Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc15774/doc.pdf Дата изменения: Mon Oct 29 13:01:48 2007 Дата индексирования: Mon Oct 1 20:36:45 2012 Кодировка:

, .. () . , .. ­ .. . , , . . AN AVERAGING METHOD FOR SINGULARLY PERTURBED SYSTEMS OF SEMILINEAR DIFFERENTIAL INCLUSIONS WITH HYSTERESIS NONLINEARITIES Gudovich A.N. (Voronezh) We consider a singularly perturbed system of semilinear parabolic inclusions containing high frequency terms and hysteresis nonlinearity. For such systems we justify the averaging principle, which is an analog of classical averaging theorem of N. N. Bogolyubov ­ N. M. Krilov. Our assumptions permit to define an upper semicontinuous, compact vector operator whose fixed points determine periodic solutions of original system. The existence of fixed point of
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2. (I)

constructed multivalued operator is shown by using topological degree theory. , , (., , [2], [5]). . [1] - [3] . (. [3]). :
x' (t ) A1 x(t ) + f1 (t , x(t ), y (t ), w(t )), y ' (t ) A2 y (t ) + f 2 (t , x(t ), y (t ), w(t )), w(t ) = ( w ) x(t ), t 0,

(1)

­ , A1 A2 ­ eA1t eA2t, E1 E2, f1 f2 ­ , - (.[1]), w* ­ , E3. A1-1 A2-1 , fi (i=1,2) T­ . (1) [0,T],
x(0) = x , y (0) = y , w(0) = w


(2)

(x, y, w), , [0,T] , , E1, E2 E3,
x (t ) {g 1 (t ) : g1 (t ) = e
A1t

x + e
0

t

A1 (t - s )

v1 ( s ) ds, v1 :[0, T ] E1,

(3)

v1 - , v1 ( s) f1 ( s, x ( s), y ( s ), w ( s )) . . s [0, T ]},
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y (t ) {g 2 (t ) : g 2 (t ) = e

A2t

y + e
0

t

A2 ( t - s )

v 2 ( s ) ds, v 2 :[0, T ] E

2,

(4)

v 2 - , v 2 ( s ) f 2 ( s, x ( s ), y ( s ), w ( s )) . . s [0, T ]},

w (t ) = ( w ) x (t ),

t [0, T ].

(5)

(1) , [0,) T­ (x, y, w), (1) [0,). , , >0 UT: E1вE2вE3E1вE2вE3, (x0, y0, w0) x(0)=A1-x0, y(0)=A2-y0 w(0)=w0 T­ (1). UT , , , R­ (. [3]). eA1t eA2t : ||eAit|| e-dit, t0, di>0, i=1,2. fi: RвE1вE2вE3Ei, i =1,2, Ai : F1) (t, x, y, w)fi(t, A1-x, A2-y, w) RвE1вE2вE3 Kv(Ei) R>0 R, [0,T] ||fi(t, A1-x, A2-y, w)||R (1+||x||E1+||y||E2), (||w||R), i=1,2. Kv(Ei) , , Ei. F2) (x,y,w)E1вE2вE3 fi(·, A1-x, A2-y, w): [0,T]Kv(Ei), i=1,2. F3) t[0,T] fi(t, A1139


2. (I)

·, A2-·, ·): E1вE2вE3Kv(Ei), i=1,2 . : C1) t[0,T] : E3вC([0,t],E1) C([0,t],E3). 2) (w)u(t)=((w)u(t1))u(t), 0t1t, t0. C3) . C4) , , KE3, wK, (w)x(t) K t0, xC([0,T],E1). , D1) xE1, wK, [0,1] T- y(t) y'(t)A2y(t)+A2f2(t, A1-x, A2-y(t),w). Yx,w. D2) E1 { Yx,w }, x, wK, [0,1] . . V={v:[0,T]E1, v ­ , v(t)f1(t,A1-x, A2-y(t),w) [0,T], yY1x,w}. : E1вK E1вK, (x,w)=(F(x,w),*(x,w)), T 1 F(x, w) = co{- A1-1+ v( s)ds, v V }, T0
( x, w) = ( w) A1- u , u ([0, T ], E1 ), u (t ) x



, . P E1вK. ME1вE3. MP=MP. . A)-D2) P MP, (-, MP) 0. >0 (1) 140


.. -- -10, 2002, .137-141

(x, y, w)CT(E1)в CT(E2)в CT(E3), n, , (xn, yn, wn) (A1-x*, A20 y , w*), (x*, w*) ­ , y0Y1x*,w*. . 1. 1. . . / . . , .. . -.: , 1983. -272. 2. Friedman A. Periodic solution to a parabolic equation with hysteresis / A. Friedman, L.-S. Jiang // Math. ­ 1987. ­ Z.194. ­ P.61-70. 3. Kamenskii M. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces / M. Kamenskii, V. Obukhovskii, P. Zecca. -Walter de Gruyter, Almagne, 2001. ­ 345p. 4. Kamenski M. An averaging method for singularly perturbed systems of semilinear differential inclusions with analytic semigroups / M. Kamenski, P. Nistri // Nonlinear Analysis. ­ 2002. ­ P.1-14. 5. Seidman T. Some problems with thermostat nonlinearities / T. Seidman // Proc. 27th IEEE Conf. Decision and Control. ­ 1988. ­ P.1255-1259.

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