. Inst. Math. Sci., 10(1):147--207, 1974/75.
[104] J. Kisy'nski. A proof of the Trotter­Kato theorem on approximation of semi­groups. Colloq. Math.,
18:181--184, 1967.
[105] Y. Kobayashi. Difference approximation of Cauchy problems for quasi­dissipative operators and
generation of nonlinear semigroups. J. Math. Soc. Japan, 27(4):640--665, 1975.
[106] Y. Kobayashi. Difference approximation of evolution equations and generation of nonlinear semi­
groups. Proc. Japan Acad., 51(6):406--410, 1975.
[107] S. G. Krein. Linear differential equations in Banach space. American Mathematical Society, Prov­
idence, R.I., 1971. Translated from the Russian by J. M. Danskin, Translations of Mathematical
Monographs, Vol. 29.
[108] P. Kunstmann and L. Weis. Perturbation theorems for maximal regularity. submitted.
[109] P.Ch. Kunstmann Abstrakte Cauchy probleme und Distributionshalbgruppen. Dissertation, Univ.
Kiel, 1995.
[110] P.Ch. Kunstmann. Regularization of semigroups that are strongly continuous for t ? 0: Proc.
Amer. Math. Soc., 1998, 126, N9, p. 2721­2724.
[111] T. G. Kurtz. Extensions of Trotter's operator semigroup approximation theorems. J. Functional
Analysis, 3:354--375, 1969.
[112] T. G. Kurtz. A general theorem on the convergence of operator semigroups. Trans. Amer. Math.
Soc., 148:23--32, 1970.
[113] T. G. Kurtz and P. E. Protter. Weak convergence of stochastic integrals and differential equations.
II. Infinite­dimensional case. In Probabilistic models for nonlinear partial differential equations
(Montecatini Terme, 1995), volume 1627 of Lecture Notes in Math., pages 197--285. Springer,
Berlin, 1996.
[114] R. Labbas. Some results on the sum of linear operators with nondense domains. Ann. Mat. Pura
Appl. (4), 154:91--97, 1989.
[115] A. Largillier. A numerical quadrature for some weakly singular integral operators. Appl. Math.
Lett., 8(1):11--14, 1995.
[116] S. Larsson and S. Yu. Pilyugin. Numerical shadowing near the global attractor for a semilinear
parabolic equation. Preprint 21, Department of Mathematics, Chalmers University of Technology,
1998.
[117] S. Larsson and J. M. Sanz­Serna. The behavior of finite element solutions of semilinear parabolic
problems near stationary points. SIAM J. Numer. Anal., 31(4):1000--1018, 1994.
[118] S. Larsson and J.­M. Sanz­Serna. A shadowing result with applications to finite element approxi­
mation of reaction­diffusion equations. Math. Comp., 68(225):55--72, 1999.
[119] S. Larsson, V. Thom'ee, and L. B. Wahlbin. Numerical solution of parabolic integro­differential
equations by the discontinuous Galerkin method. Math. Comp., 67(221):45--71, 1998.
41

[120] S. Larsson, V. Thom'ee, and S. Z. Zhou. On multigrid methods for parabolic problems. J. Comput.
Math., 13(3):193--205, 1995.
[121] I. Lasiecka and A. Manitius. Differentiability and convergence rates of approximating semigroups
for retarded functional­differential equations. SIAM J. Numer. Anal., 25(4):883--907, 1988.
[122] P. D. Lax and R. D. Richtmyer. Survey of the stability of linear finite difference equations. Comm.
Pure Appl. Math., 9:267--293, 1956.
[123] R. D. Lazarov, V. L. Makarov, and A. A. Samarskii. Application of exact difference schemes
for constructing and investigating difference schemes on generalized solutions. Mat. Sb. (N.S.),
117(159)(4):469--480, 559, 1982.
[124] C. Le Merdy. Counterexamples on L p ­maximal regularity. Math. Z., 230:47--62, 1999.
[125] M. Le Roux and V. Thom'ee. Numerical solution of semilinear integrodifferential equations of
parabolic type with nonsmooth data. SIAM J. Numer. Anal., 26(6):1291--1309, 1989.
[126] T. D. Lee. Difference equations and conservation laws. J. Statist. Phys., 46(5­6):843--860, 1987.
[127] H. Linden. Starke Konvergenz im verallgemeinerten Sinne und Spektra. Math. Z., 134:205--213,
1973.
[128] H. Linden. ¨
Uber die Stabilit¨at von Eigenwerten. Math. Ann., 203:215--220, 1973.
[129] C. Lizama. On an extension of the Trotter­Kato theorem for resolvent families of operators. J.
Integral Equations Appl., 2(2):269--280, 1990.
[130] C. Lizama. On the convergence and approximation of integrated semigroups. J. Math. Anal. Appl.,
181(1):89--103, 1994.
[131] G. J. Lord and A. M. Stuart. Discrete Gevrey regularity, attractors and upper­semicontinuity for
a finite difference approximation to the Ginzburg­Landau equation. Numer. Funct. Anal. Optim.,
16(7­8):1003--1047, 1995.
[132] C. Lubich and A. Ostermann. Runge­Kutta methods for parabolic equations and convolution
quadrature. Math. Comp., 60(201):105--131, 1993.
[133] C. Lubich and A. Ostermann. Runge­Kutta approximation of quasi­linear parabolic equations.
Math. Comp., 64(210):601--627, 1995.
[134] C. Lubich and A. Ostermann. Runge­Kutta time discretization of reaction­diffusion and Navier­
Stokes equations: nonsmooth­data error estimates and applications to long­time behaviour. Appl.
Numer. Math., 22(1­3):279--292, 1996. Special issue celebrating the centenary of Runge­Kutta
methods.
[135] Ch. Lubich, I. H. Sloan, and V. Thom'ee. Nonsmooth data error estimates for approximations of
an evolution equation with a positive­type memory term. Math. Comp., 65(213):1--17, 1996.
[136] A. Lunardi. Analytic semigroups and optimal regularity in parabolic problems, volume 16 of Progress
in Nonlinear Differential Equations and their Applications. Birkh¨auser Verlag, Basel, 1995.
42

[137] I. V. Mel'nikova and S. V. Bochkareva. C­semigroups and the regularization of an ill­posed Cauchy
problem. Dokl. Akad. Nauk, 329(3):270--273, 1993.
[138] M. MiklavŸciŸc. Approximations for weakly nonlinear evolution equations. Math. Comp.,
53(188):471--484, 1989.
[139] Jr. Mills, W. H. The resolvent stability condition for spectra convergence with application to the
finite element approximation of noncompact operators. SIAM J. Numer. Anal., 16:695--703, 1979.
[140] I. Miyadera and N. Tanaka. Exponentially bounded C­semigroups and generation of semigroups.
J. Math. Anal. Appl., 143(2):358--378, 1989.
[141] J. K. Mountain. The lax equivalence theorem for linear, inhomogeneous equations in l 2 spaces. J.
Approx. Theory, 33:126--130, 1981.
[142] M. T. Nair. On strongly stable approximations. J. Austral. Math. Soc. Ser. A, 52(2):251--260,
1992.
[143] E. Nakaguchi and A. Yagi. Error estimate of implicit runge­kutta methods for quasilinear abstract
equations of parabolic type in banach space. Preprint, Department of Applied Physics, Graduate
School of Engineering, Osaka University, 1996. apanese Journal of Mathematics.
[144] O. Nevanlinna. On the convergence of difference approximations to nonlinear contraction semi­
groups in Hilbert spaces. Math. Comp., 32(142):321--334, 1978.
[145] O. Nevanlinna and G. Vainikko. Limit spectrum of discretely converging operators. Numer. Funct.
Anal. Optim., 17(7­8):797--808, 1996.
[146] A. Ostermann and M. Roche. Runge­Kutta methods for partial differential equations and fractional
orders of convergence. Math. Comp., 59(200):403--420, 1992.
[147] C. Palencia. A stability result for sectorial operators in Banach spaces. SIAM J. Numer. Anal.,
30(5):1373--1384, 1993.
[148] C. Palencia. On the stability of variable stepsize rational approximations of holomorphic semi­
groups. Math. Comp., 62(205):93--103, 1994.
[149] C. Palencia. Stability of rational multistep approximations of holomorphic semigroups. Math.
Comp., 64(210):591--599, 1995.
[150] C. Palencia. Maximum norm analysis of completely discrete finite element methods for parabolic
problems. SIAM J. Numer. Anal., 33(4):1654--1668, 1996.
[151] C. Palencia and B. Garc'ia­Archilla. Stability of linear multistep methods for sectorial operators in
Banach spaces. Appl. Numer. Math., 12(6):503--520, 1993.
[152] C. Palencia and S. Piskarev. On multiplicative perturbations of c 0 ­groups and c 0 ­cosine operator
functions. Technical report, Department of Mathematics, Universidad de Valladolid, Valladolid,
Spain, 1998.
[153] C. Palencia and J. M. Sanz­Serna. Equivalence theorems for incomplete spaces: an appraisal. IMA
J. Numer. Anal., 4(1):109--115, 1984.
43

[154] C. Palencia and J. M. Sanz­Serna. An extension of the Lax­Richtmyer theory. Numer. Math.,
44(2):279--283, 1984.
[155] A. K. Pani, V. Thom'ee, and L. B. Wahlbin. Numerical methods for hyperbolic and parabolic
integro­differential equations. J. Integral Equations Appl., 4(4):533--584, 1992.
[156] S. V. Parter. On the roles of ''stability'' and ''convergence'' in semidiscrete projection methods for
initial­value problems. Math. Comp., 34(149):127--154, 1980.
[157] D. Pfeifer. Some general probabilistic estimations for the rate of convergence in operator semigroup
representations. Appl. Anal., 23(1­2):111--118, 1986.
[158] S. Piskarev. Approximation of holomorphic semigroups. Tartu Riikl. ¨
Ul. Toimetised, (492):3--14,
1979.
[159] S. Piskarev. Error estimates in the approximation of semigroups of operators by Pad'e fractions.
Izv. Vyssh. Uchebn. Zaved. Mat., (4):33--38, 1979.
[160] S. Piskarev. Stability of difference schemes in Cauchy problems with almost periodic solutions.
Differentsial' nye Uravneniya, 20(4):689--695, 1984.
[161] S. Piskarev. Principles of discretization methods III. Report ak ­3410, Acoustic Institute, Academy
of Scince USSR, 1986. 87 pages.
[162] S. Piskarev. Estimates for the rate of convergence in the solution of ill­posed problems for evolution
equations. Izv. Akad. Nauk SSSR Ser. Mat., 51(3):676--687, 1987.
[163] S. Piskarev. Convergence of difference schemes in the solution of nonlinear parabolic equations.
Mat. Zametki, 44(1):112--123, 156, 1988.
[164] S. Piskarev. A method of increasing the acuuracy of the solution of initial value problems in Banach
space. USSR Comput. Math. Phys., 28(6):211--212, 1988.
[165] S. Piskarev. Approximation of positive C 0 ­semigroups of operators. Differentsial' nye Uravneniya,
27(7):1245--1250, 1287, 1991.
[166] S. Piskarev. On the approximation of perturbed C 0 ­semigroups. Differentsial'nye Uravneniya,
30(2):339--341, 367, 1994.
[167] R. D. Richtmyer and K. W. Morton. Difference methods for initial­value problems. Interscience
Publishers John Wiley & Sons, Inc., New York­London­Sydney, 1967. Second edition. Interscience
Tracts in Pure and Applied Mathematics, No. 4.
[168] A. V. Romanov. Conditions for asymptotic k­dimensionality of semilinear parabolic equations.
Uspekhi Mat. Nauk, 46(1(277)):213--214, 1991.
[169] A. V. Romanov. Sharp estimates for the dimension of inertial manifolds for nonlinear parabolic
equations. Izv. Ross. Akad. Nauk Ser. Mat., 57(4):36--54, 1993.
[170] A. V. Romanov. On the limit dynamics of evolution equations. Uspekhi Mat. Nauk, 51(2(308)):173--
174, 1996.
44

[171] E. B. Saff, A. Sch¨onhage, and R. S. Varga. Geometric convergence to e \Gammaz by rational functions
with real poles. Numer. Math., 25(3):307--322, 1975/76.
[172] E. B. Saff and R. S. Varga. On the zeros and poles of Pad'e approximants to e z . Numer. Math.,
25(1):1--14, 1975/76.
[173] E. B. Saff and R. S. Varga. Geometric overconvergence of rational functions in unbounded domains.
Pacific J. Math., 62(2):523--549, 1976.
[174] E. B. Saff and R. S. Varga. On the zeros and poles of Pad'e approximants to e z . II. In Pad'e and
rational approximation (Proc. Internat. Sympos., Univ. South Florida, Tampa, Fla., 1976), pages
195--213. Academic Press, New York, 1977.
[175] E. B. Saff and R. S. Varga. On the zeros and poles of Pad'e approximants to e z . III. Numer. Math.,
30(3):241--266, 1978.
[176] A. Samarskii and V. Andr'eev. M'ethodes aux diff'erences pour 'equations elliptiques. '
Editions Mir,
Moscow, 1978. Traduit du russe par Djilali Embarek.
[177] A. Samarskii and A.V. Gulin. Stability of difference schemes. Nauka, Moscow, 1973. (in Russian).
[178] J. M. Sanz­Serna and C. Palencia. A general equivalence theorem in the theory of discretization
methods. Math. Comp., 45(171):143--152, 1985.
[179] S.­Y. Shaw and Ch.­Ch. Kuo. Local C­semigrous and the Cauchy problems. Preprint, Departmeent
of Mathematics, 1996. National Central University, Taiwan.
[180] P. E. Sobolevskii. The coercive solvability of difference equations. Dokl. Akad. Nauk SSSR,
201:1063--1066, 1971.
[181] P. E. Sobolevskii. The theory of semigroups and the stability of difference schemes. In Operator
theory in function spaces (Proc. School, Novosibirsk, 1975) (Russian), pages 304--337, 344. Izdat.
``Nauka'' Sibirsk. Otdel., Novosibirsk, 1977.
[182] P. E. Sobolevskii and Ho'ang V–an Lai. Difference schemes of optimal type for the approximate
solution of parabolic equations (the Banach case). Ukrain. Mat. Zh., 33:39--46, 1981.
[183] Z. Strkalj and L. Weis. On operator­valued fourier multiplier theorems. submitted.
[184] F. Stummel. Diskrete Konvergenz linearer Operatoren. III. In Linear operators and approximation
(Proc. Conf., Oberwolfach, 1971), pages 196--216. Internat. Ser. Numer. Math., Vol. 20. Birkh¨auser,
Basel, 1972.
[185] T. Takahashi. Convergence of difference approximation of nonlinear evolution equations and gen­
eration of semigroups. J. Math. Soc. Japan, 28(1):96--113, 1976.
[186] N. Tanaka. Approximation of integrated semigroups by ``integrated'' discrete parameter semi­
groups. Semigroup Forum, 55(1):57--67, 1997.
[187] N. Tanaka and I. Miyadera. Some remarks on C­semigroups and integrated semigroups. Proc.
Japan Acad. Ser. A Math. Sci., 63(5):139--142, 1987.
45

[188] N. Tanaka and N. Okazawa. Local C­semigroups and local integrated semigroups. Proc. London
Math. Soc. (3), 61(1):63--90, 1990.
[189] V. Thom'ee. Finite difference methods for linear parabolic equations. In Handbook of numerical
analysis, Vol. I, pages 5--196. North­Holland, Amsterdam, 1990.
[190] V. Thom'ee. Approximate solution of O.D.E.s in Banach space---rational approximation of semi­
groups. In Hellenic research in mathematics and informatics '92 (Athens, 1992), pages 409--420.
Hellenic Math. Soc., Athens, 1992.
[191] V. Thom'ee. Finite element methods for parabolic problems---some steps in the evolution. In Finite
element methods (Jyv¨askyl¨a, 1993), volume 164 of Lecture Notes in Pure and Appl. Math., pages
433--442. Dekker, New York, 1994.
[192] A. N. Tihonov and V. Ja. Arsenin. Methods for the solution of ill­posed problems. Nauka, Moscow,
1979.
[193] A. N. Tikhonov and V. Y. Arsenin. Solutions of ill­posed problems. V. H. Winston & Sons,
Washington, D.C.: John Wiley & Sons, New York, 1977. Translated from the Russian, Preface by
translation editor Fritz John, Scripta Series in Mathematics.
[194] H. F. Trotter. Approximation of semi­groups of operators. Pacific J. Math., 8:887--919, 1958.
[195] H. F. Trotter. On the product of semi­groups of operators. Proc. Amer. Math. Soc., 10:545--551,
1959.
[196] H. F. Trotter. Approximation and perturbation of semigroups. In Linear operators and approxi­
mation, II (Proc. Conf., Oberwolfach Math. Res. Inst., Oberwolfach, 1974), pages 3--21. Internat.
Ser. Numer. Math., Vol. 25. Birkh¨auser, Basel, 1974.
[197] E. H. Twizell. Computational methods for partial differential equations. Ellis Horwood Ltd.,
Chichester, 1984.
[198] E. H. Twizell and A. Q. M. Khaliq. Multiderivative methods for periodic initial value problems.
SIAM J. Numer. Anal., 21(1):111--122, 1984.
[199] T. Ushijima. On the abstract Cauchy problems and semi­groups of linear operators in locally
convex spaces. Sci. Papers College Gen. Ed. Univ. Tokyo, 21:93--122, 1971.
[200] T. Ushijima. Approximation theory for semi­groups of linear operators and its application to
approximation of wave equations. Japan. J. Math. (N.S.), 1(1):185--224, 1975/76.
[201] T. Ushijima. The approximation of semi­groups of linear operators and the finite element method.
In Boundary value problems for linear evolution: partial differential equations (Proc. NATO Ad­
vanced Study Inst., Li`ege, 1976), volume 29 of NATO Advanced Study Inst. Ser., Ser. C: Math.
and Phys. Sci., pages 351--362. Reidel, Dordrecht, 1977.
[202] T. Ushijima. Approximation of semigroups and the finite element method. S“ugaku, 32:133--148,
1980.
[203] T. Ushijima and M. Matsuki. Fully discrete approximation of a second order linear evolution
equation related to the water wave problem. In Functional analysis and related topics, 1991 (Kyoto),
volume 1540 of Lecture Notes in Math., pages 361--380. Springer, Berlin, 1993.
46

[204] G. Vainikko. Funktionalanalysis der Diskretisierungsmethoden. B. G. Teubner Verlag, Leipzig,
1976. Mit Englischen und Russischen Zusammenfassungen, Teubner­Texte zur Mathematik.
[205] G. Vainikko. ¨
Uber die Konvergenz und Divergenz von N¨aherungsmethoden bei Eigenwertproble­
men. Math. Nachr., 78:145--164, 1977.
[206] G. Vainikko. ¨
Uber Konvergenzbegriffe f¨ur lineare Operatoren in der numerischen Mathematik.
Math. Nachr., 78:165--183, 1977.
[207] G. Vainikko. Approximative methods for nonlinear equations (two approaches to the convergence
problem). Nonlinear Anal., 2:647--687, 1978.
[208] G. Vainikko. Foundations of finite difference method for eigenvalue problems. In The use of finite
element method and finite difference method in geophysics (Proc. Summer School, Liblice, 1977),
pages 173--192. Ÿ
Cesk. Akad. VŸed, Prague, 1978.
[209] G. Vainikko and O. Karma. An estimate for the rate of convergence of eigenvalues in approximation
methods. Tartu Riikl. ¨
Ul. Toimetised, (833):75--83, 1988.
[210] G. Vainikko and S. Piskarev. Regularly compatible operators. Izv. VysŸs. UŸcebn. Zaved. Matematika,
10(185):25--36, 1977.
[211] J. van Casteren, S. Piskarev, and S.­Y. Shaw. Discretisation of C­semigroups. submitted to J. of
Inverse and Ill­posed Problem.
[212] J. van Casteren, S. Piskarev, and S.­Y. Shaw. Discretization of C­semigroups. Technical report,
Proceedings of the Conference on inverse and ill­posed problems, Moscow State University, 1998.
18.
[213] V. V. Vasilev, S. G. Krein, and S. Piskarev. Operator semigroups, cosine operator functions, and
linear differential equations. In Mathematical analysis, Vol. 28 (Russian), Itogi Nauki i Tekhniki,
pages 87--202, 204. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990.
Translated in J. Soviet Math. 54 (1991), no. 4, 1042--1129.
[214] V.V. Vasiliev and S. Piskarev. Diffeential equations in Banach spaces II. Cosine­operator functions.
Translated in J. Soviet Math.
[215] V.V. Vasiliev and S. Piskarev. Diffeential equations in Banach spaces I. Semigroup theory. Moscow
State University Publishing House, 1996. in Russian, 164 pages.
[216] V. V. Vasilev, S.I. Piskarev. Bibliography on Differential Equations in Abstract Spaces. Moscow
State University, 2001, Electronic journal.
( http://www.srcc.msu.su=num anal=list wrk=page 5u.htm )
PostScript (1115,2 Kb) PostScript.zip (336,1 Kb)
[217] J. Voigt. On the convex compactness property for the strong operator topology. Note Mat.,
12:259--269, 1992. Dedicated to the memory of Professor Gottfried K¨othe.
[218] L. Weis. A new approach to maximal L p regularity. Proc of the 6th Internat. Conf on Evolution
Equations, Marcel Dekker 2000.
47

[219] L. Weis. Operator­valued multiplier theorems and maximal L p regularity. Mathematische Annalen,
to appear.
[220] R. Wolf. ¨
Uber lineare approximationsregul¨are Operatoren. Math. Nachr., 59:325--341, 1974.
[221] L. Wu. The semigroup stability of the difference approximations for initial­boundary value prob­
lems. Math. Comp., 64(209):71--88, 1995.
[222] Q. Zheng and Y. S. Lei. Exponentially bounded C­semigroup and integrated semigroup with
nondensely defined generators. I. Approximation. Acta Math. Sci. (English Ed.), 13(3):251--260,
1993.
Davide Guidetti
Department of Mathematics, University of Bologna
Piazza di Porta S. Donato, 5, 40127 Bologna, Italy
guidetti@dm.unibo.it
B¨ulent Karas¨ozen
Department of Mathematics, Middle East Technical University
06531 Ankara, Turkey
bulent@metu.edu.tr
Serguei Piskarev
Scientific Research Computer Center, Moscow State University
Vorobjevy Gory, Moscow 119899, Russia
serguei@piskarev.srcc.msu.su
48