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Viability of a Time Dependent Closed Set with Respect to a Semilinear Delay Evolution Inclusion

Viability of a Time Dependent Closed Set with Respect to a Semilinear Delay Evolution Inclusion We prove a sufficient condition for a time-dependent closed seo be viable with respeco a delay evolution inclusion governed by a strongly-weakly u.s.c. perturbation of an infinitesimal generator of a C0 -semigroup. This condition is expressed in terms of a natural concept involving tangent sets, generalizing tangent vectors in the sense of Bouligand and Severi. Mathematics Subject Classification 2010: 34K09, 34G25, 34A60, 47D06. Key words: differential delay evolution inclusion, locally closed graph, tangent set, tangency condition, multi-valued mapping, viability. 1. Introduction Let X be a real Banach space, I = [a, b) R be a nonempty and bounded interval and let A : D(A) X X be the infinitesimal generator of a C0 semigroup {S(t); t 0} of type (M, ), i.e., S(t) M et for each t 0. Let > 0 and C = C([-, 0]; X) endowed with the norm = sup{ (t) ; t [-, 0]}. If u C([ - , T ], X), then for each t [, T ] we denote by ut C the function given by ut (s) = u(t + s) for s [-, 0]. * Supported by a grant of the Romanian National Authority for Scientific Research, CNCS­UEFISCDI, project number PN-II-ID-PCE-2011-3-0052. MIHAI NECULA and MARIUS POPESCU Let K : I values, where X and F : K X be two multi-functions with nonempty K = {(t, ) I × C ; (0) K(t)}. Our aim here is to prove some new necessary and sufficient conditions in order that K be viable with respeco A + F . To be more precise, let (, ) K and consider the Cauchy Problem (1.1) u (t) Au(t) + F (t, ut ), u = . t Definition 1.1. By a mild solution of (1.1) on [, T ] I, we mean a function u C([ - , T ]; X) satisfying (t, ut ) K for t [, T ], u(t) = (t - ) for t [ - , ] and for which there exists f L1 (, T ; X) with f (t) F (t, ut ) a.e. for t [, T ] and (1.2) for each t [, T ]. Definition 1.2. We say that K is mild viable with respeco A + F , where F : K X, if for each (, ) K, there exists T > , such that [, T ] I and (1.1) has at least one mild solution u : [ - , T ] X. If T (, sup I) can be taken arbitrary, we say that K is globally mild viable with respeco A + F . The existence of solutions for functional differential equations governed or not by linear and nonlinear operators in Banach spaces has been studied extensively in many papers. The first viability results for (1.1) in the case A = 0 and F single valued have been proved in the papers Lakshmikantham, Leela and Moauro [11] and Leela and Moauro [12]. The case when A = 0, X is a finite dimensional space and F is upper semicontinuous and with convex compact values has been studied by Haddad [9, 10]. The Haddad's result has been extended by Gavioli and Malaguti [8] to the infinite dimensional setting. The case when A is the infinitesimal generator of C0 semigroup and F is a continuous single-valued function has been studied by Pavel and Iacob [17] and the corresponding multivalued case, with F a Carath`odory multifunction with nonconvex values, has been considered e by Lupulescu and Necula [13]. As concern the case of delay evolution inclusions subjected to nonlocal initial conditions, a very recent research direction, we mention here the results established by Vrabie [20, 21, 22, 23], Burlica and Rosu [2] and ¸ , Rosu and Vrabie [3]. Burlica ¸ In this paper we shall use a concept of tangent sets that extends the notion of tangent vectors in the sense of Bouligand and Severi. This concept was introduced, in the specific case in which K does not depend on t, by ^ Carja, Necula and Vrabie [4, 5], and it was adapted for the t-dependent case in Necula, Popescu and Vrabie [15, 16]. 2. Tangency concepts If (Y, d) is a metric space, y Y and r > 0, D(y, r) denotes the closed ball with center y and radius r > 0, i.e. D(y, r) = {x Y ; d(y, x) r}, while S(y, r) denotes the open ball with center y and radius r > 0, i.e. S(y, r) = {x Y ; d(y, x) < r}. If B Y and C Y , we denote by dist(y, C) = inf{d(y, z); z C} and by dist(B, C) = inf{d(x, y); x B, y C}. Also B(Y ) denotes the family of all nonempty and bounded subsets of Y . Definition 2.1. Let Y X be nonempty. The function Y : B(X) R+ , defined by n() Y (B) := inf > 0; x1 , x2 , . . . , xn() Y, B i=1 D(xi , ) , is called the Hausdorff measure of noncompactness on X subordinated to Y . If Y = X, we simply denote X by , and we simply call ihe Hausdorff measure of noncompactness on X. Remark 2.1. We have the following properties: (i) for each B B(X) and r > 0 with B D(0, r), we have (B) r ; (ii) (B) = 0 if and only if B is relatively compact ; MIHAI NECULA and MARIUS POPESCU (iii) the restriction of Y to B(Y ) coincides with the Hausdorff measure of noncompactness on Y ; (iv) for each B B(Y ) we have (B) Y (B) 2 (B). ¨ The next lemma is due to Monch [14]. Lemma 2.1. Let X be a separable Banach space and {fn ; n N} a subset in L1 (, T ; X) for which there exists L1 (, T ; R+ ) such that fn (s) (s) for each n N and a.e. for s [ , T ]. Then the mapping s ({fn (s); n N}) is integrable on [ , T ] and, for each t [ , T ], we have (2.1) fn (s) ds; n N ({fn (s); n N}) ds. For further details on the Haussdorf measure of noncompactness see ^ Carja, Necula and Vrabie [4], Section 2.7, pp. 4853. Definition 2.2. Let (Y, d) be a metric space and let F : Y X a multi-function. We say that F is u.s.c. at y Y if for each open set U X with F (y) U there exists an open set V Y with y V and F (V ) U . We say that F is u.s.c. on Y if it is u.s.c. at each y Y . Throughout, K is endowed with the metric, d, defined by d((, ), (, )) = max{| - |, - }, for all (, ), (, ) K. Furthermore, whenever we will use the term strongly-weakly u.s.c. we will mean thahe domain of the multi-function in question is equipped with the strong topology, while the range is equipped with the weak topology. The term u.s.c. refers to the case in which both domain and range are endowed with the strong topology. Definition 2.3. The multi-function F : K X is called locally bounded if, for each (, ) K, there exist > 0, > 0, and M0 > 0 such that for all (t, ) ([ - , + ] × D(, )) K, we have F (t, ) M0 . Let (, ) K, let X and let E B(X). Definition 2.4. We say that (i) is A-right-tangeno K at (, ) if (2.2) lim inf S(h - s) ds, K( + h) = 0; (ii) E is A-right-tangeno K at (, ) if (2.3) lim inf S(h - s)E ds, K( + h) = 0; (iii) E is A-right-quasi-tangeno K at (, ) if (2.4) where FE = f L1 (R; X); f (s) E a.e. for s R . loc Throughout, we denote by: A (i) TK (, ) the set of all A-right-tangent vectors to K at (, ); (ii) TSA (, ) the set of all A-right-tangent sets to K at (, ); K (iii) QTSA (, ) the set of all A-right-quasi-tangent sets to K at (, ). K If K is constant, E is A-right-tangeno K at (, ) if and only if it is A^ tangeno K at = (0) in the sense of Carja, Necula and Vrabie [4, 5], A i.e., if and only if E TSK (), which means (2.5) lim inf lim inf S(h - s)FE ds, K( + h) = 0, 1 dist S(h) + h S(h - s)E ds, K Similarly, if K is constant, E is right-quasi-tangeno K at (, ) if ^ and only if it is A-quasi-tangeno K at = (0) in the sense of Carja, A Necula and Vrabie [4, 5], i.e., if and only if E QTSK ()), which means (2.6) lim inf 1 dist S(h) + h S(h - s)FE ds, K By identifying vectors with singleton sets and constants with locally integrable functions, we have (2.7) A TK (, ) TSA (, ) QTSA (, ), K K and it may happen that, even in the simplest case A 0, both inclusions ^ to be strict. See Example 2.4.1, p. 36 in Carja, Necula and Vrabie [4]. MIHAI NECULA and MARIUS POPESCU 3. Necessary conditions for viability For the proof of the main result of this section we need the following simple lemma: Lemma 3.1. Let f : [, T ] X be a measurable function and B, C X two nonempty sets such that f (t) B+C a.e. for t [, T ]. Then, for every > 0 there exishree measurable functions, b : [, T ] B, c : [, T ] C and r : [, T ] S(0, ) such that f (t) = b(t) + c(t) + r(t) a.e. for t [, T ]. Proof. Let > 0 be fixed. Let f : [, T ] X be countably valued and such that f (t) - f (t) < 2 a.e. for t [, T ]. So, we have f (t) B + C + S(0, ) a.e. for t [, T ]. 2 Then, there exishree countably valued functions b : [, T ] B, c : [, T ] C and r : [, T ] S(0, 2 ) such that f (t) = b(t) + c(t) + r(t) a.e. for t [, T ]. The proof is complete once we take r(t) = r(t) + f (t) - f (t). Theorem 3.1. If F : K X is u.s.c. and K is mild viable with respeco A + F then, for all (, ) K we have 1 lim dist S(h)(0) + h h 0 S(h - s)FF (,) ds, K( + h) Proof. Let (, ) K and u : [ - , T ] X be a mild solution of (1.1). Hence there exists f L1 (, T ; X) such that f (s) F (s, us ) a.e. for s [, T ] and for all t [, T ]. Let > 0 be arbitrary but fixed. Since F is u.s.c. at (, ) and limt ut = u = in C , we may find > 0 such that f (s) F (s, us ) F (, ) + S(0, ) a.e. for s [, + ]. From Lemma 3.1, we deduce thahere exiswo integrable functions g : [, + ] F (, ) and r : [, + ] S(0, 2) such that f (s) = g(s) + r(s) a.e. for s [, + ]. Since u( + h) K( + h), we deduce that for each 0 < h < + S( + h - s)FF (,) ds, K( + h) S( + h - s)FF (,) ds, S(h)(0) S( + h - s)f (s)ds 1 dist h 1 h S( + h - s)g(s)ds, S( + h - s)f (s)ds S( + h - s)(g(s) - f (s)) ds 2M e(T - ) . Passing to lim sup in the inequality above we get 1 lim sup dist S(h)(0) + h 2M e(T - ) . As > 0 was arbitrary, we deduce that F (, ) QTSA (, ). K A simple consequence of Theorem 3.1 is: Theorem 3.2. If F : K X is u.s.c. and K is mild viable with respeco A + F then F (, ) QTSA (, ) for all (, ) K. K 4. Sufficient conditions for viability Definition 4.1. We say thahe multi-function K : I X is : S( + h - s)FF (,) ds, K( + h) (i) closed from the left on I if for any sequence ((tn , xn ))n1 from I × X, with xn K(tn ) and (tn )n nondecreasing, limn tn = t I and limn xn = x, we have x K(t). (ii) locally closed from the left if for each (, ) I ×X with K( ) there exis > and > 0 such thahe multi-function t K(t) D(, ) is closed from the left on [, T ]. MIHAI NECULA and MARIUS POPESCU Definition 4.2. By a Carath´odory uniqueness function we mean a e function : I × R+ R+ such that: (i) for each x R+ , t (t, x) is locally integrable; (ii) for a.e. t I, x (t, x) is continuous, nondecreasing; (iii) for each I, the only absolutely continuous solution of the Cauchy problem x (t) = (t, x(t)) u( ) = 0 is x 0. Remark 4.1. If : I ×R+ R+ is a Carath´odory uniqueness function e and x : [, T ] X is a measurable and bounded function which satisfies x(t) (s, x(s))ds, for all t [, T ], then x(t) = 0 for all t [, T ]. ^ See Problem 1.8.2. in Carja, Necula and Vrabie [4]. Definition 4.3. We say that A + F is -compact if for all (, ) K there exist > 0, > 0, a Carath´odory uniqueness function, : I × e R+ R+ and a continuous function m : [0, ) [0, ), such that, for all B D (, ), all t (0, ) and a.e. for s [ - , + ] we have (4.1) lim (S(t)F (([s - h, s] × B) K)) m(t)(s, (B(0))) where B(0) = {(0); B}. Remark 4.2. (1) If {S(t) : X X; t 0} is compact and F is locally bounded, then A + F is -compact; (2) If F maps bounded subsets of K into relatively compact subsets of X, then A + F is -compact; (3) If F is u.s.c., has compact values and K is locally compact, then A+F is -compact. Theorem 4.1. Let K be locally closed from the left and let F : K X be nonempty, convex and weakly compact valued. If F is strongly-weakly u.s.c., locally bounded and A + F is -compact, then a sufficient condition in order that K be mild viable with respeco A + F is (4.2) F (, ) QTSA (, ) for all (, ) K. K If, in addition, F is u.s.c., then (4.2) is also necessary in order that K be mild-viable with respeco A + F . The necessity follows from Theorem 3.2, while the sufficiency will be proved later. From Theorem 4.1, under the additional assumption that K is even closed from the left, by Brezis-Browder Ordering Principle ­ see Brezis and Browder [1] ­, we easily deduce the global mild viability result below. Theorem 4.2. Let K be closed from the left and let F : K X be nonempty, convex and weakly compact valued. If F is strongly-weakly u.s.c., locally bounded and A + F is -compact, then a sufficient condition in order that K be globally mild viable with respeco A + F is (4.2). If, in addition, F is u.s.c., then (4.2) is also necessary in order that K be mild-viable with respeco A + F . ^ The next lemma, inspired from Carja and Vrabie [7], is the main step through the proof of Theorem 4.1. Lemma 4.1. Let K : I X be locally closed from the left, F : K X be locally bounded and let (, ) K. Let us assume that (4.2) is satisfied. Let > 0, T > and M0 > 0 be such that: (1) the multi-function t K(t) D((0), ) is closed from the left on [ , T ); (2) F (t, ) M0 for all t [, T ] and all D (, ) with (t, ) K; (3) supt[,T ] S(t - )(0) - (0) + sup|t-s|T - (t) - (s) + M e(T - ) (T - )(M0 + 1) < . Then, for each (0, 1), there exist a family PT = {[tm , sm ); m } of disjoint intervals, with finite or at most countable, and four functions: f, r L1 (, T ; X), : {(t, s); s t T } (0, T - ] measurable, and u C([ - , T ]; X) such that: MIHAI NECULA and MARIUS POPESCU (i) [tm , sm ) = [, T ) and sm - tm , for all m ; (ii) u(tm ) K(tm ), for all m and u(T ) K(T ) ; (iii) (t, s) t - s for s t T ; t (t, s) is nonexpansive on (s, T ] and, for each t (, T ], s (t, s) is measurable on [, t) ; (iv) f (s) F (tm , utm ) a.e. for s [tm , sm ) and f (s) M0 a.e. for s [, T ] ; (v) r(s) a.e. for s [, T ] ; (t - ), t [ - , ] (vi) t S(t - s)f (s)ds t + S((t, s))r(s)ds, t [, T ]; ut - (vii) < for all t [, T ] ; (viii) u(t) - u(tm ) for all t [tm , sm ) and all m . Proof. First, let us observe that, if (i)(vi) are satisfied, then (vii) is satisfied too, i.e. u(t + s) - (s) < for all t [, T ] and all s [-, 0]. Indeed, if t + s then u(t + s) - (s) = (t + s - ) - (s) sup |t1 -t2 |T - (t1 ) - (t2 ) < . If t + s > then |s| < T - and from (3) and (vi) we get u(t + s) - (s) S(t + s - )(0) - (s) + S((t, s))r(s) ds S(t + s - )(0) - (0) + (0) - (s) + M e(T - ) M0 (T - ) + M e(T - ) (T - ) < . Let (0, 1) be arbitrary, but fixed. We will show thahere exist = () in (, T ) and P ,f ,, r, u such that (i)(vii) hold true with instead of T . Using the tangency condition (4.2), we deduce thahere exist hn 0, gn FF (,) and pn X, with pn 0, such that n S(hn )(0) + S( + hn - s)gn (s) ds + pn hn K( + hn ) for every n N, n 1. Let n0 N and = + hn0 such that (, T ), hn0 < and pn0 < . Let us define P = {[, )}, f (s) = gn0 (s), (t, s) = 0 for s t , r(s) = pn0 for s [, ] and let u : [, ] X given by (vi). One may easily see that (i)(vi) are satisfied. Moreover, we may diminish > (increase n0 ), if necessary, in order to (viii) be satisfied too. Let U = {(P , f, , r, u); (, T ] and (i)(vii) are satisfied with instead of T }. As we already have shown, U = . On U we define a partial order by (P1 , f1 , 1 , r1 , u1 ) (P2 , f2 , 2 , r2 , u2 ), if 1 2 , P1 P2 , f (s) = f (s), r (s) = r (s) a.e. for s [, ], 1 2 1 2 1 1 (t, s) = 2 (t, s) for s t 1 , u (s) = u (s) for all s [, ]. 1 2 1 We will prove that each nondecreasing sequence in U is bounded from above. Let ((Pj , fj , j , rj , uj ))j1 be a nondecreasing sequence in U and let = supj1 j . If there exists j0 N such that j0 = , then (Pj0 , fj0 , j0 , rj0 , uj0 ) is an upper bound for the sequence. So, let us assume that j < , for all j 1. Obviously, (, T ]. We define P = j1 Pj , f (s) = fj (s), (t, s) = j (t, s) for s t j and r(s) = rj (s)for all j and all s [, j ). Clearly, f, r L1 (, ; X). Since |j (j , s)-i (i , s)| |j -i | for all i, j 1 and s < min{i , j }, we may define (, s) = limj j (j , s) for all s < . It follows that satisfies (iii). Next, we define u : [, ] X by + S((t, s))r(s) ds, for all t [, ]. We have u C([, ]; X) and u(s) = uj (s), for all j 1 and all s [, j ]. Since u() = limt u(t) = limj u(j ) = limj uj (j ), and uj (j ) D((0), ) K(j ) and the latter is closed MIHAI NECULA and MARIUS POPESCU from the left, we deduce that u() D((0), ) K(). The rest of conditions in lemma being obviously satisfied, it follows that (P , f, , r, u) is an upper bound for the sequence. Thus, the partially ordered set (U, ) and the function N : (U, ) R, defined by N(P , f, , r, u) = , for each (P , f, , r, u) U, satisfy the hypotheses of the Brezis-Browder Ordering ^ Principle, i.e. Theorem 2.1.1, p. 30 in Carja, Necula and Vrabie [4]. Accordingly, there exists an N-maximal element in U. This means thahere exists (P , f , , r , u ) U such that, whenever (P , f , , r , u ) (P , f , , r, u), we necessarily have N(P , f , , r , u ) = N(P , f , , r, u). We will show that = T . To this aim, let us assume by contradiction that < T . Since ( , u ) K and using the tangency condition (4.2) we deduce thahere exist hn 0, gn FF ( ,u ) and pn X, with pn 0, such that S(hn )u ( ) + n S( + hn - s)gn (s) ds + hn pn K( + hn ), for every n N, n 1. Let n0 N and = + hn0 such that ( , T ), hn0 < and pn0 < . Let us define P = P {[ , ]}, (t, s), s t (t, s) = t - + ( , s), s < < t < , 0, s < t f (s) = f (s), s [, ] gn0 (s), s ( , ] , r(s) = r (s), s [, ] pn0 , s ( , ] , u (t), t [, ] u(t) = Since, by (vii) we have u S (, ) and using the relation (2), it follows that f (s) M0 a.e. for s (, ). Clearly (i)(vii) are satisfied, and we can diminish (increase n0 ) in order that (viii) be satisfied too. (P , f , , r, u), but < So, (P , f , , r, u) U, (P , f , , r , u ) which contradicts the maximality of (P , f , , r , u ). Hence = T , and P , f , , r and u satisfy all the conditions (i)(vii). The proof is complete. S(t - )u ( ) + S(t - s)gn0 (s) ds + (t - )pn0 , t ( , ]. Definition 4.4. Let >0. A quintuple (PT , f, , r, u) satisfying (i)(viii) in Lemma 4.1, is called an -approximate solution of (1.1). Remark 4.3. Lemma 4.1 offers a sufficient condition in order that, for each > 0 to exists at least one -approximate solution of (1.1). Next, let us prove Theorem 4.1. Proof. Since the necessity follows from Theorem 3.2, we will confine ourselves only to the proof of the sufficiency. Let > 0 and T > and M0 be as in Lemma 4.1. Let n (0, 1), with n 0. Let ((Pn , fn , n , rn , un ))n be a sequence of n -approximate solutions of T (1.1), sequence given by Lemma 4.1. If Pn = {[tn , sn ); m n } with m m T n finite or at most countable, we denote by an : [, T ) [, T ) the step function, defined by an (s) = tn for each s [tn , sn ). Clearly m m m (4.3) uniformly for s [, T ). In view of (vi), we have lim an (s) = s (4.4) un (t) = S(t - )(0) + S(t - s)fn (s) ds + S(n (t, s))rn (s) ds for each n N and t [, T ]. We will show that, on a subsequence at least, (un )n is uniformly convergent on [, T ] to some function u which will turn ouo be a mild solution for the problem (1.1). We analyze firshe case when X is separable. Using (v) from Lema 4.1 we deduce that, for each t [, T ], we have (4.5) ({ S(n (t, s))rn (s) ds; n 1}) From (iv) we get (4.6) fn (t) M0 for all n 1 and a.e. for t [, T ]. Using (viii) we deduce limn un (an (t)) - un (t) = 0 uniformly for t [, T ) and from here we get (4.7) ({un (an (t)); n 1}) = ({un (t); n 1}). MIHAI NECULA and MARIUS POPESCU Le [, T ). Denote by x(t) = ({un (t); n 1}) and Bt = {(un )an (t) ; n 1}. By applying the function in (4.4), we obtain x(t) ({ S(t - s)fn (s) ds; n 1}) + ({ S(n (t, s))rn (s) ds; n 1}) ({S(t - s)fn (s); n 1}) ds. From (iv) in Lemma 4.1, we deduce that, for all n, k N, n k and a.e. for s [, T ), we have fn (s)F (an (s), (un )an (s) ) F (([s-n , s]×Bs )K) F (([s-k , s]×Bs )K) From here and the fachat A + F is -compact we gehat, a.e. for s [, t) ({S(t - s)fn (s); n 1}) = lim ({S(t - s)fn (s); n k}) lim (S(t - s)F (([s - k , s] × Bs ) K)) m(t - s)(s, (Bs (0))). Let 0 = (sups[0,T - ] m(s)). It follows that x(t) 0 (s, ({un (an (s)); n 1})) ds = 0 (s, x(s)) ds. Since 0 is a Carath´odory uniqueness function, too, we deduce that e x(t) = 0 for all t [, T ). So, {un (t); n 1} is relatively compact for all t [, T ). By Theorem 8.4.1, p. 194, in Vrabie [19], we conclude that (un )n has at least one uniformly convergent subsequence to some function u, subsequence denoted, for simplicity, again by (un )n . Since an (t) t, limn un (an (t)) = u(t), uniformly for t [, T ) and t K(t) D((0), ) is closed from the left, we gehat u(t) K(t) for all t [, T ]. But limn (un )an (t) = ut in C , uniformly for t [, T ). Hence, the set C = {(an (t), (un )an (t) ); n 1, t [, T )} is compact and C K. Since F is strongly-weakly u.s.c. and has weakly ^ compact values, by Lemma 2.6.1, p. 47, in Carja, Necula and Vrabie [4], it follows thahe set B = conv n1 t[,T ) F (an (t), (un )an (t) ) is weakly compact. We notice that fn (s) B for every n 1 and a.e. for ^ s [, T ], hence, by Diestel's Theorem 1.3.8, p.10, in Carja, Necula and Vrabie [4], it follows that, on a subsequence at least, limn fn = f weakly in L1 (, T ; X). As F is strongly-weakly u.s.c. with closed and convex values, and, by Lemma 4.1, for each n 1, we have fn (s) F (an (s), (un )an (s) ) a.e. for s [, T ], from Theorem 3.1.2, p. 88, in Vrabie [18], we conclude that f (s) F (s, us ) a.e. for s [, T ]. Finally, passing to the limit both sides in (4.4), for n , we ge , for each t [, T ]. If X is not separable, we have to observe thahere exists a separable and closed subspace Y X such thahe families: {S(·)fn (·); n 1}, {S(·)un (·); n 1} and {S(·)rn (·); n 1} are Y -valued. Then, to complete the proof, it suffices to follows the very same arguments as before and to make use of (iv) in Remark 2.1. 5. A comparison result Let X be a real Banach space, let C X be a closed convex cone and C (-C) = {0}, let " " be the partial order on X defined by C, i.e., x y if and only if y - x C. Let A : D(A) X X be the infinitesimal generator of a C0 -semigroup {S(t); t 0} and let a : I X be a continuous function. Let K : I X be defined by K(t) = {x X; a(t) x} for each t I. Let K = {(t, ) I × C ; (0) K(t)} and F : K X be a given multi-function. We are interested in finding sufficient conditions in order that K be mild viable with respeco A + F , i.e., in order that, for each (, ) K, to exists at least one mild solution u : [, T ] X of the problem u (t) Au(t) + F (t, ut ) (5.1) u = a(t) u(t) for each t [, T ]. The next lemma is, essentially, Lemma 8.1 in Necula, Popescu and ^ Vrabie [16]. See also Carja, Necula and Vrabie [6]. 1,1 Lemma 5.1. Let a Wloc (I; X) and let K be as above. Let I be a point of differentiability from the right for a, with a( ) D(A), let C and let E B(X). Let us assume that S(t)C C for each t 0. Then, the following two conditions : MIHAI NECULA and MARIUS POPESCU (i) E TSA (, a + ) K (ii) Aa( ) - a ( ) + E TSA () (see relation(2.5)) C are equivalent. Moreover, (iii) dist(Aa( ) - a ( ) + E; C) = 0 implies both (i) and (ii). The nexheorem is obtained from Theorem 4.1 and the above lemma. Theorem 5.1. Let A : D(A) X X be the infinitesimal generator of a C0 -semigroup {S(t); t 0}, let a : I D(A), a C 1 (I; X), K be as above and F : K X be a nonempty, convex and weakly compact valued multi-function. Let us suppose that S(t)C C for every t > 0, F is strongly-weakly u.s.c., locally bounded and A + F is -compact. Then, any of the nexwo conditions is a sufficient condition in order that K be mild viable with respeco A + F . (5.2) Aa( ) - a ( ) + F (, ) TSA ((0) - a( )) C for all (, ) K; (5.3) for all (, ) K. Proof. From Lemma , we deduce that (5.3) implies (5.2) which, in turn, implies that F (, ) TSA (, a + (0) - a( )) = TSA (, ). The conclusion K K follows from Theorem 4.1. Remark 5.1. Since F has convex and weakly compact values and C is closed and convex, it results that (5.3) is equivalent with: for all (, ) K, there exists F (, ) such that (5.4) a ( ) - Aa( ) . dist(Aa( ) - a ( ) + F (, ); C) = 0 If, in addition, F has compact values, then (5.2) is equivalent with: (5.5) A [Aa( ) - a ( ) + F (, )] TC ((0) - a( )) = for all (, ) K. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

Viability of a Time Dependent Closed Set with Respect to a Semilinear Delay Evolution Inclusion

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Abstract

We prove a sufficient condition for a time-dependent closed seo be viable with respeco a delay evolution inclusion governed by a strongly-weakly u.s.c. perturbation of an infinitesimal generator of a C0 -semigroup. This condition is expressed in terms of a natural concept involving tangent sets, generalizing tangent vectors in the sense of Bouligand and Severi. Mathematics Subject Classification 2010: 34K09, 34G25, 34A60, 47D06. Key words: differential delay evolution inclusion, locally closed graph, tangent set, tangency condition, multi-valued mapping, viability. 1. Introduction Let X be a real Banach space, I = [a, b) R be a nonempty and bounded interval and let A : D(A) X X be the infinitesimal generator of a C0 semigroup {S(t); t 0} of type (M, ), i.e., S(t) M et for each t 0. Let > 0 and C = C([-, 0]; X) endowed with the norm = sup{ (t) ; t [-, 0]}. If u C([ - , T ], X), then for each t [, T ] we denote by ut C the function given by ut (s) = u(t + s) for s [-, 0]. * Supported by a grant of the Romanian National Authority for Scientific Research, CNCS­UEFISCDI, project number PN-II-ID-PCE-2011-3-0052. MIHAI NECULA and MARIUS POPESCU Let K : I values, where X and F : K X be two multi-functions with nonempty K = {(t, ) I × C ; (0) K(t)}. Our aim here is to prove some new necessary and sufficient conditions in order that K be viable with respeco A + F . To be more precise, let (, ) K and consider the Cauchy Problem (1.1) u (t) Au(t) + F (t, ut ), u = . t Definition 1.1. By a mild solution of (1.1) on [, T ] I, we mean a function u C([ - , T ]; X) satisfying (t, ut ) K for t [, T ], u(t) = (t - ) for t [ - , ] and for which there exists f L1 (, T ; X) with f (t) F (t, ut ) a.e. for t [, T ] and (1.2) for each t [, T ]. Definition 1.2. We say that K is mild viable with respeco A + F , where F : K X, if for each (, ) K, there exists T > , such that [, T ] I and (1.1) has at least one mild solution u : [ - , T ] X. If T (, sup I) can be taken arbitrary, we say that K is globally mild viable with respeco A + F . The existence of solutions for functional differential equations governed or not by linear and nonlinear operators in Banach spaces has been studied extensively in many papers. The first viability results for (1.1) in the case A = 0 and F single valued have been proved in the papers Lakshmikantham, Leela and Moauro [11] and Leela and Moauro [12]. The case when A = 0, X is a finite dimensional space and F is upper semicontinuous and with convex compact values has been studied by Haddad [9, 10]. The Haddad's result has been extended by Gavioli and Malaguti [8] to the infinite dimensional setting. The case when A is the infinitesimal generator of C0 semigroup and F is a continuous single-valued function has been studied by Pavel and Iacob [17] and the corresponding multivalued case, with F a Carath`odory multifunction with nonconvex values, has been considered e by Lupulescu and Necula [13]. As concern the case of delay evolution inclusions subjected to nonlocal initial conditions, a very recent research direction, we mention here the results established by Vrabie [20, 21, 22, 23], Burlica and Rosu [2] and ¸ , Rosu and Vrabie [3]. Burlica ¸ In this paper we shall use a concept of tangent sets that extends the notion of tangent vectors in the sense of Bouligand and Severi. This concept was introduced, in the specific case in which K does not depend on t, by ^ Carja, Necula and Vrabie [4, 5], and it was adapted for the t-dependent case in Necula, Popescu and Vrabie [15, 16]. 2. Tangency concepts If (Y, d) is a metric space, y Y and r > 0, D(y, r) denotes the closed ball with center y and radius r > 0, i.e. D(y, r) = {x Y ; d(y, x) r}, while S(y, r) denotes the open ball with center y and radius r > 0, i.e. S(y, r) = {x Y ; d(y, x) < r}. If B Y and C Y , we denote by dist(y, C) = inf{d(y, z); z C} and by dist(B, C) = inf{d(x, y); x B, y C}. Also B(Y ) denotes the family of all nonempty and bounded subsets of Y . Definition 2.1. Let Y X be nonempty. The function Y : B(X) R+ , defined by n() Y (B) := inf > 0; x1 , x2 , . . . , xn() Y, B i=1 D(xi , ) , is called the Hausdorff measure of noncompactness on X subordinated to Y . If Y = X, we simply denote X by , and we simply call ihe Hausdorff measure of noncompactness on X. Remark 2.1. We have the following properties: (i) for each B B(X) and r > 0 with B D(0, r), we have (B) r ; (ii) (B) = 0 if and only if B is relatively compact ; MIHAI NECULA and MARIUS POPESCU (iii) the restriction of Y to B(Y ) coincides with the Hausdorff measure of noncompactness on Y ; (iv) for each B B(Y ) we have (B) Y (B) 2 (B). ¨ The next lemma is due to Monch [14]. Lemma 2.1. Let X be a separable Banach space and {fn ; n N} a subset in L1 (, T ; X) for which there exists L1 (, T ; R+ ) such that fn (s) (s) for each n N and a.e. for s [ , T ]. Then the mapping s ({fn (s); n N}) is integrable on [ , T ] and, for each t [ , T ], we have (2.1) fn (s) ds; n N ({fn (s); n N}) ds. For further details on the Haussdorf measure of noncompactness see ^ Carja, Necula and Vrabie [4], Section 2.7, pp. 4853. Definition 2.2. Let (Y, d) be a metric space and let F : Y X a multi-function. We say that F is u.s.c. at y Y if for each open set U X with F (y) U there exists an open set V Y with y V and F (V ) U . We say that F is u.s.c. on Y if it is u.s.c. at each y Y . Throughout, K is endowed with the metric, d, defined by d((, ), (, )) = max{| - |, - }, for all (, ), (, ) K. Furthermore, whenever we will use the term strongly-weakly u.s.c. we will mean thahe domain of the multi-function in question is equipped with the strong topology, while the range is equipped with the weak topology. The term u.s.c. refers to the case in which both domain and range are endowed with the strong topology. Definition 2.3. The multi-function F : K X is called locally bounded if, for each (, ) K, there exist > 0, > 0, and M0 > 0 such that for all (t, ) ([ - , + ] × D(, )) K, we have F (t, ) M0 . Let (, ) K, let X and let E B(X). Definition 2.4. We say that (i) is A-right-tangeno K at (, ) if (2.2) lim inf S(h - s) ds, K( + h) = 0; (ii) E is A-right-tangeno K at (, ) if (2.3) lim inf S(h - s)E ds, K( + h) = 0; (iii) E is A-right-quasi-tangeno K at (, ) if (2.4) where FE = f L1 (R; X); f (s) E a.e. for s R . loc Throughout, we denote by: A (i) TK (, ) the set of all A-right-tangent vectors to K at (, ); (ii) TSA (, ) the set of all A-right-tangent sets to K at (, ); K (iii) QTSA (, ) the set of all A-right-quasi-tangent sets to K at (, ). K If K is constant, E is A-right-tangeno K at (, ) if and only if it is A^ tangeno K at = (0) in the sense of Carja, Necula and Vrabie [4, 5], A i.e., if and only if E TSK (), which means (2.5) lim inf lim inf S(h - s)FE ds, K( + h) = 0, 1 dist S(h) + h S(h - s)E ds, K Similarly, if K is constant, E is right-quasi-tangeno K at (, ) if ^ and only if it is A-quasi-tangeno K at = (0) in the sense of Carja, A Necula and Vrabie [4, 5], i.e., if and only if E QTSK ()), which means (2.6) lim inf 1 dist S(h) + h S(h - s)FE ds, K By identifying vectors with singleton sets and constants with locally integrable functions, we have (2.7) A TK (, ) TSA (, ) QTSA (, ), K K and it may happen that, even in the simplest case A 0, both inclusions ^ to be strict. See Example 2.4.1, p. 36 in Carja, Necula and Vrabie [4]. MIHAI NECULA and MARIUS POPESCU 3. Necessary conditions for viability For the proof of the main result of this section we need the following simple lemma: Lemma 3.1. Let f : [, T ] X be a measurable function and B, C X two nonempty sets such that f (t) B+C a.e. for t [, T ]. Then, for every > 0 there exishree measurable functions, b : [, T ] B, c : [, T ] C and r : [, T ] S(0, ) such that f (t) = b(t) + c(t) + r(t) a.e. for t [, T ]. Proof. Let > 0 be fixed. Let f : [, T ] X be countably valued and such that f (t) - f (t) < 2 a.e. for t [, T ]. So, we have f (t) B + C + S(0, ) a.e. for t [, T ]. 2 Then, there exishree countably valued functions b : [, T ] B, c : [, T ] C and r : [, T ] S(0, 2 ) such that f (t) = b(t) + c(t) + r(t) a.e. for t [, T ]. The proof is complete once we take r(t) = r(t) + f (t) - f (t). Theorem 3.1. If F : K X is u.s.c. and K is mild viable with respeco A + F then, for all (, ) K we have 1 lim dist S(h)(0) + h h 0 S(h - s)FF (,) ds, K( + h) Proof. Let (, ) K and u : [ - , T ] X be a mild solution of (1.1). Hence there exists f L1 (, T ; X) such that f (s) F (s, us ) a.e. for s [, T ] and for all t [, T ]. Let > 0 be arbitrary but fixed. Since F is u.s.c. at (, ) and limt ut = u = in C , we may find > 0 such that f (s) F (s, us ) F (, ) + S(0, ) a.e. for s [, + ]. From Lemma 3.1, we deduce thahere exiswo integrable functions g : [, + ] F (, ) and r : [, + ] S(0, 2) such that f (s) = g(s) + r(s) a.e. for s [, + ]. Since u( + h) K( + h), we deduce that for each 0 < h < + S( + h - s)FF (,) ds, K( + h) S( + h - s)FF (,) ds, S(h)(0) S( + h - s)f (s)ds 1 dist h 1 h S( + h - s)g(s)ds, S( + h - s)f (s)ds S( + h - s)(g(s) - f (s)) ds 2M e(T - ) . Passing to lim sup in the inequality above we get 1 lim sup dist S(h)(0) + h 2M e(T - ) . As > 0 was arbitrary, we deduce that F (, ) QTSA (, ). K A simple consequence of Theorem 3.1 is: Theorem 3.2. If F : K X is u.s.c. and K is mild viable with respeco A + F then F (, ) QTSA (, ) for all (, ) K. K 4. Sufficient conditions for viability Definition 4.1. We say thahe multi-function K : I X is : S( + h - s)FF (,) ds, K( + h) (i) closed from the left on I if for any sequence ((tn , xn ))n1 from I × X, with xn K(tn ) and (tn )n nondecreasing, limn tn = t I and limn xn = x, we have x K(t). (ii) locally closed from the left if for each (, ) I ×X with K( ) there exis > and > 0 such thahe multi-function t K(t) D(, ) is closed from the left on [, T ]. MIHAI NECULA and MARIUS POPESCU Definition 4.2. By a Carath´odory uniqueness function we mean a e function : I × R+ R+ such that: (i) for each x R+ , t (t, x) is locally integrable; (ii) for a.e. t I, x (t, x) is continuous, nondecreasing; (iii) for each I, the only absolutely continuous solution of the Cauchy problem x (t) = (t, x(t)) u( ) = 0 is x 0. Remark 4.1. If : I ×R+ R+ is a Carath´odory uniqueness function e and x : [, T ] X is a measurable and bounded function which satisfies x(t) (s, x(s))ds, for all t [, T ], then x(t) = 0 for all t [, T ]. ^ See Problem 1.8.2. in Carja, Necula and Vrabie [4]. Definition 4.3. We say that A + F is -compact if for all (, ) K there exist > 0, > 0, a Carath´odory uniqueness function, : I × e R+ R+ and a continuous function m : [0, ) [0, ), such that, for all B D (, ), all t (0, ) and a.e. for s [ - , + ] we have (4.1) lim (S(t)F (([s - h, s] × B) K)) m(t)(s, (B(0))) where B(0) = {(0); B}. Remark 4.2. (1) If {S(t) : X X; t 0} is compact and F is locally bounded, then A + F is -compact; (2) If F maps bounded subsets of K into relatively compact subsets of X, then A + F is -compact; (3) If F is u.s.c., has compact values and K is locally compact, then A+F is -compact. Theorem 4.1. Let K be locally closed from the left and let F : K X be nonempty, convex and weakly compact valued. If F is strongly-weakly u.s.c., locally bounded and A + F is -compact, then a sufficient condition in order that K be mild viable with respeco A + F is (4.2) F (, ) QTSA (, ) for all (, ) K. K If, in addition, F is u.s.c., then (4.2) is also necessary in order that K be mild-viable with respeco A + F . The necessity follows from Theorem 3.2, while the sufficiency will be proved later. From Theorem 4.1, under the additional assumption that K is even closed from the left, by Brezis-Browder Ordering Principle ­ see Brezis and Browder [1] ­, we easily deduce the global mild viability result below. Theorem 4.2. Let K be closed from the left and let F : K X be nonempty, convex and weakly compact valued. If F is strongly-weakly u.s.c., locally bounded and A + F is -compact, then a sufficient condition in order that K be globally mild viable with respeco A + F is (4.2). If, in addition, F is u.s.c., then (4.2) is also necessary in order that K be mild-viable with respeco A + F . ^ The next lemma, inspired from Carja and Vrabie [7], is the main step through the proof of Theorem 4.1. Lemma 4.1. Let K : I X be locally closed from the left, F : K X be locally bounded and let (, ) K. Let us assume that (4.2) is satisfied. Let > 0, T > and M0 > 0 be such that: (1) the multi-function t K(t) D((0), ) is closed from the left on [ , T ); (2) F (t, ) M0 for all t [, T ] and all D (, ) with (t, ) K; (3) supt[,T ] S(t - )(0) - (0) + sup|t-s|T - (t) - (s) + M e(T - ) (T - )(M0 + 1) < . Then, for each (0, 1), there exist a family PT = {[tm , sm ); m } of disjoint intervals, with finite or at most countable, and four functions: f, r L1 (, T ; X), : {(t, s); s t T } (0, T - ] measurable, and u C([ - , T ]; X) such that: MIHAI NECULA and MARIUS POPESCU (i) [tm , sm ) = [, T ) and sm - tm , for all m ; (ii) u(tm ) K(tm ), for all m and u(T ) K(T ) ; (iii) (t, s) t - s for s t T ; t (t, s) is nonexpansive on (s, T ] and, for each t (, T ], s (t, s) is measurable on [, t) ; (iv) f (s) F (tm , utm ) a.e. for s [tm , sm ) and f (s) M0 a.e. for s [, T ] ; (v) r(s) a.e. for s [, T ] ; (t - ), t [ - , ] (vi) t S(t - s)f (s)ds t + S((t, s))r(s)ds, t [, T ]; ut - (vii) < for all t [, T ] ; (viii) u(t) - u(tm ) for all t [tm , sm ) and all m . Proof. First, let us observe that, if (i)(vi) are satisfied, then (vii) is satisfied too, i.e. u(t + s) - (s) < for all t [, T ] and all s [-, 0]. Indeed, if t + s then u(t + s) - (s) = (t + s - ) - (s) sup |t1 -t2 |T - (t1 ) - (t2 ) < . If t + s > then |s| < T - and from (3) and (vi) we get u(t + s) - (s) S(t + s - )(0) - (s) + S((t, s))r(s) ds S(t + s - )(0) - (0) + (0) - (s) + M e(T - ) M0 (T - ) + M e(T - ) (T - ) < . Let (0, 1) be arbitrary, but fixed. We will show thahere exist = () in (, T ) and P ,f ,, r, u such that (i)(vii) hold true with instead of T . Using the tangency condition (4.2), we deduce thahere exist hn 0, gn FF (,) and pn X, with pn 0, such that n S(hn )(0) + S( + hn - s)gn (s) ds + pn hn K( + hn ) for every n N, n 1. Let n0 N and = + hn0 such that (, T ), hn0 < and pn0 < . Let us define P = {[, )}, f (s) = gn0 (s), (t, s) = 0 for s t , r(s) = pn0 for s [, ] and let u : [, ] X given by (vi). One may easily see that (i)(vi) are satisfied. Moreover, we may diminish > (increase n0 ), if necessary, in order to (viii) be satisfied too. Let U = {(P , f, , r, u); (, T ] and (i)(vii) are satisfied with instead of T }. As we already have shown, U = . On U we define a partial order by (P1 , f1 , 1 , r1 , u1 ) (P2 , f2 , 2 , r2 , u2 ), if 1 2 , P1 P2 , f (s) = f (s), r (s) = r (s) a.e. for s [, ], 1 2 1 2 1 1 (t, s) = 2 (t, s) for s t 1 , u (s) = u (s) for all s [, ]. 1 2 1 We will prove that each nondecreasing sequence in U is bounded from above. Let ((Pj , fj , j , rj , uj ))j1 be a nondecreasing sequence in U and let = supj1 j . If there exists j0 N such that j0 = , then (Pj0 , fj0 , j0 , rj0 , uj0 ) is an upper bound for the sequence. So, let us assume that j < , for all j 1. Obviously, (, T ]. We define P = j1 Pj , f (s) = fj (s), (t, s) = j (t, s) for s t j and r(s) = rj (s)for all j and all s [, j ). Clearly, f, r L1 (, ; X). Since |j (j , s)-i (i , s)| |j -i | for all i, j 1 and s < min{i , j }, we may define (, s) = limj j (j , s) for all s < . It follows that satisfies (iii). Next, we define u : [, ] X by + S((t, s))r(s) ds, for all t [, ]. We have u C([, ]; X) and u(s) = uj (s), for all j 1 and all s [, j ]. Since u() = limt u(t) = limj u(j ) = limj uj (j ), and uj (j ) D((0), ) K(j ) and the latter is closed MIHAI NECULA and MARIUS POPESCU from the left, we deduce that u() D((0), ) K(). The rest of conditions in lemma being obviously satisfied, it follows that (P , f, , r, u) is an upper bound for the sequence. Thus, the partially ordered set (U, ) and the function N : (U, ) R, defined by N(P , f, , r, u) = , for each (P , f, , r, u) U, satisfy the hypotheses of the Brezis-Browder Ordering ^ Principle, i.e. Theorem 2.1.1, p. 30 in Carja, Necula and Vrabie [4]. Accordingly, there exists an N-maximal element in U. This means thahere exists (P , f , , r , u ) U such that, whenever (P , f , , r , u ) (P , f , , r, u), we necessarily have N(P , f , , r , u ) = N(P , f , , r, u). We will show that = T . To this aim, let us assume by contradiction that < T . Since ( , u ) K and using the tangency condition (4.2) we deduce thahere exist hn 0, gn FF ( ,u ) and pn X, with pn 0, such that S(hn )u ( ) + n S( + hn - s)gn (s) ds + hn pn K( + hn ), for every n N, n 1. Let n0 N and = + hn0 such that ( , T ), hn0 < and pn0 < . Let us define P = P {[ , ]}, (t, s), s t (t, s) = t - + ( , s), s < < t < , 0, s < t f (s) = f (s), s [, ] gn0 (s), s ( , ] , r(s) = r (s), s [, ] pn0 , s ( , ] , u (t), t [, ] u(t) = Since, by (vii) we have u S (, ) and using the relation (2), it follows that f (s) M0 a.e. for s (, ). Clearly (i)(vii) are satisfied, and we can diminish (increase n0 ) in order that (viii) be satisfied too. (P , f , , r, u), but < So, (P , f , , r, u) U, (P , f , , r , u ) which contradicts the maximality of (P , f , , r , u ). Hence = T , and P , f , , r and u satisfy all the conditions (i)(vii). The proof is complete. S(t - )u ( ) + S(t - s)gn0 (s) ds + (t - )pn0 , t ( , ]. Definition 4.4. Let >0. A quintuple (PT , f, , r, u) satisfying (i)(viii) in Lemma 4.1, is called an -approximate solution of (1.1). Remark 4.3. Lemma 4.1 offers a sufficient condition in order that, for each > 0 to exists at least one -approximate solution of (1.1). Next, let us prove Theorem 4.1. Proof. Since the necessity follows from Theorem 3.2, we will confine ourselves only to the proof of the sufficiency. Let > 0 and T > and M0 be as in Lemma 4.1. Let n (0, 1), with n 0. Let ((Pn , fn , n , rn , un ))n be a sequence of n -approximate solutions of T (1.1), sequence given by Lemma 4.1. If Pn = {[tn , sn ); m n } with m m T n finite or at most countable, we denote by an : [, T ) [, T ) the step function, defined by an (s) = tn for each s [tn , sn ). Clearly m m m (4.3) uniformly for s [, T ). In view of (vi), we have lim an (s) = s (4.4) un (t) = S(t - )(0) + S(t - s)fn (s) ds + S(n (t, s))rn (s) ds for each n N and t [, T ]. We will show that, on a subsequence at least, (un )n is uniformly convergent on [, T ] to some function u which will turn ouo be a mild solution for the problem (1.1). We analyze firshe case when X is separable. Using (v) from Lema 4.1 we deduce that, for each t [, T ], we have (4.5) ({ S(n (t, s))rn (s) ds; n 1}) From (iv) we get (4.6) fn (t) M0 for all n 1 and a.e. for t [, T ]. Using (viii) we deduce limn un (an (t)) - un (t) = 0 uniformly for t [, T ) and from here we get (4.7) ({un (an (t)); n 1}) = ({un (t); n 1}). MIHAI NECULA and MARIUS POPESCU Le [, T ). Denote by x(t) = ({un (t); n 1}) and Bt = {(un )an (t) ; n 1}. By applying the function in (4.4), we obtain x(t) ({ S(t - s)fn (s) ds; n 1}) + ({ S(n (t, s))rn (s) ds; n 1}) ({S(t - s)fn (s); n 1}) ds. From (iv) in Lemma 4.1, we deduce that, for all n, k N, n k and a.e. for s [, T ), we have fn (s)F (an (s), (un )an (s) ) F (([s-n , s]×Bs )K) F (([s-k , s]×Bs )K) From here and the fachat A + F is -compact we gehat, a.e. for s [, t) ({S(t - s)fn (s); n 1}) = lim ({S(t - s)fn (s); n k}) lim (S(t - s)F (([s - k , s] × Bs ) K)) m(t - s)(s, (Bs (0))). Let 0 = (sups[0,T - ] m(s)). It follows that x(t) 0 (s, ({un (an (s)); n 1})) ds = 0 (s, x(s)) ds. Since 0 is a Carath´odory uniqueness function, too, we deduce that e x(t) = 0 for all t [, T ). So, {un (t); n 1} is relatively compact for all t [, T ). By Theorem 8.4.1, p. 194, in Vrabie [19], we conclude that (un )n has at least one uniformly convergent subsequence to some function u, subsequence denoted, for simplicity, again by (un )n . Since an (t) t, limn un (an (t)) = u(t), uniformly for t [, T ) and t K(t) D((0), ) is closed from the left, we gehat u(t) K(t) for all t [, T ]. But limn (un )an (t) = ut in C , uniformly for t [, T ). Hence, the set C = {(an (t), (un )an (t) ); n 1, t [, T )} is compact and C K. Since F is strongly-weakly u.s.c. and has weakly ^ compact values, by Lemma 2.6.1, p. 47, in Carja, Necula and Vrabie [4], it follows thahe set B = conv n1 t[,T ) F (an (t), (un )an (t) ) is weakly compact. We notice that fn (s) B for every n 1 and a.e. for ^ s [, T ], hence, by Diestel's Theorem 1.3.8, p.10, in Carja, Necula and Vrabie [4], it follows that, on a subsequence at least, limn fn = f weakly in L1 (, T ; X). As F is strongly-weakly u.s.c. with closed and convex values, and, by Lemma 4.1, for each n 1, we have fn (s) F (an (s), (un )an (s) ) a.e. for s [, T ], from Theorem 3.1.2, p. 88, in Vrabie [18], we conclude that f (s) F (s, us ) a.e. for s [, T ]. Finally, passing to the limit both sides in (4.4), for n , we ge , for each t [, T ]. If X is not separable, we have to observe thahere exists a separable and closed subspace Y X such thahe families: {S(·)fn (·); n 1}, {S(·)un (·); n 1} and {S(·)rn (·); n 1} are Y -valued. Then, to complete the proof, it suffices to follows the very same arguments as before and to make use of (iv) in Remark 2.1. 5. A comparison result Let X be a real Banach space, let C X be a closed convex cone and C (-C) = {0}, let " " be the partial order on X defined by C, i.e., x y if and only if y - x C. Let A : D(A) X X be the infinitesimal generator of a C0 -semigroup {S(t); t 0} and let a : I X be a continuous function. Let K : I X be defined by K(t) = {x X; a(t) x} for each t I. Let K = {(t, ) I × C ; (0) K(t)} and F : K X be a given multi-function. We are interested in finding sufficient conditions in order that K be mild viable with respeco A + F , i.e., in order that, for each (, ) K, to exists at least one mild solution u : [, T ] X of the problem u (t) Au(t) + F (t, ut ) (5.1) u = a(t) u(t) for each t [, T ]. The next lemma is, essentially, Lemma 8.1 in Necula, Popescu and ^ Vrabie [16]. See also Carja, Necula and Vrabie [6]. 1,1 Lemma 5.1. Let a Wloc (I; X) and let K be as above. Let I be a point of differentiability from the right for a, with a( ) D(A), let C and let E B(X). Let us assume that S(t)C C for each t 0. Then, the following two conditions : MIHAI NECULA and MARIUS POPESCU (i) E TSA (, a + ) K (ii) Aa( ) - a ( ) + E TSA () (see relation(2.5)) C are equivalent. Moreover, (iii) dist(Aa( ) - a ( ) + E; C) = 0 implies both (i) and (ii). The nexheorem is obtained from Theorem 4.1 and the above lemma. Theorem 5.1. Let A : D(A) X X be the infinitesimal generator of a C0 -semigroup {S(t); t 0}, let a : I D(A), a C 1 (I; X), K be as above and F : K X be a nonempty, convex and weakly compact valued multi-function. Let us suppose that S(t)C C for every t > 0, F is strongly-weakly u.s.c., locally bounded and A + F is -compact. Then, any of the nexwo conditions is a sufficient condition in order that K be mild viable with respeco A + F . (5.2) Aa( ) - a ( ) + F (, ) TSA ((0) - a( )) C for all (, ) K; (5.3) for all (, ) K. Proof. From Lemma , we deduce that (5.3) implies (5.2) which, in turn, implies that F (, ) TSA (, a + (0) - a( )) = TSA (, ). The conclusion K K follows from Theorem 4.1. Remark 5.1. Since F has convex and weakly compact values and C is closed and convex, it results that (5.3) is equivalent with: for all (, ) K, there exists F (, ) such that (5.4) a ( ) - Aa( ) . dist(Aa( ) - a ( ) + F (, ); C) = 0 If, in addition, F has compact values, then (5.2) is equivalent with: (5.5) A [Aa( ) - a ( ) + F (, )] TC ((0) - a( )) = for all (, ) K.

Journal

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

Published: Jan 1, 2015

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