ON SECOND-ORDER MULTIVALUED IMPULSIVE FUNCTIONAL DIFFERENTIAL INCLUSIONS IN BANACH SPACES

Differential equations arise in many real world problems such as physics, population dynamics, ecology, biological systems, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. Much has been done under the assumption that the state variables and system parameters change continuously. However, one may easily visualize situations in nature where abrupt changes such as shock, harvesting, and disasters may occur. These phenomena are shortterm perturbations whose duration is negligible in comparison with the duration of the whole evolution process. Consequently, it is natural to assume, in modelling these problems, that these perturbations act instantaneously, that is, in the form of impulses. For more details on this theory and on its applications we refer to the monographs of Baı̆nov and Simeonov [2], Lakshmikantham, Baı̆nov, and Simeonov [19], and Samoilenko and Perestyuk [24]. However, very few results are available for impulsive differential inclusions; see for instance, the papers of Benchohra and Boucherif [4, 5], Erbe and Krawcewicz [12], and Frigon and O’Regan [14]. Very recently an extension to functional differential equations of first order with impulsive effects has been done by Yujun [10] by using the coincidence degree theory, and by Benchohra and Ntouyas [7] with the aid of Schaefer’s theorem. These results have been also generalized to the multivalued case by the authors in [6] by combining the a priori bounds and the Leray-Schauder


Introduction
Differential equations arise in many real world problems such as physics, population dynamics, ecology, biological systems, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth.Much has been done under the assumption that the state variables and system parameters change continuously.However, one may easily visualize situations in nature where abrupt changes such as shock, harvesting, and disasters may occur.These phenomena are shortterm perturbations whose duration is negligible in comparison with the duration of the whole evolution process.Consequently, it is natural to assume, in modelling these problems, that these perturbations act instantaneously, that is, in the form of impulses.For more details on this theory and on its applications we refer to the monographs of Baȋnov and Simeonov [2], Lakshmikantham, Baȋnov, and Simeonov [19], and Samoilenko and Perestyuk [24].However, very few results are available for impulsive differential inclusions; see for instance, the papers of Benchohra and Boucherif [4,5], Erbe and Krawcewicz [12], and Frigon and O'Regan [14].
Very recently an extension to functional differential equations of first order with impulsive effects has been done by Yujun [10] by using the coincidence degree theory, and by Benchohra and Ntouyas [7] with the aid of Schaefer's theorem.These results have been also generalized to the multivalued case by the authors in [6] by combining the a priori bounds and the Leray-Schauder nonlinear alternative for multivalued maps.For other results concerning functional differential equations, we refer the interested reader to the monographs of Erbe, Qingai, and Zhang [13], Hale [15], Henderson [16], and the survey paper of Ntouyas [23].
The fundamental tools used in the existence proofs of all the above-mentioned works are essentially fixed point arguments, nonlinear alternative, topological transversality [11], topological degree theory [22], or the monotone method combined with upper and lower solutions [18].
In this paper, we will be concerned with the existence of solutions of the second-order initial value problem for the impulsive functional differential inclusion ) ) where For any continuous function y defined on the interval [−r, T ] − {t 1 , . . ., t m } and any t ∈ J , we denote by y t the element of C([−r, 0], E) defined by Here y t (•) represents the history of the state from time t − r, up to the present time t.
In this paper, we will generalize the results of Benchohra and Ntouyas [8] considered for second-order impulsive functional differential equations to the multivalued case.Our approach is based on a fixed point theorem for condensing maps due to Martelli [21].

Preliminaries
In this section, we introduce notations, definitions, and results which are used throughout the paper.
Let ( If the multivalued 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., In the following CC(E) denotes the set of all nonempty compact, convex subsets of E.
We remark that a completely continuous multivalued map is the easiest example of a condensing map.
In order to define the solution of (1.1), (1.2), (1.3), and (1.4) we will consider the following space = {y : [−r, T ] → E : y k ∈ C(J k , E), k = 0, . . ., m and there exist y(t − k ), and y(t + k ), k = 1, . . ., m with y(t − k ) = y(t k ), y(t) = φ(t), for all t ∈ [−r, 0]} which is a Banach space with the norm where y k is the restriction of y to J k = [t k , t k+1 ], k = 0, . . ., m.We will also consider the set 1 = {y : [−r, T ] → E : y k ∈ W 2,1 (J k , E), k = 0, . . ., m and there exist y(t − k ) and y(t is the Sobolev space of functions y : J k → E such that y and y are absolutely continuous, and y ∈ L 1 (J k , E).The set 1 is a Banach space with the norm (2.6) Let I be a compact real interval.For any y ∈ C(I, E) we define the set The following lemmas are crucial in the proof of our main theorem.
Lemma 2.3 [20].Let I be a compact real interval and X a Banach space.Let F be a multivalued map satisfying the Carathéodory conditions with the set of L 1 -selections S F is nonempty, and let be a linear continuous mapping from L 1 (I, X) to C(I, X).Then the operator Lemma 2.4 [21].Let G : X → CC(X) be an u.s.c.condensing map.If the set ᏹ := y ∈ X : λy ∈ G(y) for some λ > 1 (2.9) is bounded, then G has a fixed point.
We introduce the following hypotheses: (H1)

Main result
where We will show that G satisfies the assumptions of Lemma 2.4.The proof will be given in several steps.
Step 1. G(y) is convex for each y ∈ .
Indeed, if h 1 , h 2 belong to G(y), then there exist g 1 , g 2 ∈ S F,y such that for each t ∈ J we have Step 2. G maps bounded sets into bounded sets in .Indeed, it is enough to show that there exists a positive constant such that for each h ∈ G(y) with y ∈ B q = {y ∈ : y ∞ ≤ q} one has h ∞ ≤ .If h ∈ G(y), then there exists g ∈ S F,y such that for each t ∈ J we have (3.6)By (H2) and (H3) we have for each t ∈ J h(t) ≤ φ + t y 0 + t 0 (t − s) g(s) ds (3.8) Step 3. G maps bounded sets into equicontinuous sets of .
Let r 1 , r 2 ∈ J , r 1 < r 2 and B q = {y ∈ : y ∞ ≤ q} a bounded set of .For each y ∈ B q and h ∈ G(y), there exists g ∈ S F,y such that (3.9) Thus As r 2 → r 1 the right-hand side of the above inequality tends to zero.
The equicontinuity for the cases r 1 < r 2 ≤ 0 and r 1 ≤ 0 ≤ r 2 are obvious. Step Consider the linear continuous operator ) for some g * ∈ S F,y * .
Step 5. Now it remains to show that the set is bounded.
[a, b] denote a real compact interval of R. Let C([a, b], E) be the Banach space of continuous functions from [a, b] into E with norm y ∞ = sup |y(t)| : t ∈ [a, b] ∀y ∈ C [a, b], E .
4. G has a closed graph.Let y n → y * , h n ∈ G(y n ), and h n → h * .We will prove that h * ∈ G(y * ).h n ∈ G(y n ) means that there exists g n ∈ S F,y n such that for each t ∈ J k y n t k + t − t k Īk y n t k .We must prove that there exists g * ∈ S F,y * such that for each t ∈ J h * (t) = φ(0) + ty 0 + t k + t − t k Īk y * t k .Clearly, since I k and Īk , k = 1, . . ., m are continuous we have I *