NEUMANN BOUNDARY VALUE PROBLEMS FOR IMPULSIVE DIFFERENTIAL INCLUSIONS

In this paper, we investigate the existence of solutions for a class of second order impulsive differential inclusions with Neumann boundary conditions. By using suitable fixed point theorems, we study the case when the right hand side has convex as well as nonconvex values.

In the literature there are few papers dealing with the existence of solutions for Neumann boundary value problems; see [15], [16] and the references therein.Recently in [14], the authors studied Neumann boundary value problems with impulse actions.
Motivated by the work above, this paper attempts to study existence results for impulsive Neumann boundary value problems for differential inclusions.
The aim of our paper is to present existence results for the problem (1)-( 3), when the right hand side is convex as well as nonconvex valued.The first result relies on the nonlinear alternative of Leray-Schauder type.In the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values, while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler.The methods used are standard, however their exposition in the framework of problem ( 1)-( 3) is new.It is remarkable also that the results of this paper are new, even for the special case I k = 0.
The paper is organized as follows: in Section 2 we recall some preliminary facts that we need in the sequel and in Section 3 we prove our main results.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this paper.
L 1 ([0, 1], R) denotes the Banach space of measurable functions x : [0, 1] −→ R which are Lebesgue integrable and normed by AC 1 ((0, 1), R) is the space of differentiable functions x : (0, 1) → R, whose first derivative, x ′ , is absolutely continuous.For a normed space (X, 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.A multivalued map G : [0, 1] → P cl (R) is said to be measurable if for every y ∈ R, the function is measurable.For more details on multivalued maps see the books of Aubin and Cellina [1], Aubin and Frankowska [2], Deimling [6] and Hu and Papageorgiou [9].
(iii) for each q > 0, there exists Let E be a Banach space, X a nonempty closed subset of E and G : X → P(E) a multivalued operator with nonempty closed values.G is lower semi-continuous (l.s.c.) if the set {x ∈ X : belongs to the σ-algebra generated by all sets of the form J × D, where J is Lebesgue measurable in [0, 1] and D is Borel measurable in R. A subset A of L 1 ([0, 1], R) is decomposable if for all u, v ∈ A and J ⊂ [0, 1] measurable, the function uχ J + vχ J−J ∈ A, where χ J stands for the characteristic function of J .Definition 2.2 Let Y be a separable metric space and let N : Y → P(L 1 ([0, 1], R)) be a multivalued operator.We say N has property (BC) if 1) N is lower semi-continuous (l.s.c.);

2) N has nonempty closed and decomposable values.
Let F : [0, 1] × R → P(R) be a multivalued map with nonempty compact values.Assign to F the multivalued operator The operator F is called the Nymetzki operator associated with F.  d(a, b).Then (P b,cl (X), H d ) is a metric space and (P cl (X), H d ) is a generalized metric space (see [12]).
The following lemmas will be used in the sequel.
Lemma 2.1 [13].Let X be a Banach space.Let F : [0, 1] × R −→ P cp,c (X) be an L 1 -Carathéodory multivalued map and let Γ be a linear continuous mapping from Lemma 2.2 [3].Let Y be a separable metric space and let N : be a multivalued operator which has property (BC).Then N has a continuous selection; i.e., there exists a continuous function (single-valued) g :

Main Results
In this section, we are concerned with the existence of solutions for the problem (1)-( 3) when the right hand side has convex as well as nonconvex values.Initially, we assume that F is a compact and convex valued multivalued map.
In the following, we introduce first the space a.e. on J ′ , and for k = 1, . . ., m the function , and the boundary conditions x ′ (0) = 0 = x ′ (1).
We need the following modified version of Lemma 2.3 from [14].
has a unique solution x ∈ AC 1 ((0, 1), R) with the representation where G(t, s) is the Green function associated to the correspondinh homogeneous problem given by It is easy to prove the following properties of the Green's function: (I) G(t, s) ≥ 0 for any (t, s) Theorem 3.1 Suppose that: (H2) there exist a continuous non-decreasing function ψ : [0, ∞) −→ (0, ∞) and a function p ∈ L 1 ([0, 1], R + ) such that for each (t, u) ∈ [0, 1] × R; (H3) there exists a continuous non-decreasing function (H4) there exists a number M > 0 such that Then the BVP ( 1)-( 3) has at least one solution.
Proof.Consider the operator We shall show that N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.The proof will be given in several steps.
Indeed, if h 1 , h 2 belong to N(x), then there exist v 1 , v 2 ∈ S F,x such that for each t ∈ [0, 1] we have Let 0 ≤ d ≤ 1.Then, for each t ∈ [0, 1], we have Since S F,x is convex (because F has convex values), then Step 2: N maps bounded sets into bounded sets in C([0, 1], R).
) and x ∈ B q .Then for each h ∈ N(x), there exists v ∈ S F,x such that Then we have Step 3: N maps bounded sets into equicontinuous sets of C([0, 1], R).
Let r 1 , r 2 ∈ [0, 1], r 1 < r 2 and B q be a bounded set of C([0, 1], R) as in Step 2 and The right hand side tends to zero as r 2 − r 1 → 0. As a consequence of Steps 1 to 3 together with the Arzelá-Ascoli Theorem, we can conclude that N : Step 4: N has a closed graph.
We must show that there exists h * ∈ S F,x * such that, for each t ∈ [0, 1], Clearly we have Consider the continuous linear operator From Lemma 2.1, it follows that Γ • S F is a closed graph operator.Moreover, we have for some v * ∈ S F,x * .
Step 5: A priori bounds on solutions.
The operator N : U → P(C([0, 1], R)) is upper semicontinuous and completely continuous.From the choice of U, there is no x ∈ ∂U such that x ∈ λN(x) for some λ ∈ (0, 1).As a consequence of the nonlinear alternative of Leray-Schauder type [7], we deduce that N has a fixed point x in U which is a solution of the problem (1)-( 3).This completes the proof.
Next, we study the case where F is not necessarily convex valued.Our approach here is based on the nonlinear alternative of Leray Schauder type combined with the selection theorem of Bresssan and Colombo for lower semi-continuous maps with decomposable values.Theorem 3.2 Suppose that: (H6) for each ρ > 0, there exists for all u ∞ ≤ ρ and for a.e.t ∈ [0, 1].
Proof.Note that (H5) and (H6) imply that F is of l.s.c.type.Then from Lemma 2.2, there exists a continuous function f : Consider the problem x It is clear that if ) is a solution of ( 4)-( 6), then x is a solution to the problem (1)- (3).Transform the problem ( 4)-( 6) into a fixed point theorem.Consider the operator N defined by We can easily show that N is continuous and completely continuous.The remainder of the proof is similar to that of Theorem 3.1.
We present now a result for the problem ( 1)-( 3) with a nonconvex valued right hand side.Our considerations are based on the fixed point theorem for multivalued map given by Covitz and Nadler [5].(H9) there exist constants c k such that then the BVP ( 1)-( 3) has at least one solution.
Proof.We shall show that N satisfies the assumptions of Lemma 2.3.The proof will be given in two steps.
Step 1: N ], R) and there exists v n ∈ S F,x such that, for each t ∈ [0, 1], Using the fact that F has compact values and from (H5), we may pass to a subsequence if necessary to get that v n converges to v in L 1 ([0, 1], R) and hence v ∈ S F,x .Then, for each t ∈ [0, 1], x So, x ∈ N(x).
Let x, x ∈ C([0, 1], R) and h 1 ∈ N(x).Then, there exists v 1 (t) ∈ F (t, x(t)) such that for each t ∈ [0, 1]  Thus By an analogous relation, obtained by interchanging the roles of x and x, it follows that So, N is a contraction and thus, by Lemma 2.3, N has a fixed point x which is solution to (1)-( 3).The proof is complete.
EJQTDE Spec.Ed.I, 2009 No. 22 Definition 2.3 Let F : [0, 1] × R → P(R) be a multivalued function with nonempty compact values.We say F is of lower semi-continuous type (l.s.c.type) if its associated Nymetzki operator F is lower semi-continuous and has nonempty closed and decomposable values.Let (X, d) be a metric space induced from the normed space (X, | • |).Consider H d : P(X) × P(X) −→ R + ∪ {∞} given by H d (A, B) = max sup a∈A d(a, B), sup b∈B d(A, b) , where d(A, b) = inf a∈A d(a, b), d(a, B) = inf b∈B EJQTDE Spec.Ed.I, 2009 No. 22 EJQTDE Spec.Ed.I, 2009 No. 22
and if for each open set N of X containing G(x 0 ), there exists an open neighborhood 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 ) [4]ce the multivalued operatorV (t) = U(t) ∩ F (t, x(t)) is measurable (see Proposition III.4 in[4]), there exists a function v 2 (t) which is a measurable selection for V .So, v 2 (t) ∈ F (t, x(t)), and for each t ∈ [0, 1],|v 1 (t) − v 2 (t)| ≤ l(t)|x(t) − x(t)|.