FUNCTIONAL EVOLUTION EQUATIONS WITH NONCONVEX LOWER SEMICONTINUOUS MULTIVALUED PERTURBATIONS

In this paper we prove some existence theorems concerning the solutions and integral solution for functional (delay) evolution equations with nonconvex lower semicontinuous multivalued perturbations ,

Problem (P) for which E is reflexive, A(t,.) is an m-accretive multivalued operator and F is a Lipschitz single-valued function cf Tanaka [2] Problems (P) and (Q) without delay cf Cichon [3], [4], Ibrahim [5] and the references therein 2. NOTATIONS AND DEFINITIONS Let E* be the dual of E, Eo the Banach space E endowed with the weak topology a(E, E') If B is a multivalued operator from E to 2 E then B is said to be accretive if for each A > O, zl,z2 E D(B) (the domain of B), Yx E B(:r,x) and Y2 B(z2) we have We say that B is m-accretive ifB is accretive and ifthere exists A > 0 such that R(I + AB) E, where I is the identity map It is known that if B is m-accretive, then for every ,k > 0 the resolvant JaB (I + AB) -1 and the Yosida approximation of B; B, (I-JaB)/,k, are defined everywhere The generalized domain of B is defined by D'(B) {zEE'[B(x)[ =li_,rn[[Bx[, < oo}.
lira If E is metrizable then lower semicontinuity and lower semicontinuity in the Kuratowski sense are equivalent (cf [8], [9]) The following known result will be used in the sequel LEMMA .1 [6].For every I, let A(t, .)be an m-accretive multivalued operator from E to 2 E {} satisfying the following condition: (C1) There exist A0 > 0, a continuous function h I E and a nondecreasing continuous function L-[0, oo)--+ [0, oo)such that for all A $ (0, A0)and for almost , s I, Then D* (A(t, .))and D(A(t, .))are independent oft So if A is as in Lemma 2.1 we may write D'(A):= D'(A(,.)) and D(A):= D(A(, .));I respectively LEMMA .[10].Let E be a Banach space and M a compact metric space If T is a lower semicontinuous multivalued function on M and with nonempty closed decomposable values in L](I), then T has a continuous selection.on I--r, 0] and u(t) b(O) + f(s)ds; f E K is nonempty and convex, where K z {f LIE(I) If(t)[ _< 5 a e on I} If E is reflexive then X s compact subset of C6o ([ r, T]) If, in addition, E is separable then X is metrizable PROOF.It is obvious that X is nonempty, convex and equicontinuous and that the set {u(t) u X}; t I, is bounded So, ifE is reflexive then, X is relatively compact in CEo([r,T]) by Ascoli's theorem Let us verify that X is closed in CE ([-r, T]) Let (u,) be a sequence in X converging to u CEo ([-r, T]) Then u on [r, 0] and for each n > 1 there exists f,, K such that un(t)-b(0)+ f f(s)ds; I Since E is reflexive, K; is weakly compact in LE(I) Hence, the sequence (f,) has a subsequence, denoted again by (fn), converging weakly to f Ko Let G be a multivalued function from Eo to the nonempty closed subsets of E such that G is lower semicontinuous in the Kuratowski sense.If (xn) is a sequence converging to z in Eo, then for every z E, PROOF.Let y lim,_,ooinfG(x) Then there exists a sequence (y,) such that y G(xn)'n >_ and y, y as n c For any z E E we have which proves the first inequality The second inequality follows from the lower semicontinuity of G TttEOREM 3.1.Let E be a reflexive separable Banach space Let A(t,.)" I be an m-accretive multivalued operator from E to 26-{} satisfying condition (C) together with the following conditions (C2) There exist z > 0 such that for all x E E, the function w (I + A(t, .))-1belongs to L2E(I) (C3) For all r > 0 there exists 6(r) > 0 such that for all A > 0 and all x (A) with [[x[[ < r, IIJA(O,) 11 < ().
Let F be a measurable multivalued function from I x C([r, 0]) to P(E) satisfying the following conditions (F1) There exists a > 0 such that sup{]lull y F(t, u)} _< a, V(t, u) I x CE([r, 0]).
(F2) For all I,F(t, .) is lower semicontinuous in the sense of Kuratowski from CE([r, 0]) to (Fa) For all u Cs([ r, 0]) the multivalued function F(t, stu) admits a measurable selection Then for every CE([r, 0]) with @(0) E D*(A), the problem (P) has a solution.
PROOF.We split the proof into the following three steps (1) Let f E Ko { LE(I) II(t)ll _< aa.e on I}.Since A satisfies conditions (C), (C) and (C3), then by Theorem 4 of [5], there exists a unique absolutely continuous function uf :I E such that (i) u'l(t A(t, u(t)) + f(t) a e. on 1, u,(0) (0), (ii) Iluz(t)l _< 1 (O -[-1)T --[--L(r)suptezllh(t)[[ + 5(r),Vt I, where r (x + L(I](0)II)) + [A(0, x0)l, A G IBRAHIM (iii) the function f u/is continuous from Ko to CE (I)   (2) Set X1 (U CE([-r, T]), u -= on [-r, 0] and u(t) (0) + ff(s)ds, f K s } By Lemma 3 1, X1 is a compact subset of Co([-r, T]) and is metrizable.Define a multivalued function T on X1 by Tl(u) {f Ka f(t) F(t, stu) a e on I} In this step we prove that T1 has a continuous selection V'X Ko For this purpose, we show that T satisfies the conditions of Lemma 2 2 Condition (F3) assures that the values of T1 are nonempty Moreover, if D is a measurable subset of I and gx, g2 Tx (u) for some u X1, then the function g Nogl + Nt-Dg2 belongs to T1 (u), where N is the characteristic function.Then the values of T1 are decomposable It remains to prove that T1 is lower semicontinuous Since X1 is compact metrizable in CEo([-r, T]), it suffices to show that T is lower semicontinuous in the Kuratowski sense So, let (u,) be a sequence in X converging to u X l, with respect to the topology on Co([r, T]) and let g Tl(U) Since F is measurable, then for all n > 1 the multivalued function B.(t)= {z F(t, stun)'llg(t)-zll-d(g(t),F(t, stu.))} has a measurable selection g I E. Thus, by Lemma 3 2, for all I, lira Ilg(t) g(t.)ll < lim supd(g(t),F(t, stu.)) <_ d(g(t),lirninf F(t, stu)) This means that T1 is lower semicontinuous and hence there exists a continuous function V X Ko such that V (x) T(x), V x X (3) Define a function O'Xl--X by O(z)= ul,f V(x) By (iii) of the first step, 0 s continuous Hence, by Tichonoffs fixed point theorem, there exists u X such that u u,f Vx(u) T(u) This means that u'(t) A(t,u(t)) + f(t) and f(t) F(t, stu) ae on I The theorem is thus proved.TItEOREM 3.2.Let H be a Hilbert space and F be a measurable multivalued function from 1 x CH([r, 0]) to P(H) satisfying conditions (F), (F) and (F3) Let F be a multivalued funcuon from I to the family of nonempty closed convex subsets of H, with compact graph G and satisfies the following conditions.
(F1) There exists -), > 0 such that IIx-projr(t)xll <_ ( t) for all (t,x) G and all z I, (t < -) (F2) The function (t,x) 5(x,F(t)) sup{(x,y) "y F(t)} is lower semicontinuous on I x Bo, where Bo is the relative weak topology Then for all CE([r, 0]) with b(0) F(0), the problem (Q) has a solution PROOF.We split the proof into the following three steps (1) Let f Ko Since F has a compact graph and satisfies conditions (F) and (F) then by Theorem 3 11 ], there exists a unique absolutely continuous function u/ I H such that (i) u(t) Nr(tl(u(t)) + f(t) a.e. on I, (ii) u/(0) (0), ul(t) r(t), v e z, (iii) Iluy(t)ll <_ T(7 + a),'t I and the function f ul is continuous from Ko tOCH (2) Set X2 {u CH([-r, T])" u on [-r, 0] and u(t) (0) + ff(s)ds, f K& and define a multivalued function T2 on X2 by T(u) {f Ko f(t) F(t, stu) a.e on I} As in the second step of the proof of Theorem 3.1 we can show that T2 has a continuous selection (3) Define the function 0 X2 X by 0(x) ul, f V2(x) As in the third step ofthe proof of Theorem 3.1, we can show that there exists a unique u X2 such that u u I, f T2(u) Clearly u is a solution of (Q) where f LI(I) By an integral solution of (P') we mean a continuous function u I D(A) with u(0) x0 such that Itu(t)-zll _< Ilu(s)-zll + [u(r)-z, f(r)-y]+dr, for each z D(A), y A(z) and 0 <_ s <_ < T, where It is known that [7] if A is an m-accretive operator then for each (xo, f) D(A) x LE(I), the problem (P*) has a umque integral solution uf, such that the function f u I is continuous In this section we are concerned with the existence of integral solutions of the functional evolution equation where F is a multivalued function from I CE([-r, 0]) to 2 E {), q; > 0 is the operator of translation defined in section and is a given function, belongs to CE([r, 0]) with b(0) D(A) By an integral solution of (P**) we mean a continuous function u-[-r, T] E with u on [-r,]0, such that u is an integral solution of the evolution equation u' (t) -A (u / f(t), u (0) (0), where f LE(I) and f(t) F(t, su), a e on I We say that the operator A E 2 E {} has the (M)-property ( [7], [12]) if for each xo D (A)   and each uniformly integrable subset Q of LIE(I), the set {u s g Q} is a relatively compact subset of CE(I) where u s is the unique integral solution of the evolution equation u'() -A(u(t)) + g(t) a e on I; u(0) x0.It is well known that ( [7], [12]) if the proper operator -A generates a compact semigroup (via Crandall-Liggett's exponential formula [3], 13 ]), then A has the property (M) TiIEOREM 4.1.Let E be a Banach space and A an m-accretive multivalued operator from E to 2 E {} having the (M)-property.Let F be a measurable multivalued function from I x CE([-r, 0]) to the non-empty closed subsets of E satisfying the condition (Fs) together with the following conditions (F4) There exists a function h L (I) such that sup{llzll z e F(t, )} < h(t), V (t, ,) e C([-, 0]).
(Fh) For all I, F(t, .)CE([ r, 0]) E is lower semicontinuous in the Kuratowski sense Then for all CE([r, 0]) with q.,(0) D(A), the problem (P**) has an integral solution PROOF.Consider the set Q {f LE(I) [If(t)[[ _< h(t) a e. on I} One can easily show that Q is nonempty and uniformly integrable subset of LE(I) As mentioned above, for each f Q there exists a unique continuous function u I I D(A) such that u I is the unique integral solution of the evolution equation u'(t) A(u(t)) + f(t), u(0) q.,(0) and the function f uf is continuous from Q to CE(I).Let X" ={u,}CE([-r,T])'fQ}, where u)= on I-r, 0] and u" l=uI on I Since a has the property (M), X" is compact in the metric space CE([-r,T]) Now, define a multivalued function T on X:" by T(x) {f LE(I) f(t) F(t, sx) a e on I} As in the second step of the proof of Theorem 3 1, we can show that T has a continuous selection V X* LE(I) Also, define a function # "X* X*, #(z) u), f V(z) The function # is clearly continuous and hence has a fixed point a: E X* It is obvious that z is the desired solution 5. EXAMPLES In this section we give some examples illustrating the scope of the results developed in sections 3 and EXAMPLE 1.Let for all I, A(t) B-h(t) where h :I E is integrable and B is an m- accretive operator on E Clearly A(t) is m-accretive for all I Let .k> 0, s, I and z E Then 1 A(t,x) A(s,x)[] <_ -JA(t,x) JA(s,x)[ <_ [Ih(t) h(s) [.Hence condition (C1) of Lemma 2.1 holds EXAMPLE 2. In [6] there are several examples for operators A such that for every I, A(t) is m-accretive and satisfies condition (C1) EXAMPLE 3. Let H be a real Hilbert space with inner product (., .)and let #:H H be a proper lower semicontinuous convex function.The set i)#(x) {z H :#(x) <_ #(y) + (x-y, z} for each y H} is called the subdifferential of # at the point x We recall that D(0#)= {x H 0#(x) is nonempty}.Now if we define an operator A:D(A) DO(#) 2 H by A(x) O#(x), then A is m-accretive and the following conditions are equivalent [7]   (i) For each ,k > 0, the resolvent JA is a compact operator (ii) The function # is of compact type (iii) The semigroup generated by the operator A is compact EXAMPLE 4. Take E L([0, zr]) and let us define A" D(A) C_ E E by Au u()(t) for each u D(A) where D(a) {u E u (/ E E, u(0) u(Tr) 0} The operator A is m-accretive and the semigroup {S(t) > 0} generated by -A(S(t) limn-oo (I+-A) -n) is compact [7]

3 .
EXISTENCE OF SOLUTIONS FOR THE PROBLEMS (P) AND () To prove our results we need the following lemmas LEMMA 3.1.Let b be an element of CE([ r, 0]) and/3 be a positive real number The set { Io ) X= u E CE([-r,O]) u

4 .
EXISTENCE OF INTEGRAL SOLUTIONS FOR TI'IE PROBLEM (P) WI'IEN TI:IE OPERATOR A IS INDEPENDENT OF TIME In this section A denotes a multivalued operator from E to 2 E-{8} Consider the evolution equation '(t) -A(u(t.))+f()ae onI (P') u(O) xo D(A),