FILIPPOV WAŻEWSKI THEOREMS AND STRUCTURE OF SOLUTION SETS FOR FIRST ORDER IMPULSIVE SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS

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1 Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 32, 28, FILIPPOV WAŻEWSKI THEOREMS AND STRUCTURE OF SOLUTION SETS FOR FIRST ORDER IMPULSIVE SEMILINEAR FUNCTIONAL DIFFERENTIAL INCLUSIONS Smaïl Djebali Lech Górniewicz Abdelghani Ouahab Abstract. In this paper, we first present an impulsive version of Filippov s Theorem for first-order semilinear functional differential inclusions of the form: 8 >< >: (y Ay) F (t, y t) a.e. t J \ {t 1,..., t m}, y(t + k ) y(t k ) = I k(y(t k )) for k = 1,..., m, y(t) = φ(t) for t [ r, ], where J = [, b], A is the infinitesimal generator of a C -semigroup on a separable Banach space E and F is a set-valued map. The functions I k characterize the jump of the solutions at impulse points t k (k = 1,..., m). Then the convexified problem is considered and a Filippov Ważewski result is proved. Further to several existence results, the topological structure of solution sets closeness and compactness is also investigated. Some results from topological fixed point theory together with notions of measure on noncompactness are used. Finally, some geometric properties of solution sets, AR, R δ -contractibility and acyclicity, corresponding to Aronszajn Browder Gupta type results, are obtained. 2 Mathematics Subject Classification. 34A37, 34A6, 34K3, 34K45, 47D6, 47H9, 47H1. Key words and phrases. Impulsive functional differential inclusions, mild solution, Filippov s theorem, relaxation, solution set, compactness, AR, R δ, contractibility, acyclicity. c 28 Juliusz Schauder Center for Nonlinear Studies 261

2 262 S. Djebali L. Górniewicz A. Ouahab 1. Introduction Differential equations with impulses were considered for the first time by Milman and Myshkis [43] and then followed by a period of active research which culminated with the monograph by Halanay and Wexler [31]. The dynamics of many processes in physics, population dynamics, biology, medicine may be subject to abrupt changes such that shocks, perturbations (see for instance [1], [4] and the references therein). These perturbations may be seen as impulses. For instance, in the periodic treatment of some diseases, impulses correspond to the administration of a drug treatment. In environmental sciences, impulses correspond to seasonal changes of the water level of artificial reservoirs. Their models are described by impulsive differential equations and inclusions. Important contributions to the study of the mathematical aspects of such equations have been undertaken in [8], [41], [49], [51] among others. Functional differential equations with impulsive effects with fixed moments have been recently addressed by Yujun and Erxin [57] and Yujun [56]. For further readings on functional differential equations, we recommend the monographs by Azbelez et. al [7] or by J. Hale and S. M. V. Lunel [3]. Some existence results on impulsive functional differential equations with finite or infinite delay may be found in [9], [11], [45], [46] too. During the last couple of years, impulsive ordinary differential inclusions and functional differential inclusions with different conditions have been intensively studied (see the book by Aubin [4] and also [12], [23], [32], [33], [52] and the references therein). Given a real separable Banach space E with norm, we will consider in this paper the impulsive problem for first-order semilinear differential inclusions (1.1) (y Ay)(t) F (t, y t ) for a.e. t J \ {t 1,..., t m }, y t=tk = I k (y(t k )) for k = 1,..., m, y(t) = φ(t) for t [ r, ], where < r <, = t < t 1 <... < t m < t m+1 = b, J = [, b]. F : J D P(E) is a multifunction, and φ D where D = {ψ: [ r, ] E: ψ is continuous everywhere except for a finite number of points t at which ψ(t ) and ψ(t + ) exist and satisfy ψ(t ) = ψ(t)}. The operator A is the infinitesimal generator of a C -semigroup {T (t)} t (see Section 2), I k C(E, E) (k = 1,..., m) and y t=tk = y(t + k ) y(t k ). The notations y(t + k ) = lim h + y(t k + h) and y(t k ) = lim h + y(t k h) stand for the right and the left limits of the function y at t = t k, respectively. For any function y defined on [ r, b] and any t J, y t refers to the element of D such that y t (θ) = y(t + θ), θ [ r, ];

3 Semilinear Differential Inclusions 263 thus the function y t represents the history of the state from time t r up to the present time t. Some auxiliary results needed in this paper are gathered together in Section 2. In this work, we shall be mainly concerned with the Filippov s theorem for first order impulsive semilinear functional differential inclusions in a Banach space. This is presented and developed in Section 3. In Section 4, we discuss the relaxed problem associated to problem (1.1), that is the problem when we consider the closure of the convex hull of the right-hand side instead. This corresponds to a Filippov Ważewski approach; we prove that the solution set of problem (1.1) is dense in that of the convexified problem. Then some topological properties of the operator solution and of the solution sets (closeness and compactness) are provided in Section 5. In addition, some existence results are obtained. Finally, Section 6 is devoted to proving some geometric properties of solution sets such that acyclicity, AR, R δ contractibility, and R δ -contractibility. We end the paper with some concluding remarks and a rich bibliography. 2. Preliminaries In this section, we recall some notations, definitions, and preliminary facts which will be used throughout. Let [, b] be a interval in R and C([, b], E) be the Banach space of all continuous functions from [, b] into E with the norm y = sup{ y(t) : t b}. B(E) refers to the Banach space of linear bounded operators from E into E with norm N B(E) = sup{ N(y) : y = 1}. A function y: J E is called measurable provided for every open subset U E, the set y 1 (U) = {t J: y(t) U} is Lebesgue measurable. A measurable function y: J E is Bochner integrable if y is Lebesgue integrable. For properties of the Bochner integral, see e.g. Yosida [55]. In what follows, L 1 (J, E) denotes the Banach space of functions y: J E, which are Bochner integrable with norm y L 1 = b y(t) dt Multivalued analysis. Denote by P(E) = {Y E : Y }, P cl (E) = {Y P(E) : Y closed}, P b (E) = {Y P(E) : Y bounded}, P cv (E) = {Y P(E) : Y convex}, P cp (E) = {Y P(E) : Y compact}, and P wkcp (E) = {Y P(E) : Y weakly compact}. Consider the Hausdorff pseudo-metric distance H d : P(E) P(E) R + { }

4 264 S. Djebali L. Górniewicz A. Ouahab defined by { H d (A, B) = max sup a A } d(a, B), sup d(a, b) b B where d(a, b) = inf a A d(a, b) and d(a, B) = inf b B d(a, b). Then (P b,cl (E), H d ) is a metric space and (P cl (X), H d ) is a generalized metric space (see [39]). In particular, H d satisfies the triangle inequality. Definition 2.1. A multivalued operator N: E P cl (E) is called (a) γ-lipschitz if there exists γ > such that H d (N(x), N(y)) γd(x, y), for each x, y E, (b) a contraction if it is γ-lipschitz with γ < 1. Notice that if N is γ-lipschitz, then for every γ > γ, N(x) N(y) + γ d(x, y)b(, 1), for all x, y A. Let (X, d) and (Y, ρ) be two metric spaces and G: X P cl (Y ) be a multivalued mapping. A single-valued map g: X Y is said to be a selection of G and we write g G whenever g(x) G(x) for every x X. G is called upper semi-continuous (u.s.c. for short) on X if for each x X the set G(x ) is a nonempty, closed subset of X, and if for each open set N of Y containing G(x ), there exists an open neighbourhood M of x such that G(M) Y. That is, if the set G 1 (V ) = {x X : G(x) V } is closed for any closed set V in Y. Equivalently, G is u.s.c. if the set G +1 (V ) = {x X : G(x) V } is open for any open set V in Y. G is said to be completely continuous if it is u.s.c. and, for every bounded subset A X, G(A) is relatively compact, i.e. there exists a relatively compact set K = K(A) X such that G(A) = {G(x) : x A} K. G is compact if G(X) is relatively compact. It is called locally compact if, for each x X, there exists U V(x) such that G(U) is relatively compact. We denote the graph of G to be the set Gr(G) = {(x, y) X Y : y G(x)} and recall Lemma 2.2 ([18, Proposition 1.2]). If G: X P cl (Y ) is u.s.c. then Gr(G) is a closed subset of X Y, i.e. for every sequences (x n ) n N X and (y n ) n N Y, if when n, x n x, y n y and y n G(x n ), then y G(x ). Conversely, if G has nonempty compact values, is locally compact and has a closed graph, then it is u.s.c. The following two lemmas are concerned with measurability of multi-functions; they will be needed in this paper. The first one is the celebrated Kuratowski Ryll Nardzewski selection theorem.

5 Semilinear Differential Inclusions 265 Lemma 2.3 (see [27, Theorem 19.7]). Let E be a separable metric space and G a measurable multi-valued map with nonempty closed values. Then G has a measurable selection. Lemma 2.4 (see [58, Lemma 3.2]). Let G: [, b] P(E) be a measurable multifunction and u: [, b] E a measurable function. Then for any measurable v: [, b] R + there exists a measurable selection g of G such that for almost every t [, b], u(t) g(t) d(u(t), G(t)) + v(t). Finally, for a multi-valued function G: J D P(E), denote G(t, z) P := sup{ v : v G(t, z)}. Definition 2.5. G is called a multi-valued Carathéodory function if (a) the function t G(t, z) is measurable for each z D, (b) for almost every t J, the map z G(t, z) is upper semi-continuous. It is further an L 1 -Carathéodory if it is locally integrably bounded, i.e. for each positive real number r, there exists some h r L 1 (J, R + ) such that G(t, z) P h r (t) for a.e. t J and all z D r. For further readings and details on multivalued analysis, we refer to the books by Andres and Górniewicz [2], Aubin and Celina [5], Aubin and Frankowska [6], Deimling [18], Górniewicz [27], Hu and Papageorgiou [36], Kamenskiĭ [38], and Tolstonogov [54] C -semigroups. Definition 2.6. A semigroup of class C (or C -semigroup) is a one parameter family {T (t) : t } B(E) satisfying the conditions: (a) T (t) T (s) = T (t + s), for t, s, (b) T () = I. Here I denotes the identity operator in E. Definition 2.7. A semigroup T (t) is uniformly continuous if lim t + T (t) T () B(E) =, that is if lim t s T (t) T (s) B(E) =. Definition 2.8. We say that the semigroup {T (t) t } is strongly continuous (or a C -semigroup) if the map t T (t)(x) is strongly continuous, for each x E, i.e. lim t + T (t)x = T ()x, for all x E.

6 266 S. Djebali L. Górniewicz A. Ouahab Definition 2.9. Let T (t) be a C -semigroup defined on E. The infinitesimal generator A B(E) of T (t) is the linear operator defined by T (t)(x) T ()x A(x) = lim, for x D(A), t + t where D(A) = {x E : lim t +(T (t)(x) x)/t exists in E}. The following properties are classical (see Pazy [5], Engel and Nagel [2], Hill and Philips [35]). Proposition 2.1. Let {T (t)} t be a uniformly continuous semigroup of bounded linear operators. Then there exists some constant ω such that T (t) B(E) exp(ωt), for t. Proposition (a) If {T (t)} t is a C -semigroup of bounded linear operators, then there exist constants ω and M 1 such that T (t) B(E) M exp(ωt), for t. (b) If A is the infinitesimal generator of a C -semigroup {T (t)} t, then D(A), the domain of A, is dense in X and A is a closed linear operator Mild solutions. Let J k = (t k, t k+1 ], k =,..., m, and let y k be the restriction of a function y to J k. In order to define mild solutions for problem (1.1), consider the space P C = {y: [, b] E, y k C(J k, E), k =,..., m, such that y(t k ) and y(t+ k ) exist and satisfy y(t k ) = y(t k) for k = 1,..., m}. Endowed with the norm y P C = max{ y k, k =,..., m}, P C is a Banach space. Moreover, if Ω = {y: [ r, b] E, y P C D}, then Ω is a Banach space with the norm y Ω = max{ y P C, y D }, where y D = sup t [ r,] y(t). Throughout this paper, A is an infinitesimal generator of a C -semigroup {T (t)} t and the constants M > and ω are as introduced in Proposition A fundamental notion for the definition of solutions to problem (1.1) is given by Definition A function y Ω is said to be a mild solution of problem (1.1) if there exists v L 1 (J, E) such that v(t) F (t, y t ) almost everywhere on J, y(t) = φ(t), t [ r, ] and y(t) = T (t)φ() + T (t s)v(s) ds + <t k <t T (t t k )I k (y(t k )).

7 Semilinear Differential Inclusions Filippov s Theorem Regarding differential equations and inclusions, some existence results for problem (1.1) can be found in [47]. Further results will be given subsequently in this paper. In this section, we are mainly concerned with a Filippov s result for problem (1.1). Such results are of great importance in stability and control theory. In the finite dimensional case, the problem was investigated by Filippov [21] in 1967 for first-order differential inclusions and later by Frankowska [23] in 199 for first-order semilinear differential inclusions; see e.g. also Aubin and Cellina [5, Theorem 1, p. 12], Aubin and Frankowska [6, Theorem 1.4.1, p. 41]). When E is not necessarily separable, interesting results are given in [58]. Filippov s Theorem yields an estimate of the distance of a given solution to the solution set of a problem providing a kind of Gronwall s inequality (see also [53, Theorem 4.5, p. 91]) Filippov s Theorem on a bounded interval. Let ψ D, g L 1 (J, E) and let x Ω be a mild solution of the impulsive differential problem with semi-linear equation: (3.1) x (t) Ax(t) = g(t) for a.e. t J \ {t 1,..., t m }, x t=tk = I k (x(t k )) for k = 1,..., m, x(t) = ψ(t) for t [ r, ]. We will consider the following two assumptions (H 1 ) The function F : J D P cl (E) is such that (a) for all y D the map t F (t, y) is measurable, (b) the map γ: t d(g(t), F (t, x t )) is integrable. (H 2 ) There exist a function p L 1 (J, R + ) and a positive constant β > such that H d (F (t, z 1 ), F (t, z 2 )) p(t) z 1 z 2 D, for all z 1, z 2 B(x t, β), where B(x t, β) is the closed ball of D centered in x t with radius β. Remark 3.1. From Assumptions (H 1 )(a) and (H 2 ), it follows that the multi-function t F (t, x t ) is measurable and by Lemmas 1.4, 1.5 from [23], we deduce that γ(t) = d(g(t), F (t, x t )) is measurable (see also Remark p. 4 in [6]). Let P (t) = p(s) ds and δ be a positive constant. Define the family of functions (η k (t)) t (k =,..., m) by η (t) = Me ωt1 δ + Me ωt1 [Me ωt1 H (s)p (s) + γ(s)] ds, t (, t 1 ]

8 268 S. Djebali L. Górniewicz A. Ouahab where H (t) = δm exp(me ωt+p (t) ) + M and for k = 1,..., m where η k (t) = Me ωt k+1 H k (t) = δ exp(me ω(t t k)+p (t) ) + γ(s) exp(me ωt+p (t) P (s) ) ds, t k [Me ωt H k (s)p (s) + γ(s)] ds, t (t k, t k+1 ], t k γ(s) exp(me ω(t t k)+p (t) P (s) ) ds. Theorem 3.2. Let γ k := γ Jk and assume that η k 1 (t k ) β for k = 1,..., m. Then, for every φ D with φ ψ D δ, problem (1.1) has at least one solution y satisfying, for almost every t [, b], the estimates y t x t D M e ω(t tk) [ y(t k ) x(t k ) + I k (y(t k )) I k (x(t k )) ] and <t k <t y (t) Ay(t) g(t) Mp(t) <t k <t e ωt k+1 H k (t) + <t k <t + <t k <t γ k (t). η k (t), Proof. We are going to study problem (1.1) respectively in the intervals [ r, t 1 ], (t 1, t 2 ],..., (t m, b]. The proof will be given in three steps and then continued by induction. Let φ D be such that φ ψ D δ. Step 1. In this first step, we construct a sequence of function (y n ) n N which will be shown to converge to some solution of problem (1.1) on the interval [ r, t 1 ], namely to { (y (t) Ay(t)) F (t, y t ) for t J = (, t 1 ], y(t) = φ(t) for t [ r, ]. Let f = g on [ r, t 1 ] and y (t) = x(t), t [, t 1 ), i.e. ψ(t) for t [ r, ], y (t) = T (t)ψ() + T (t s)f (s) ds for t (, t 1 ], Then define the multi-valued map U 1 : [, t 1 ] P(E) by U 1 (t) = F (t, yt ) B(g(t), γ(t)). Since g and γ are measurable, Theorem III.4.1 in [16] tells us that the ball B(g(t), γ(t)) is measurable. Moreover F (t, yt ) is measurable (see Remark 3.1) and U 1 is nonempty. Indeed, since v = is a measurable function,

9 Semilinear Differential Inclusions 269 from Lemma 2.4, there exists a function u which is a measurable selection of F (t, y t ) and such that u(t) g(t) d(g(t), F (t, y t )) = γ(t). Then u U 1 (t), proving our claim. We deduce that the intersection multivalued operator U 1 (t) is measurable (see [6], [16], [27]). By Lemma 2.3 (Kuratowski Ryll Nardzewski selection theorem), there exists a function t f 1 (t) which is a measurable selection for U 1. Consider φ(t) for t [ r, ], y 1 (t) = T (t)φ() + T (t s)f 1 (s) ds for t [, t 1 ]. For each t [, t 1 ], we have by Proposition 2.11 (3.2) Hence and then y 1 (t) y (t) Me ωt φ() ψ() + M sup t [ r,t 1] y 1 t y t D = e ω(t s) f (s) f 1 (s) ds Me ωt1 φ ψ D + Me ωt1 f (s) f 1 (s) ds. { y 1 (t) y (t) } Me ωt1 δ + Me ωt1 γ(s) ds sup yt 1 (θ) yt (θ) = sup y 1 (t + θ) y (t + θ) θ [ r,] θ [ r,] = sup y 1 (s) y (s) η (t 1 ) β. θ [ r+t,t] Then Lemma 1.4 in [23] tells us that F (t, y 1 t ) is measurable. The ball B(f 1 (t), p(t) y 1 t y t D ) is also measurable by Theorem III.4.1 in [16]. The set U 2 (t) = F (t, y 1 t ) B(f 1 (t), p(t) y 1 t y t D ) is nonempty. Indeed, since f 1 is a measurable function, Lemma 2.4 yields a measurable selection u of F (t, y 1 t ) such that u(t) f 1 (t) d(f 1 (t), F (t, y 1 t ). Moreover, y 1 t y t D η (t 1 ) β. Then using (H 2 ), we get u(t) f 1 (t) d(f 1 (t), F (t, y 1 t )) H d (F (t, y t ), F (t, y 1 t )) p(t) y t y 1 t D, i.e. u U 2 (t), proving our claim Now, since the intersection multi-valued operator U 2 defined above is measurable (see [6], [16], [27]), there exists a measurable selection f 2 (t) U 2 (t). Hence (3.3) f 1 (t) f 2 (t) p(t) y 1 t y t D.

10 27 S. Djebali L. Górniewicz A. Ouahab Define φ(t) for t [ r, ], y 2 (t) = T (t)φ() + T (t s)f 2 (s) ds for t (, t 1 ]. Using (3.2) and (3.3), a simple integration by parts yields the following estimates, valid for every t [ r, t 1 ], y 2 (t) y 1 (t) T (t s) f 2 (s) f 1 (s) ds s ) Me ωt1 p(s) (Me ωt1 δ + Me ωt1 γ(u) du ds = M 2 e 2ωt1 (δ M 2 e 2ωt1 (δ M 2 e 2ωt1 (δe P (t) + p(s) ds + p(s)e P (s) ds + p(s) ds s ) γ(u) du p(s)e P (s) ds ) γ(s)e P (t) P (s) ds. s ) e P (u) γ(u) du Let U 3 (t) = F (t, y 2 t ) B(f 2 (t), p(t) y 2 t y 1 t D ). Arguing as for U 2, we can prove that U 3 is a measurable multi-valued map with nonempty values; so there exists a measurable selection f 3 (t) U 3 (t). This allows us to define φ(t) for t [ r, ], y 3 (t) = T (t)φ() + T (t s)f 3 (s) ds for t (, t 1 ]. For t [, t 1 ], we have y 3 (t) y 2 (t) Me ωt1 f 2 (s) f 3 (s) ds Me ωt1 p(s) ys 2 ys 1 D ds. However y 2 s y 1 s D = sup ys(θ) 2 ys(θ) 1 = sup y 2 (s + θ) y 1 (s + θ), θ [ r,] θ [ r,] and from the estimates above, for θ [ r, ], we have y 2 (s + θ) y 1 (s + θ) M 2 e 2ωt1 (δe P (θ+s) + M 2 e 2ωt1 (δe P (s) + s θ+s ) γ(u)e P (θ+s) P (u) du ). γ(u)e P (θ+s) P (u) du

11 Semilinear Differential Inclusions 271 Performing an integration by parts, we obtain, since P is a nondecreasing function, the following estimates y 3 (t) y 2 (t) M 3 e 3ωt1 2 M 3 e 3ωt1 2 M 3 e 3ωt1 2 M 3 e 3ωt1 2 M 3 e 3ωt1 2 ( 2p(s) δe 2P (s) + ( δe 2P (t) + ( δe 2P (t) + s 2p(s) ds s (e 2P (s) ) ds ) γ(u)e P (s) P (u) du ds s (δe 2P (t) + e 2P (t) γ(s)e 2P (s) ds ( δe 2P (t) + ) γ(s)e 2(P (t) P (s)) ds, ) γ(u)e 2(P (s) P (u)) du ) γ(u)e 2P (u)) du ) γ(s) ds for t [ r, t 1 ]. Let U 4 (t) = F (t, yt 3 ) B(f 3 (t), p(t) yt 3 yt 2 D ). Then, arguing again as for U 1, U 2, U 3, we show that U 4 is a measurable multi-valued map with nonempty values and that there exists a measurable selection f 4 (t) in U 4 (t). Define φ(t) for t [ r, ], y 4 (t) = T (t)φ() + T (t s)f 4 (s) ds for t (, t 1 ]. For t [, t 1 ], we have y 4 (t) y 3 (t) Me ωt1 f 4 (s) f 3 (s) ds Me ωt1 p(s) ys 3 ys 2 D ds M 4 e 4ωt1 t ( s ) p(s) δe 2P (s) + γ(s)e 2(P (s) P (u)) du ds 2 M 4 e 4ωt1 δ 6 M 4 e 4ωt1 6 3p(s)e 3P (s) ds + M 4 e 4ωt1 3p(s)e 3P (s) ds 6 ) γ(s)e 3(P (t) P (s)) ds. ( δe 3P (t) + s γ(s)e 3P (u) du Repeating the process for n =, 1,..., we arrive at the following bound (3.4) y n (t) y n 1 (t) M n e nωt1 (n 1)! γ(s)e (n 1)(P (t) P (s)) ds + M n e nωt1 (n 1)! δe(n 1)P (t), for t [ r, t 1 ]. By induction, suppose that (3.4) holds for some n and check it for n + 1. Let U n+1 (t) = F (t, y n (t)) B(f n, p(t) yt n yt n 1 D ). As in the above arguing, U n+1 is a nonempty measurable set, then has a measurable selection

12 272 S. Djebali L. Górniewicz A. Ouahab f n+1 (t) U n+1 (t); this allows us to define for n N φ(t) for t [ r, ], (3.5) y n+1 (t) = T (t)φ() + T (t s)f n+1 (s) ds for t (, t 1 ]. Therefore, for almost every t [, t 1 ], we have y n+1 (t) y n (t) Me ωt1 f n+1 (s) f n (s) ds M n+1 e (n+1)ωt1 (n 1)! M n+1 e (n+1)ωt1 (n 1)! M n+1 e (n+1)ωt1 n! + M n+1 e (n+1)ωt1 n! p(s) ys n ys n 1 D ds ( p(s) ds δe (n 1)P (s) + δnp(s)e np (s) ds Again, an integration by parts yields np(s)e np (s) ds y n+1 (t) y n (t) M (n+1) e (n+1)ωt1 n! s s ) γ(u)e (n 1)(P (s) P (u)) du γ(u)e np (u) du. γ(s)e n(p (t) P (s)) ds + M (n+1) e (n+1)ωt1 δe np (t). n! Consequently, (3.4) holds true for all n N. We infer that {y n } is a Cauchy sequence in Ω 1, converging uniformly to a limit function y Ω 1, where Ω 1 = C([, t 1 ], E) D. Moreover, from the definition of {U n }, we have f n+1 (t) f n (t) p(t) y n t y n 1 t D, for a.e. t [, t 1 ]. Hence, for almost every t [, t 1 ], {f n (t)} is also a Cauchy sequence in E and then converges almost everywhere to some measurable function f( ) in E. In addition, since f = g, we have, for almost every t [, t 1 ] Hence f n (t) n p(t) f k (t) f k 1 (t) + f (t) k=1 n p(t) y k 1 (t) y k 2 (t) + γ(t) + g(t) k=1 p(t) y k (t) y k 1 (t) + γ(t) + g(t). k=1 f n (t) Me ωt1 H (t)p(t) + γ(t) + g(t),

13 where (3.6) H (t) := δm exp(me ωt+p (t) ) + M Semilinear Differential Inclusions 273 γ(s) exp(me ωt+p (t) P (s) ) ds. From the Lebesgue dominated convergence theorem, we deduce that {f n } converges to f in L 1 ([, t 1 ], E). Passing to the limit in (3.5), we find that the function φ(t) for t [ r, ], y(t) = y(t) = T (t)φ() + T (t s)f(s) ds, for t (, t 1 ] is solution to problem (1.1) on [, t 1 ]; thus y S [,t1](φ). Moreover, for almost every t [, t 1 ], we have x(t) y(t) = T (t)φ() + T (t s)g(s) ds T (t)ψ() T (t s)f(s) ds Me ωt1 φ() ψ() + Me ωt1 f(s) f (s) ds Me ωt1 φ ψ D + Me ωt1 f(s) f n (s) ds + Me ωt1 f n (s) f (s) ds Me ωt1 φ ψ D + Me ωt1 f(s) f n (s) ds + Me ωt1 (Me ωt1 H(s)P (s) + γ(s)) ds. Passing to the limit as n, we get (3.7) x(t) y(t) η (t) for a.e. t [ r, t 1 ] with η (t) := Me ωt1 δ + Me ωt1 (Me ωt1 H (s)p (s) + γ(s)) ds. Next, we give an estimate for y (t) Ay(t) g(t) for t [, t 1 ]. We have y (t) Ay(t) g(t) = f(t) f (t) f n (t) f (t) + f n (t) f(t) p(t) y k+1 (t) y k (t) + γ(t) + f n (t) f(t). k=1 Arguing as in (3.6) and passing to the limit as n, we deduce that y (t) Ay(t) g(t) Me ωt1 H (t)p(t) + γ(t), t [, t 1 ]. The obtained solution is thus denoted by y 1 := y [ r,t1].

14 274 S. Djebali L. Górniewicz A. Ouahab Step 2. Consider now problem (1.1) on the second interval [t 1 r, t 2 ], i.e. (y (t) Ay(t)) F (t, y t ) for a.e. t (t 1, t 2 ], (3.8) y(t + 1 ) = y 1(t 1 ) + I 1 (y 1 (t 1 )), y(t) = y 1 (t) for t [t 1 r, t 1 ]. Let f = g and set y 1 (t) for t [t 1 r, t 1 ], y (t) = T (t t 1 )[x(t 1 ) I 1 (x(t 1 ))] + T (t s)f (s) ds for t (t 1, t 2 ]. t 1 Notice that (3.7) allows us to use Assumption (H 2 ), apply again Lemma 1.4 in [23] and argue as in Step 1 to prove that the multi-valued map U 1 : [t 1, t 2 ] P(E) defined by U 1 (t) = F (t, yt ) B(g(t), γ(t)) is U 1 (t) is measurable. Hence, there exists a function t f 1 (t) which is a measurable selection for U 1. Define y 1 (t) for t [t 1 r, t 1 ], y 1 (t)= T (t t 1 )[y 1 (t 1 ) + I 1 (y 1 (t 1 ))]+ T (t s)f 1 (s) ds for t (t 1, t 2 ]. t 1 Next define the measurable multi-valued map U 2 (t) = F (t, yt 1 ) B(f 1 (t), p(t) yt 1 yt D ). It has a measurable selection f 2 (t) U 2 (t) by the Kuratowski Ryll Nardzewski selection theorem. Repeating the process of selection as in Step 1, we can define by induction a sequence of multi-valued maps U n (t) = F (t, yt n 1 ) B(f n 1 (t), p(t) yt n 1 yt n 2 D ) where {f n } U n and (y n ) n N is as defined by y 1 (t) for t [t 1 r, t 1 ], y n (t)= T (t t 1 )[y 1 (t 1 )+I 1 (y 1 (t 1 ))]+ T (t s)f n (s) ds for t (t 1, t 2 ]. t 1 Let Ω 2 = {y: y D C[, t 1 ] C(t 1, t 2 ] and y(t + 1 ) exists}. As in Step 1, we can prove that the sequence {y n } converges to some y Ω 2 solution to problem (3.8) such that, for almost every t (t 1, t 2 ], we have and x(t) y(t) η (t 1 ) + Me ω(t t1) I 1 (x(t 1 )) I 1 (y 1 (t 1 )) where + Me ω(t t1) x 1 (t 1 ) y 1 (t 1 ) + Me ωt2 t 1 (M ωt2 H 1 (s)p (s) + γ(s)) ds y (t) Ay(t) g(t) Me ωt2 H 1 (t)p(t) + γ(t), H 1 (t) := δ exp(me ω(t t1)+p (t) ) + Denote the restriction y [t1,t 2] by y 2. t 1 γ(s) exp(me ω(t t1)+p (t) P (s) ) ds.

15 Semilinear Differential Inclusions 275 Step 3. We continue this process until we arrive at the function y m+1 := y [tm r,t m] (t m,b] solution of the problem (y (t) Ay(t)) F (t, y t ) for a.e. t (t m, b], y(t + m) = y m 1 (t m ) + I m (y m 1 (t m )), y(t) = y m 1 (t) for t [t m r, t m ]. Then, for almost every t (t m, b], the following estimates are easily derived and x(t) y(t) Me ω(t tm) [ I m (x(t m )) I m (y(t m )) + y m (t m ) x(t m ) ] + Me ωtm t m (Me ωtm H m (s)p (s) + γ(s)) ds y (t) Ay(t) g(t) Me ωb H m (t)p (t) + γ(t)). Step 4. Summarizing, a solution y of problem (1.1) can be defined as follows y 1 (t) if t [ r, t 1 ], y 2 (t) if t (t 1, t 2 ], y(t) = y m+1 (t) if t (t m, b]. From Steps 1 3, we have that, for almost every t [ r, t 1 ], x(t) y(t) η (t) and y (t) Ay(t) g(t) Me ωt1 H (t)p(t) + γ(t), as well as the following estimates, valid for t (t 1, b] x(t) y(t) Similarly m+1 k=2 x(t) y k (t) M + M <t k <t y (t) Ay(t) g(t) Mp(t) <t k <t e ω(t t k) x(t k ) y k (t k ) e ω(t t k) I k (x(t k )) I k (y k (t k )) + <t k <t e ωt k+1 H k (t) + where γ k := γ Jk. The proof of Theorem 3.2 is complete. <t k <t m η k (t). k= γ k (t), 3.2. Filippov s Theorem on the half line. We may consider Filippov s problem on the half-line (y Ay)(t) F (t, y t ) for a.e. t J \ {t 1,... }, (3.1) y t=tk = I k (y(t k )) for k = 1, 2..., y(t) = φ(t) for t [ r, ],

16 276 S. Djebali L. Górniewicz A. Ouahab where J = [, ), < r <, = t < t 1 <... < t m <..., lim m t m =, F : J D P(E) is a multifunction, and φ D where D = g{ψ: [ r, ] E : ψ is continuous everywhere exept for a finite number of points t at which ψ(t ) and ψ(t + ) exist, ψ(t ) = ψ(t) and sup θ [ r,] ψ(θ) < }. Let x be the solution of problem (3.1) on the half-line. We will consider the following assumptions: ( H 1 ) The function F : J D P cl (E) is such that: (a) for all y D the map t F (t, y) is measurable, (b) the map t γ(t) = d(g(t), F (t, x t )) L 1 ([, ), R + ). ( H 2 ) There exist a function p L 1 ([, ), R + ) and a positive constant β > such that H d (F (t, z 1 ), F (t, z 2 )) p(t) z 1 z 2 D, for all z 1, z 2 B(x t, β), where B(x t, β) is the closed ball of D centered in x t with radius β. ( H 3 ) For every x E, I k (x) <. k=1 Then we can extend Filippov s Theorem to the half-line. We have Theorem 3.3. Let γ k := γ Jk and assume ( H 1 ) ( H 3 ) hold together with lim sup η k 1(t k ) β. k Then, for every φ D with φ ψ D δ, problem (3.1) has at least one solution y satisfying, for t [, ), the estimates y t x t D M e ω(t tk) [ y(t k ) x(t k ) + I k (y(t k )) I k (x(t k )) ] + η k (t), <t k <t <t k <t and y (t) Ay(t) g(t) Mp(t) e ωt k+1 H k (t) + γ k (t). <t k <t <t k <t Proof. The solution will be sought in the space P C = {y: [, ) E : y k C(J k, E), k =, 1,... such that y(t k ) and y(t+ k ) exist and satisfy y(t k ) = y(t k) for k = 1, 2,... },

17 Semilinear Differential Inclusions 277 where y k is the restriction of y to J k = (t k, t k+1 ], k. Theorem 3.2 yields estimates of y k on each one of the bounded intervals J = [ r, t 1 ], and J k = [t k 1 r, t k ], k = 2, 3,.... Let y be solution of problem (1.1) on J with x t y t D η (t 1 ) β. Then, consider the following problem (y (t) Ay(t)) F (t, y t ) for a.e. t (t 1, t 2 ], y(t + 1 ) = y (t 1 ) + I 1 (y (t 1 )), y(t) = y (t) for t [t 1 r, t 1 ]. From Theorem 3.2, this problem has a solution y 1. We continue this process taking into account that y m := y [tm,b] is a solution to the problem (y (t) Ay(t)) F (t, y t ) for a.e. t (t m, b], y(t + m) = y m 1 (t m ) + I m (y m 1 (t m)), y(t) = y m 1 (t) for t [t m 1 r, t m ]. Then a solution y of problem (3.1) may be rewritten as y 1 (t) if t [ r, t 1 ], y 2 (t) if t (t 1, t 2 ], y(t) = y m (t) if t (t m, t m+1 ], The relaxed problem In this section, we examine whether the solutions of the nonconvex problem are dense in those of the convexified one, that is the problem where the right-hand side is replaced by its convex hull. Such a result is important in optimal control theory whether the relaxed optimal state can be approximated by original states; the relaxed problems are generally much simpler to build. For the problem for first-order differential inclusions, we refer e.g. to [5, Theorm 2, p. 124] or [6, Theorem 1.4.4, p. 42]. More precisely, in this section, we compare trajectories of the following problem (4.1) (y (t) Ay(t)) F (t, y t ) for a.e. t J \ {t 1,..., t m }, y(t + k ) y(t k) = I k (y(t k )) for k = 1,..., m, y(t) = φ(t) for t [ r, ],

18 278 S. Djebali L. Górniewicz A. Ouahab and those of the convexified problem (x (t) Ax(t)) co F (t, x t ) for a.e. t J \ {t 1,..., t m }, (4.2) x(t + k ) x(t k) = I k (x(t k )) for k = 1,..., m, x(t) = φ(t) for t [ r, ], where co A refers to the closure of the convex hull of the set A. We will need the following auxiliary results in order to prove our main relaxation theorem. The first two are concerned with measurability of multi-valued mappings. The third one is due to Mazur, 1933 while the last one is a classical fixed point theorem. Lemma 4.1 ([34]). Let E be a separable Banach space, U: [, b] P cl (E) be a measurable, integrably bounded set-valued map and let t d(, U(t)) be an integrable map. Then the integral b U(t) dt is convex, the map t co U(t) is measurable and, for every ε >, and every measurable selection u of co U(t), there exists a measurable selection u of U such that sup u(s) ds u(s) ds ε and b t [,b] co U(t) dt = b U(t) dt = b co U(t) dt. Lemma 4.2 ([23]). Let E be a separable Banach space and G: [, b] P cl (E) be a measurable, integrably bounded multifunction; then so is s T (b s)g(s). Moreover, if f(s) T (b s)g(s) almost everywhere in [, b], then there exists a measurable selection g(s) G(s) such that f(s) = T (b s)g(s) almost everywhere in [, b]. Lemma 4.3 (Mazur s Lemma, [44, Theorem 21.4]). Let E be a normed space and {x k } k N E be a sequence weakly converging to a limit x E. Then there exists a sequence of convex combinations y m = m k=1 α mkx k, where α mk > for k = 1,..., m and m k=1 α mk = 1, which converges strongly to x. Lemma 4.4 (Covitz Nadler, [16]). Let (X, d) be a complete metric space. If N: X P cl (X) is a contraction, then Fix N. The following hypotheses will be assumed in this section: (H 1 ) The function F : J D P cl (E) satisfies (a) for all y D, the map t F (t, y) is measurable, (b) the map t d(, F (t, )) is integrable. (H 2 ) There exist a function p L 1 (J, R + ) such that H d (F (t, z 1 ), F (t, z 2 )) p(t) z 1 z 2 D for each z 1, z 2 D.

19 Semilinear Differential Inclusions 279 (H 3 ) there exist constants c k such that I k (u 1 ) I k (u 2 ) c k u 1 u 2, for each u 1, u 2 E. Also either E will be, in this section, a reflexive Banach or F : J D P wkcp (E). Then our main contribution is the following Theorem 4.5. Assume that (H 1 ) (H 3 ) hold. Then problem (4.2) has at least one solution. In addition, for all ε > and every solution x of problem (4.2), problem (4.1) has a solution y defined on [, b] satisfying x(t) = y(t), t [ r, ] and x y P C ε. In particular S co [ r,b] (φ) = S [ r,b](φ) where S[ r,b] co = {y : y is a solution to (4.2) on [ r, b] and y(t) = φ(t), t [ r, ]}. Remark 4.6. Notice that the multi-valued map t co F (t, ) also satisfies condition (H 2 ). Proof. Part 1. S[ r,b] co. For this, we first transform problem (4.2) into a fixed point problem and then make use of Lemma 4.4. Consider the problem on the interval [ r, t 1 ], that is (4.3) { (y (t) Ay(t)) co F (t, y t ) for a.e. t [, t 1 ], y(t) = φ(t) for t [ r, ]. It is clear that all solutions of problem (4.3) are fixed points of the multivalued operator N: Ω([ r, t 1 ]) P(Ω[ r, t 1 ]) defined by N(y) := h Ω([ r, t 1]) : φ(t) for t [ r, ], h(t) = T (t)φ() + T (t s)g(s) ds for t (, t 1 ], where g S cof,y = {g L 1 ([, t 1 ], E) : g(t) co F (t, y t ) for a.e. t (, t 1 ]} and Ω([ r, t 1 ]) = D C([, t 1 ], E). To show that N satisfies the assumptions of Lemma 4.4, the proof will be given in two steps. In Steps 3, 4, we study the problem on the intervals (t k, t k+1 ] for k = 1,..., m 1. Step 1. N(y) P cl (Ω([ r, t 1 ]) for each y Ω([ r, t 1 ]). Indeed, let {y n } N(y) be such that y n ỹ in Ω([ r, t 1 ], as n. Then ỹ Ω([ r, t 1 ]) and there exists a sequence g n S co F,y such that y n (t) = φ(t) for t [ r, ] and y n (t) = T (t)φ() + T (t s)g n (s) ds, t (, t 1 ].

20 28 S. Djebali L. Górniewicz A. Ouahab Then {g n } is integrably bounded. Since F (, ) has closed values, let w( ) F (, ) be such that w(t) = d(, F (t, )). From (H 1 ) and (H 2 ), we infer that for almost every t [, t 1 ] g n (t) g n (t) w(t) + w(t) p(t) y P C + d(, F (t, )) := M(t), for all n N, that is g n (t) M(t)B(, 1), for a.e. t [, t 1 ]. Since B(, 1) is weakly compact in the reflexive Banach space E, there exists a subsequence still denoted {g n } which converges weakly to g by the Dunford Pettis theorem. By Mazur s Lemma 4.3, there exists a second subsequence which converges strongly to g in E, hence almost everywhere (see [19, p. 15]). Then the Lebesgue dominated convergence theorem implies that, as n, g n g L 1 and thus y n (t) ỹ(t) with ỹ(t) = φ(t), for almost every t [ r, ] and proving that ỹ N(y). ỹ(t) = T (t)φ() + T (t s)g(s) ds, t (, t 1 ], Step 2. There exists γ < 1 such that H d (N(y), N(y)) γ y y [ r,t1] for each y, y Ω([ r, t 1 ]) where the norm y y [ r,t1] will be chosen conveniently. Indeed, let y, y Ω([ r, t 1 ]) and h 1 N(y). Then there exists g 1 (t) co F (t, y t ) such that for each t [, t 1 ] Since, for each t [ r, t 1 ], h 1 (t) = T (t)φ() + T (t s)g 1 (s) ds. H d (co F (t, y t ), co F (t, y t )) p(t) y t y t D, then there exists some w(t) co F (t, y t ) such that g 1 (t) w(t) p(t) y t y t D, t [, t 1 ]. Consider the multi-map U 1 : [, t 1 ] P(E) defined by U 1 (t) = {w E : g 1 (t) w p(t) y t y t D }. As in the proof of Theorem 3.2, we can show that the multi-valued operator V 1 (t) = U 1 (t) co F (t, y t ) is measurable and takes nonempty values. Then there exists a function g 2 (t), which is a measurable selection for V 1. Thus, g 2 (t) co F (t, y t ) and g 1 (t) g 2 (t) p(t) y y D, for a.e. t [, t 1 ].

21 Semilinear Differential Inclusions 281 For each t [, t 1 ], let h 2 (t) = T (t)φ() + Therefore, for each t (, t 1 ], we have Hence where h 1 (t) h 2 (t) = = = g 1 (s) g 2 (s) ds p(s) T (t s) y s (θ) y s (θ) D ds ( p(s)me ω(t s) sup Mp(s)e ωt e ωs ( Mp(s)e ωt ( Me ωt1 Meωt1 τ r θ sup s r z s sup r θ p(s)e τ R s p(u)du ( T (t s)g 2 (s) ds. ) y s (θ) y s (θ) ds ) y(s + θ) y(s + θ) ds ) y(z) y(z) ds sup e τ r z t 1 (e τ R s p(u)du ) y y [ r,t1] ds. h 1 h 2 [ r,t1] Meωt1 y y [ r,t1], τ ) R z p(u)du y(z) y(z) ds y [ r,t1] = sup{e τ R t p(s) ds y(t) : t [ r, t 1 ], τ > Me ωt1 }. By an analogous relation, obtained by interchanging the roles of y and y, we find that H d (N(y), N(y)) Meωt1 y y [ r,t1]. τ Then N is a contraction and hence, by Lemma 4.4, N has a fixed point y, which is solution to problem (4.3). (4.4) Step 2. Let y 2 := y [t1,t 2] be a possible solution to the problem (y (t) Ay(t)) co F (t, y t ) for t (t 1, t 2 ], y(t + 1 ) = y (t 1 ) + I 1 (y (t 1 )), y(t) = y (t) for t [t 1 r, t 1 ]. Then y 2 is a fixed point of the multivalued operator N: Ω([t 1 r, t 2 ]) P(Ω([t 1 r, t 2 ]))

22 282 S. Djebali L. Górniewicz A. Ouahab defined by N(y) := h Ω([t 1 r, t 2 ]) : y (t) for t [t 1 r, t 1 ], h(t) = T (t t 1 )[y (t 1 ) + I 1 (y (t 1 ))] + T (t s)g(s) ds for t (t 1, t 2 ], t 1 where g S co F,y = {g L 1 ([t 1, t 2 ], E) : g(t) co F (t, y t ) for a.e. t [t 1, t 2 ]}. Again, we show that N satisfies the assumptions of Lemma 4.4. Clearly, N(y) P cl (Ω([t 1 r, t 2 ])) for each y Ω([t 1 r, t 2 ]). It remains to show that there exists < γ < 1 such that H d (N(y), N(y)) γ y y [t1 r,t 2] for each y, y Ω([t 1 r, t 2 ]). For this purpose, let y, y Ω([t 1 r, t 2 ]) and h 1 N(y). Then there exists g 1 (t) co F (t, y t ) such that, for each t [t 1 r, t 2 ], h 1 (t) = Since from (H 2 ) t 1 T (t s)g 1 (s) ds + T (t t 1 )[y (t 1 ) + I 1 (y (t 1 ))]. H d (co F (t, y t ), co F (t, y t )) p(t) y t y t D, t [t 1, t 2 ], then there is a w( ) co F (, y ) such that g 1 (t) w(t) p(t) y t y t D, t [t 1, t 2 ]. Consider the multi-valued map U 2 : [t 1, t 2 ] P(E) defined by U 2 (t) = {w E : g 1 (t) w p(t) y t y t D }. As in the above arguments, we can show that the multivalued operator V 2 (t) = U 2 (t) co F (t, y t ) is measurable with nonempty values; hence there exists g 2 (t) which is a measurable selection for V 2. Then g 2 (t) co F (t, y t ) and g 1 (t) g 2 (t) p(t) y t y t D, for a.e. t [t 1, t 2 ]. For almost every t [t 1, t 2 ], define h 2 (t) = T (t s)g 2 (s) ds + T (t t 1 )[y (t 1 ) + I 1 (y (t 1 ))].

23 For some τ > Me wt2, we have the estimates h 1 (t) h 2 (t) Semilinear Differential Inclusions 283 t 1 T (t s) g 1 (s) g 2 (s) ds p(s)me ω(t s) y 1s y 2s D ds t 1 ( ) Me ωt2 p(s) sup y 1s (θ) y 2s (θ) ds t 1 r θ Meωt2 y y [t1,t τ 2]. By an analogous relation, obtained by interchanging the roles of y and y, we obtain where H d (N(y), N(y)) Meωt2 y y [t1,t τ 2], y [t1 r,t 2] = sup{e τ R t t 1 p(s) ds y(t) : t [t 1 r, t 2 ]}. Therefore N is a contraction and thus, by Lemma 4.4, N has a fixed point y 2 solution of problem (4.4). Step 3. We continue this process taking into account that y m := y [tm,b] is a solution of the following problem (y (t) Ay(t)) F (t, y t ) for t (t m, b], y(t + m) = y m 1 (t m ) + I m (y m 1 (t m)), y(t) = y m 1 (t) for t [t m 1 r, t m ]. Then a solution y of problem (4.2) may be defined by y 1 (t) if t [ r, t 1 ], y 2 (t) if t (t 1, t 2 ], y(t) = y m (t) if t (t m, b]. Part 2. Let x be a solution of problem (4.2). Then, there exists g S co F,x such that φ(t) for t [ r, ], t x(t) = T (t)φ() + T (t s)g(s) ds + T (t t k )I k (x(t k )) for t [, b], <t k <t

24 284 S. Djebali L. Górniewicz A. Ouahab i.e. x is a mild solution of the problem x (t) Ax(t) = g(t) for a.e. t [, b] \ {t 1,..., t m }, x(t + k ) x(t k) = I k (x(t k )) for k = 1,..., m, x(t) = φ(t) for t [ r, ]. Let ε > and δ > be given by the relation Me ωb ε = δl m k=1 R k where L and R k, for k =,..., m, will be defined later on. From Lemmas 4.1 and 4.2, there exists a measurable selection f of t F (t, x t ) such that sup T (t s)g(s) ds T (t s)f (s) ds δ. Let t [,b] φ(t) for t [ r, ], t z(t) = T (t)φ() + T (t s)f (s) ds + T (t t k )I k (x(t k )) for t (, b]. <t k <t Hence, for each t [ r, b], x t z t D δ. With assumption (H 2 ), we infer that, for all u co F (t, z t ), γ(t) := d(g(t), F (t, x t )) d(g(t), u) + H d (F (t, z t ), F (t, x t )), H d (co F (t, x t ), co F (t, z t )) + H d (F (t, z t ), F (t, x t )) 2p(t) x t z t D 2δp(t). Since, under (H 1 (a)) and (H 2 ), γ is measurable (see [6] or [23, Lemma 1.5]), by the above inequality, we deduce that γ L 1 (J, E). From Theorem 3.2, problem (4.1) has a solution y which satisfies y(t) x(t) η (t), t [, t 1 ], Also for t (t 1, t 2 ], we have the estimates y(t) x(t) Me ω(t2 t1) (1 + c 1 )η (t 1 ) + η 1 (t 2 ) = L 1 [η (t 1 ) + η 1 (t 2 )], where L 1 = Me ω(t2 t1) (1 + c 1 ). And for t (t 2, t 3 ], we have y(t) x(t) Me ω(t3 t2) y(t 2 ) x(t 2 ) + Mc 2 e ω(t3 t2) y(t 2 ) x(t 2 ) + η 3 (t 3 ) M 2 e ω(t3 t1) (1 + c 1 )η (t 1 ) + Me ω(t3 t2) c 2 η 2 (t 2 ) + η 3 (t 3 ) L 2 [η (t 1 ) + η 2 (t 2 ) + η 2 (t 3 )], where L 2 = M 2 e ω(t3 t1) (1 + max(c 1, c 2 )). We continue this process until we arrive at k+1 y(t) x(t) L k η i (t i ), t (t k, t k+1 ], i=

25 Semilinear Differential Inclusions 285 where L k = M k e ω(t k+1 t k ) (1 + max 1 i k c i ). Thus for all t [ r, b], it holds that m m y(t) x(t) L η k (t k ) Lδ where for t [, t 1 ], and k= k= 1 ) η (t) δ (Me ωt1 + Me ωt1 (MH P (s) + p(s)) ds := δr, H = exp(me ωt1+p (t1) ) + while for k = 1,..., m, where and η k (t) δme ωt k+1 k+1 1 R k p(s) exp(me ωt1+p (t1) P (s) ) ds, t k (MH k P (s) + p(s)) ds := δr k, k+1 H k = M exp(me ω(t k+1 t k ) ) + M2p(s) ds t k L = M m e ωb (1 + max 1 i m c i). Using the definition of δ, we obtain the upper bound y x Ω Me ωb ε. Since ε is arbitrary, y x Ω ε, showing the density relation S co [,b] (φ) = S [,b](φ). 5. Topological structure of the solution sets 5.1. Closeness of the set of solutions. Let us introduce the following hypotheses: (A 1 ) For fixed y, the multi-function t F (t, y) is measurable. (A 2 ) There exists p L 1 ([, b], R + ) such that H d (F (t, z 1 ), F (t, z 2 )) p(t) z 1 z 2 D for all z 1, z 2 D, < d(, F (t, )) p(t) for a.e. t J. Theorem 5.1. Under assumptions (A 1 ) (A 2 ), the operator solution S [ r,b] of problem (1.1) has nonempty, closed valued and a closed graph. Proof. Let S [ r,b] : D P(E) be the operator solution of problem (1.1) defined by S [ r,b] (φ) = {y Ω: y solution of problem (1.1)}. With assumptions (A 1 ) (A 2 ), we may use Covitz Nadler fixed point theorem (Lemma 4.4), as in the proof of Theorem 4.5, to prove that S [ r,b] (φ) for every φ D.

26 286 S. Djebali L. Górniewicz A. Ouahab Thus we only show the closeness of both the values and the graph of S [ r,b]. Step 1. S [ r,b] ( ) P cl (E). For this, let φ D and let y n S [ r,b] (φ), n N be a sequence which converges to some limit y in Ω. Then φ(t) for t [ r, ], t y n (t) = T (t)φ() + T (t s)v n (s) ds + T (t t k )I k ((y n (t k )) for t [, b], <t k <t where v n {v L 1 ([, b], E): v( ) F (, (y ) n )}. Since F (t, ) is p-lipschitz and {y n } converges to y, there exists M > such that y n Ω M and for every ε, there exists n = n (ε) such that, for every n n v n (t) F (t, (y t ) n ) F (t, (y ) t ) + εp(t)b(, 1), for almost every t [, b]. Since F (, ) has compact values, there exists a subsequence v nm ( ) such that v nm ( ) v( ) as m, and v(t) F (t, (y ) t ) for almost every t [, b]. Since F (, ) has closed values, let w( ) F (, ) be such that w(t) = d(, F (t, )). Then w(t) p(t) for almost every t [, b] and Hence v nm (t) v nm (t) w(t) + w(t) p(t) (y nm ) t + p(t) (1 + M )p(t). b m y nm (t) z(t) Me ωb v nm (s) v(s) ds+me ωb I k ((y n (t k )) I k ((y (t k )) where k=1 φ(t) for t [ r, ], t z(t) = T (t)φ() + T (t s)v(s) ds + T (t t k )I k ((y (t k )) for t [, b]. <t k <t Using the continuity of I k and the Lebesgue dominated convergence theorem, we conclude that y = z. Step 2. S [ r,b] has a closed graph. Let φ n φ, y n S [ r,b] (φ n ) and y n y. y n S [ r,b] (φ n ) means that there exists g n L 1 such that, for each t [ r, ], y n (t) = φ n (t) and for t [, b], y n (t) = T (t)φ n () + T (t s)g n (s) ds + <t k <t T (t t k )I k (y n (t k )).

27 Semilinear Differential Inclusions 287 We prove that y S [ r,b] (φ ), i.e. there exists g S F,y such that for each t J y (t) = T (t)φ () + T (t s)g (s) ds + <t k <t T (t t k )I k (y (t k )). Using the fact that F has compact values and is an L 1 -Carathéodory function, we may pass to a subsequence, if necessary, to get that {g n } converges to some limit g in L 1 (J, E). Since the functions I k, k = 1,..., m, are continuous, we obtain the estimates y (t) T (t)φ () T (t t k )I k (y (t k )) T (t s)g (s) ds <t k <t ( y n (t) T (t)φ n () ( y (t) T (t)φ () y n y Ω + Me ωb m k=1 <t k <t <t k <t T (t t k )I k (y n (t k )) ) T (t s)g n (s) ds T (t t k )I k (y (t k )) T (t s)g (s) ds) I k (y n (t k )) I k (y (t k )) b + T (t) B(E) φ n () φ () + Me ωb g n (s) g (s) ds. The right-hand side terms tend to, as n, proving our claim Compactness of the set of solutions Auxiliary results. First, we collect some definitions and properties about measures of noncompactness in Banach spaces. More details can be found in [38]. Definition 5.1. Let E be a Banach space and (A, ) a partially ordered set. A map β: P(E) A is called a measure of noncompactness on E (MNC for short) if, for every subset Ω P(E), it satisfies β(co Ω) = β(ω). Notice that if D is dense in Ω, then co Ω = co D and hence β(ω) = β(d). Definition 5.2. A measure of noncompactness β is called (a) Monotone if Ω, Ω 1 P(E), Ω Ω 1 implies β(ω ) β(ω 1 ). (b) Nonsingular if β({a} Ω) = β(ω) for every a E, Ω P(E). (c) Invariant with respect to the union with compact sets if β(k Ω) = β(ω) for every relatively compact set K E and Ω P(E). (d) Real if A = R + = [, ] and β(ω) < for every bounded Ω. (e) Regular if the condition β(ω) = is equivalent to the relative compactness of Ω.

28 288 S. Djebali L. Górniewicz A. Ouahab As example of an MNC, one may consider the Hausdorff measure: χ(ω) = inf{ε > : Ω has a finite ε-net}. Recall that a bounded set A E has a finite ε-net if there exits a finite subset S E such that A S + εb where B is a closed ball in E. Definition 5.3. Let M be a closed subset of a Banach space E, (A, ) a partially ordered set and β: P(E) (A, ) an MNC on E. A multimap F: M P cp (E) is said to be β-condensing if for every bounded Ω M, the inequality β(ω) β(f(ω)), implies the relative compactness of Ω. Definition 5.4. A sequence {v n } n N L 1 ([, b], E) is said to be semicompact if (a) it is integrably bounded, i.e. if there exists ψ L 1 ([, b], R + ) such that v n (t) ψ(t), for a.e. t [, b] and every n N, (b) the image sequence {v n (t)} n N is relatively compact in E for almost every t [, b]. The following result follows from the Dunford Pettis theorem (see also [38, Proposition 4.2.1]) Lemma 5.5. Every semi-compact sequence is weakly compact in L 1 ([, b], E). Lemma 5.6 ([38, Theorem 5.1.1]). Let N: L 1 ([a, b], E) C([a, b], E) be an abstract operator satisfying the following conditions: (S 1 ) N is ξ-lipschitz: there exists ξ > such that for every f, g L 1 ([a, b], E) Nf(t) Ng(t) ξ b a f(s) g(s) ds, for all t [a, b]. (S 2 ) N is weakly-strongly sequentially continuous on compact subsets: for any compact K E and any sequence {f n } n=1 L 1 ([a, b], E) such that {f n (t)} n=1 K for almost every t [a, b], the weak convergence f n f implies the strong convergence N(f n ) N(f ) as n. Then for every semi-compact sequence {f n } n=1 L 1 ([, b], E), the image sequence N({f n } n=1) is relatively compact in C([a, b], E). Lemma 5.7 ([38, Theorem 5.2.2]). Let an operator N: L 1 ([a, b], E) C([a, b], E) satisfy conditions (S 1 ) (S 2 ) together with (S 3 ) There exits η L 1 ([a, b]) such that for every integrably bounded sequence {f n } n=1, we have χ({f n (t)} n=1) η(t) for almost every t [a, b], where χ is the Hausdorff MNC.

29 Semilinear Differential Inclusions 289 Then χ({n(f n )(t)} n=1) 2ξ where ξ is the constant in (S 1 ). b a η(s) ds, for all t [a, b], Finally, two useful properties of the fixed point set of β-condensing multimaps are the following (see [38]): Lemma 5.8. Let W be a convex closed subset of a Banach space E and let N: W P cp,cv (W ) be a closed β-condensing multimap where β is a nonsingular measure of noncompactness defined on subsets of W. Then Fix N. Lemma 5.9. Let W be a closed subset of a Banach space E and let N: W P cp (E) be a closed β-condensing multimap where β is a monotone MNC on E. Then Fix N is compact. The following so-called nonlinear alternative of Leray and Schauder for multivalued maps will be needed in this section. Lemma 5.1 ([28], [27]). Let (X, ) be a normed space and F : X P cl,cv (X) be a compact, u.s.c. multi-valued map. Then either one of the following conditions holds: (a) F has at least one fixed point, (b) the set M := {x X : x λf (x), λ ], 1[} is unbounded Compactness result. Let F : J D P cp,cv (E) be a Carathéodory multimap which satisfies some of the following assumptions: (B 1 ) There exist a function p L 1 (J, R + ) and a continuous nondecreasing function ρ: [, ) [, ) such that with F (t, z) p(t)ρ( z D ) for a.e. t J and each z D, b p(s) ds < 1 du ρ(u). (B 2 ) There exist constants c k > and continuous functions φ k : R + R + such that I k (x) c k φ k ( x ) for each x E, k = 1,..., m. (B 3 ) E is a reflexive Banach space and either one of the following conditions holds: (a) the semigroup T ( ) is uniformly continuous, (b) the semigroup T ( ) is compact in E.

30 29 S. Djebali L. Górniewicz A. Ouahab (B 4 ) There exists p L 1 ([, b], R + ) such that for every bounded subset D in D χ(f (t, D)) p(t) sup{χ(d(θ)) : θ [ r, ]} and there exist L k >, k =,..., m such that q k := 2Me ωt k+1 k+1 sup t [t k,t k+1 ] t k e Lk(t s) p(s) ds < 1, k =,..., m. Here χ is the Hausdorff MNC and D(θ) := {ψ(θ), ψ D}. Theorem Assume that F satisfies either (B 1 ), (B 2 ) and (B 3 ) or (B 1 ), (B 2 ) and (B 4 ). Then the set of solutions for problem (1.1) in nonempty and compact. Proof. According to the hypotheses considered, the proof is split in two parts. Part 1. Under assumptions (B 1 ) (B 3 ), the solutions set is nonempty and compact. by Step 1. S [ r,b] (φ). Consider the operator N: Ω P(Ω) defined for y Ω φ(t) for t [ r, ], t N(y) = h Ω : h(t) = T (t)φ() + T (t s)v(s) ds + T (t t k )I k (y(t k )) for t [, b], <t k <t where v S F,y = {u L 1 (J, E): u F (t, y t ), for almost every t J}. Clearly, fixed points of the operator N are mild solutions of problem (1.1). Since, for each y Ω, the nonlinearity F takes convex values, the selection set S F,y is convex and then N has convex values. As in [1], [45], [47], we can prove that N maps bounded sets into bounded sets and there exists M 1 > such that for every y solution of problem (1.1), we have y Ω M 1. Thus we only prove that N(B q ) is relatively compact in Ω, where B q = {y Ω : y Ω q}. First, N(B q ) is an equicontinuous set of Ω. To see this, let < τ 1 < τ 2 b, y B q, and h N(y). Then there exists v S F,y such that φ(t) for t [ r, ], t h(t) = T (t)φ() + T (t s)v(s) ds + T (t t k )I k (y(t k )) for t (, b]. <t k <t