Existence Results for Impulsive Semilinear Damped Differential Inclusions

In this paper we investigate the existence of mild solutions for rst and second order impulsive semilinear evolution inclusions in real separable Banach spaces. By using suitable xed point theorems, we study the case when the multivalued map has convex and nonconvex values.


Introduction
In this paper, we shall be concerned with the existence of mild solutions for first and second order impulsive semilinear damped differential inclusions in a real Banach space.Firstly, we consider the following first order impulsive semilinear differential inclusions of the form: where F : J × E → P (E) is a multivalued map (P (E) is the family of all nonempty subsets of E), A is the infinitesimal generator of a family of semigroup {T (t) : t ≥ 0}, B is a bounded linear operator from E into E, y 0 ∈ E, 0 < t Later, we study the second order impulsive semilinear evolution inclusions of the form: ∆y where F, I k , B and y 0 are as in problem ( 1)-(3), A is the infinitesimal generator of a family of cosine operators {C(t) : t ≥ 0}, I k ∈ C(E, E) and y 1 ∈ E.
The study of the dynamical buckling of the hinged extensible beam which is either stretched or compressed by axial force in a Hilbert space, can be modeled by the following hyperbolic equation where α, β, L > 0, u(t, x) is the deflection of the point x of the beam at the time t, g is a nondecreasing numerical function, and L is the length of the beam.
Equation (E) has its analogue in IR n and can be included in a general mathematical model u where A is a linear operator in a Hilbert space H and M and g are real functions.Equation (E) was studied by Patcheu [26] and the equation (E 1 ) by Matos and Pereira [25].These equations are special cases of the equations ( 4), (7).Impulsive differential and partial differential equations have become more important in recent years in some mathematical models of real phenomena, especially in control, biological or medical domains, see the mongraphs of Lakshmikantham et al [20], Samoilenko and Perestyuk [28], and the papers of Ahmed [2], Agur et al [1], Erbe et al [13], Goldbeter et al [16], Kirane and Rogovchenko [18], Liu et al [22], Liu and Zhang [23].This paper will be organized as follows.In Section 2 we will recall briefly some basic definitions and preliminary facts from multivalued analysis which will be used later.In Section 3 we shall establish two existence theorems for the problem (1)-(3) when the right hand side is convex as well as nonconvex valued.In the first case a fixed point theorem due to Bohnenblust and Karlin [8] (see also [32]) is used.A fixed point theorem for contraction multivalued maps due to Covitz and Nadler [10] is applied in EJQTDE, 2003 No. 11, p. 2 the second one.In Section 4 existence theorems for the both cases are presented for the problem ( 4)- (7) in the spirit of the analysis used in the previous section.
The special case (for B=0) of the problem (1)-( 3) was studied by Benchohra et al in [5] by using the concept of upper and lower mild solutions combined with the semigroup theory and by Benchohra and Ntouyas in [7] with the aid to a fixed point theorem due to Martelli for condensing multivalued maps [24].Notice that when the impulses are absent (i.e.I k , I k = 0, k = 1, . . ., m) the problem ( 4)-( 7) was studied by Benchohra et al in [6].Hence the results of the present paper can be seen as an extension of the problems considered in [6], [5] and [7].

Preliminaries
We will briefly recall some basic definitions and facts from multivalued analysis that we will use in the sequel.
B(E) is the Banach space of all linear bounded operator from E into E with norm A measurable function y : J → E is Bochner integrable if and only if |y| is Lebesgue integrable.(For properties of the Bochner integral, see for instance, Yosida [31]).
L 1 (J, E) denotes the Banach space of functions y : J −→ E which are Bochner integrable normed by G is called upper semi-continuous (u.s.c.) on X if for each x 0 ∈ X the set G(x 0 ) is a nonempty, closed subset of X, and if for each open set N of X containing G(x 0 ), there exists an open neighbourhood N 0 of x 0 such that G(N 0 ) ⊆ N.
G is said to be completely continuous if G(B) is relatively compact for every B ∈ P b (X).If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c.if and only if G has a closed graph (i.e.x n −→ x * , y n −→ y * , y n ∈ G(x n ) imply y * ∈ G(x * )).G has a fixed point if there is x ∈ X such that x ∈ G(x).The fixed point set of the multivalued operator G will be denoted by F ixG.

EJQTDE, 2003 No. 11, p. 3
A multivalued map N : J → P cl (E) is said to be measurable, if for every y ∈ E, the function t −→ d(y, N (t)) = inf{|y − z| : z ∈ N (t)} is measurable, where d is the metric induced by the norm of the Banach space E. For more details on multivalued maps see the books of Aubin and Cellina [3], Aubin and Frankowska [4], Deimling [11] and Hu and Papageorgiou [17] .
We say that a family {C(t) : t ∈ IR} of operators in B(E) is a strongly continuous cosine family if: (1) C(0) = I (I is the identity operator in E), (3) the map t −→ C(t)y is strongly continuous for each y ∈ E.
The strongly continuous sine family {S(t) : t ∈ IR}, associated to the given strongly continuous cosine family {C(t) : t ∈ IR}, is defined by For more details on strongly continuous cosine and sine families, we refer the reader to the books of Goldstein [15], Fattorini [14], and to the papers of Travis and Webb [29], [30].For properties of semigroup theory, we refer the interested reader to the books of Goldstein [15] and Pazy [27].
Definition 2.1 The multivalued map F : (ii) y −→ F (t, y) is upper semicontinuous for almost all t ∈ J; (iii) For each r > 0, there exists ϕ r ∈ L 1 (J, IR + ) such that for all |y| ≤ r and for a.e.t ∈ J.
For each y ∈ C(J, E), define the set of selections of F by The following Lemmas are crucial in the proof of our main results.
EJQTDE, 2003 No. 11, p. 4 Lemma 2.2 [21].Let X be a Banach space.Let F : J × X −→ P cp,c (X) be an L 1 -Carathéodory multivalued map with S F (y) = ∅ and let Γ be a linear continuous mapping from L 1 (J, X) to C(J, X), then the operator is a closed graph operator in C(J, X) × C(J, X).
Lemma 2.3 (Bohnenblust-Karlin [8], see also [32] p. 452).Let X be a Banach space and K ∈ P cl,c (X) and suppose that the operator G : Then G has a fixed point in K.

First Order Impulsive Differential Inclusions
In this section we are concerned with the existence of solutions for problem (1)-( 3) when the right hand side has convex as well as nonconvex values.Initially we assume that F : J × E → P (E) is compact and convex valued multivalued map.In order to define the solution of ( 1)-( 3) we shall consider the following space Let us introduce the following hypotheses: (H1) F : J ×E −→ P cp,c (E) is an L 1 -Carathéodory map and for each fixed y ∈ C(J, E) the set where and Opial [21]).
(ii) Assumption (H4) is satisfied if for example F satisfies the standard domination  We shall show that N satisfies the assumptions of Lemma 2.3.The proof will be given in several steps.Let where .
It is clear that K is a closed bounded convex set.Let k * = sup{ y Ω : y ∈ K}.
Step 1: Indeed, let y ∈ K and fix t ∈ J.We must show that N (y) ∈ K.There exists g ∈ S F (y) such that for each t ∈ J Thus N (y) ∈ K; So, N : K → K.
Since K is bounded and N (K) ⊂ K, it is clear that N (K) is bounded.N (K) is equicontinuous.Indeed, let τ 1 , τ 2 ∈ J, τ 1 < τ 2 and > 0 with 0 < ≤ τ 1 < τ 2 .Let y ∈ K and h ∈ N (y).Then there exists g ∈ S F (y) such that for each t ∈ J we have The right-hand side tends to zero as τ 2 − τ 1 → 0, and sufficiently small, since T (t) is a strongly continuous operator and the compactness of T (t) for t > 0 implies the continuity in the uniform operator topology.As a consequence of the Arzelá-Ascoli theorem it suffices to show that the multivalued N maps K into a precompact set in E. Let 0 < t ≤ b be fixed and let be a real number satisfying 0 < < t.For y ∈ K we define where g ∈ S F (y) .Since T (t) is a compact operator, the set H (t) = {h (t) : h ∈ N (y)} is precompact in E for every , 0 < < t.Moreover, for every h ∈ N (y) we have EJQTDE, 2003 No. 11, p. 8 Therefore there are precompact sets arbitrarily close to the set {h(t) : h ∈ N (y)}.Hence the set {h(t) : h ∈ N (y)} is precompact in E.
Step 3: N has a closed graph.
Let y n −→ y * , h n ∈ N (y n ) and h n −→ h * .We shall prove that h * ∈ N (y * ).h n ∈ N (y n ) means that there exists g n ∈ S F (yn) such that for each t ∈ J We must prove that there exists g * ∈ S F,y * such that for each Clearly since I k , k = 1, . . ., m and B are continuous we have that as n → ∞.Consider the linear continuous operator From Lemma 2.2, it follows that Γ • S F is a closed graph operator.Moreover, we have that As a consequence of Lemma 2.3 we deduce that N has a fixed point which is a mild solution of ( 1 Our considerations are based on the following fixed point theorem for contraction multivalued operators given by Covitz and Nadler in 1970 [10] (see also Deimling,[11] Theorem 11.1).Remark 3.8 For each y ∈ Ω the set S F (y) is nonempty since by (H5) F has a measurable selection (see [9], Theorem III.6).
Proof of the theorem.Transform the problem (1)-( 3) into a fixed point problem.
Let the multivalued operator N : Ω → P (Ω) defined as in Theorem 3.3.We shall show that N satisfies the assumptions of Lemma 3.6.The proof will be given in two steps.
Indeed, let (y n ) n≥0 ∈ N (y) such that y n −→ ỹ in Ω.Then ỹ ∈ Ω and there exists Using the fact that F has compact values and from (H7), we may pass to a subsequence if necessary to get that g n converges to g in L 1 (J, E) and hence g ∈ S F (y) .Then for

So ỹ ∈ N (y).
Step 2: There exists γ < 1, such that Let y, y ∈ Ω and h ∈ N (y).Then there exists g(t) ∈ F (t, y(t)) such that for each t ∈ J From (H7) it follows that Hence there is w ∈ F (t, y(t)) such that Consider U : J → P (E), given by Let us define for each t ∈ J We define on Ω an equivalent norm to • Ω by where Then Then By an analogous relation, obtained by interchanging the roles of y and y, it follows that So, N is a contraction and thus, by Lemma 3.6, N has a fixed point y, which is a mild solution to (1)- (3).EJQTDE, 2003 No. 11, p. 12 In this section we study the problem ( 4)-( 7) when the right hand side has convex and nonconvex values.We give first the definition of mild solution of the problem ( 4)-( 7) Definition 4.1 A function y ∈ Ω is said to be a mild solution of ( 4)-( 7) if there exists v ∈ L 1 (J, IR n ) such that v(t) ∈ F (t, y(t)) a.e. on J, y(0) = y 0 , y (0) = y 1 and are satisfied.Then the IVP ( 4)-( 7) has at least one mild solution.
Proof.Transform the problem ( 4)-( 7) into a fixed point problem.Consider the multivalued operator N : Ω → P (Ω) defined by: It is clear that K is a closed bounded convex set.
Step 1: Indeed, let y ∈ K 1 and fix t ∈ J.We must show that N (y) ⊂ K 1 .Let h ∈ N (y).Thus there exists v ∈ S F (y) such that for each t ∈ J h This implies by (H2) and (A1)-(A2) that for each t ∈ J we have Thus N (y) ⊂ K 1 ; So, N : As in Theorem 3.3 we can show that N (K 1 ) is relatively compact and hence by Lemma 2.3 the operator N has a least one fixed point which is a mild solution to problem (4)- (7).
In this last part we consider problem ( 4)-( 7) with a nonconvex valued right-hand side.
Proof.We define on Ω an equivalent norm by So, N is a contraction and thus, by Lemma 3.6, N has a fixed point y, which is a mild solution to ( 4)-( 7).