MULTIVALUED SEMILINEAR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH NONCONVEX-VALUED RIGHT-HAND SIDE

We investigate the existence of mild solutions on acompact interval to some classes of semilinear neutral functional 
differential inclusions. We will rely on a fixed-point theorem for contraction multivalued maps due to Covitz and Nadler and on 
Schaefer's fixed-point theorem combined with lower semicontinuous multivalued operators with decomposable values.


Introduction
This paper is concerned with the existence of mild solutions defined on a compact real interval for first-and second-order semilinear neutral functional differential inclusions (NFDIs).
In Section 3, we consider the following class of semilinear NFDIs: d dt y(t) − f t, y t ∈ Ay(t) + F t, y t , a.e.t ∈ J := [0,b], where F : J × C([−r,0],E) → ᏼ(E) is a multivalued map, A is the infinitesimal generator of a strongly continuous semigroup T(t), t ≥ 0, φ ∈ C([−r,0],E), f : J × C([−r,0],E) → E, ᏼ(E) is the family of all subsets of E, and E is a real separable Banach space with norm Section 4 is devoted to the study of the following second-order semilinear NFDIs: where F, φ, f , ᏼ(E), and E are as in problem (1.1),A is the infinitesimal generator of a strongly continuous cosine family {C(t) : t ∈ R}, and η ∈ E.
For any continuous function y defined on the interval [−r,b] and any t ∈ J, we denote by y t the element of C([−r,0],E), defined by y t (θ) = y(t + θ), θ ∈ [−r,0].(1.3)Here, y t (•) represents the history of the state from time t − r, up to the present time t.
In the last two decades, several authors have paid attention to the problem of existence of mild solutions to initial and boundary value problems for semilinear evolution equations.We refer the interested reader to the monographs by Goldstein [11], Heikkilä and Lakshmikantham [13], and Pazy [19], and to the paper of Heikkilä and Lakshmikantham [14].In [17,18], existence theorems of mild solutions for semilinear evolution inclusions are given by Papageorgiou.Recently, by means of a fixed-point argument and the semigroup theory, existence theorems of mild solutions on compact and noncompact intervals for first-and second-order semilinear NFDIs with a convex-valued right-hand side were obtained by Benchohra and Ntouyas in [1,4].Similar results for the case A = 0 are given by Benchohra and Ntouyas in [2].Here, we will extend the above results to semilinear NFDIs with a nonconvex-valued right-hand side.The method we are going to use is to reduce the existence of solutions to problems (1.1) and (1.2) to the search for fixed points of a suitable multivalued map on the Banach space C([−r,b],E).For each intial value problem (IVP), we give two results.In the first one, we use a fixed-point theorem for contraction multivalued maps due to Covitz and Nadler [7] (see also Deimling [8]).This method was applied recently by Benchohra and Ntouyas in [3], in the case when A = 0 and f ≡ 0. In the second one, we use Schaefer's theorem combined with a selection theorem of Bressan and Colombo [5] for lower semicontinuous (l.s.c) multivalued operators with decomposable values.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis, which are used throughout this paper.
We denote by ᏼ(E) the set of all subsets of E normed by • ᏼ .Let C([−r,0],E) be the Banach space of all continuous functions from [−r,0] into E with the norm A measurable function y : J → E is Bochner-integrable if and only if |y| is Lebesgueintegrable.(For properties of the Bochner-integral, see, e.g., Yosida [24].)By L 1 (J,E) we denote the Banach space of functions y : J → E which are Bochner-integrable and normed by M. Benchohra et al. 527 and by B(E) the Banach space of bounded linear operators from E to E with norm We say that a family {C(t) : t ∈ R} of operators in B(E) is a strongly continuous cosine family if The strongly continuous sine family {S(t) : t ∈ R}, associated to the given strongly continuous cosine family {C(t) : t ∈ R}, is defined by (2.5) The infinitesimal generator A : E → E of a cosine family {C(t) : t ∈ R} is defined by For more details on strongly continuous cosine and sine families, we refer the reader to the books of Fattorini [9], Goldstein [11], and to the papers of Travis and Webb [22,23].For properties of semigroup theory, we refer the interested reader to the books of Goldstein [11] and Pazy [19].
Let (X,d) be a metric space.We use the following notations: (2.7) Then (P b,cl (X),H d ) is a metric space and (P cl (X),H d ) is a generalized (complete) metric space (see [16]).
A multivalued map N : J → P cl (X) is said to be measurable if, for each x ∈ X, the function Y : J → R, defined by The multivalued operator N has a fixed point if there is x ∈ X such that x ∈ N(x).The fixed-point set of the multivalued operator N will be denoted by Fix N.
Theorem 2.2.Let (X,d) be a complete metric space.If N : Let Ꮽ be a subset of J × C([−r,0],E).The set Ꮽ is ᏸ ⊗ Ꮾ measurable if Ꮽ belongs to the σ-algebra generated by all sets of the form × Ᏸ, where is Lebesgue-measurable in J and Ᏸ is Borel-measurable in C([−r,0],E).A subset B of L 1 (J,E) is decomposable if, for all u,v ∈ B and ⊂ J measurable, the function uχ + vχ J− ∈ B, where χ denotes the characteristic function for .
Let E be a Banach space, X a nonempty closed subset of E, and G : For more details on multivalued maps, we refer to the books of Deimling [8], G órniewicz [12], Hu and Papageorgiou [15], and Tolstonogov [21].
Definition 2.3.Let Y be a separable metric space and let N : Y → ᏼ(L 1 (J,E)) be a multivalued operator.The operator N has property (BC) if it satisfies the following conditions: (1) N is l.s.c.; (2) N has nonempty, closed, and decomposable values.
Let F : J × C([−r,0],E) → ᏼ(E) be a multivalued map with nonempty compact values.Assign to F the multivalued operator by letting The operator Ᏺ is called the Niemytzki operator associated with F. Theorem 2.5 (see [5]).Let Y be separable metric space and let N : Y → ᏼ(L 1 (J,E)) be a multivalued operator which has property (BC).Then N has a continuous selection, that is, there exists a continuous function (single-valued) g : Y → L 1 (J,E) such that g(y) ∈ N(y) for every y ∈ Y .

First-order semilinear NFDIs
Now, we are able to state and prove our first theorem for the IVP (1.1).Before stating and proving this result, we give the definition of a mild solution of the IVP (1.1).
, a.e. on J, y 0 = φ, and for each t ∈ J and u,u ∈ C([−r,0],E), and where We remark that the fixed points of N are solutions to (1.1).Also, for each y ∈ C([−r,b],E), the set S F,y is nonempty since, by (H2), F has a measurable selection (see [6,Theorem III.6]).We will show that N satisfies the assumptions of Theorem 2.2.The proof will be given in two steps.
Step 1.We prove that N(y and there exists g n ∈ S F,y such that Using the fact that F has compact values and from (H3), 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 each t ∈ J, Step 2. We prove that H d (N(y 1 ),N(y (3.9) From (H3), it follows that Hence, there is w ∈ F(t, y 2t ) such that Consider U : J → ᏼ(E) given by Then we have Then By the analogous relation, obtained by interchanging the roles of y 1 and y 2 , it follows that , N is a contraction, and thus, by Theorem 2.2, it has a fixed point y which is a mild solution to (1.1).
Remark 3.3.Recall that, in the proof of Theorem 3.2, we have assumed that γ < 1.Since this assumption is hard to verify, we would like point out that using the well-known Bielecki's renorming method, it can be simplified.The technical details are omitted here.
By the help of Schaefer's fixed-point theorem, combined with the selection theorem of Bressan and Colombo for l.s.c.maps with decomposable values, we will present the second existence result for problem (1.1).Before this, we introduce the following hypotheses which are assumed hereafter: (C1) (C2) for each ρ > 0, there exists a function h ρ ∈ L 1 (J,R + ) such that In the proof of our following theorem, we will need the next auxiliary result.
(A1) There exist constants 0 ≤ c 1 < 1 and c 2 ≥ 0 such that The function f is completely continuous and, for any bounded set

E).
(A3) There exist p ∈ L 1 (J,R + ) and a continuous nondecreasing function ψ : R + → (0,∞) such that where We will show that N 1 is completely continuous.The proof will be given in several steps.

Step 1. The operator N 1 sends bounded sets into bounded sets in C([−r,b],E).
Indeed, it is enough to show that for any q > 0, there exists a positive constant l such that, for each From (A0), (C2), (A1), and (A3), we have, for each t ∈ J, N 1 (y)(t)

T(t − s) B(E) g(y)(s) ds
Then, for each h ∈ N(B q ), we have (3.26) Step 2. The operator N 1 sends bounded sets in C([−r,b],E) into equicontinuous sets.
Using (A2), it suffices to show that the operator T u 2 − s B(E) h q (s)ds. (3.28) As u 2 → u 1 , the right-hand side of the above inequality tends to zero.The equicontinuity for the cases u 1 < u 2 ≤ 0 and u 1 ≤ 0 ≤ u 2 is obvious.
Step 3. The operator N 2 is continuous.
Let {y n } be a sequence such that Since the function g is continuous and f is completely continuous, then Let y ∈ Ᏹ(N 1 ).Then y = λN 1 (y) for some 0 < λ < 1, and for t ∈ [0,b], we have This implies, by (A0), (A1), and (A3), that for each t ∈ J, we have M. Benchohra et al. 535 We consider the function µ defined by , then µ(t) = φ and inequality (3.35) holds.We take the right-hand side of inequality (3.35) as v(t); then we have Since ψ is nondecreasing, we have From this inequality, it follows that We then have This inequality implies that there exists a constant K 1 such that v(t) ≤ K 1 , t ∈ J, and hence µ(t) ≤ K 1 , t ∈ J. Since for every t ∈ J, y t ≤ µ(t), we have where K 1 depends only on b, M 1 , and M 2 and on the functions p and ψ.This shows that Ᏹ(N 1 ) is bounded.As a consequence of Schaefer's theorem (see [20]), we deduce that N 1 has a fixed point y which is a solution to problem (1.1).
such that where g ∈ S F,y .We will show that N 3 satisfies the assumptions of Theorem 2.2.The proof will be given in two steps.
Step 1.We prove that N Using the fact that F has compact values and from (H3), 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 each t ∈ J, y n (t) −→ ỹ(t) = C(t)φ(0) + S(t) η − f (0,φ) +

.8)
Then By the analogous relation, obtained by interchanging the roles of y 1 and y 2 , it follows that where Proof.Hypotheses (C1) and (C2) imply, by Lemma 3.4, that F is of l.s.c.type.Then, from Theorem 2.5, there exists a continuous function g     This inequality implies that there exists a constant K 2 such that v(t) ≤ K 2 , t ∈ J, and hence µ(t) ≤ K 2 , t ∈ J. Since for every t ∈ J, y t ≤ µ(t), we have where K 2 depends only on b, M, and on the functions p and ψ.This shows that Ᏹ(N 4 ) is bounded.
Set X := C([−r,b],E).As a consequence of Schaefer's theorem (see [20]), we deduce that N 4 has a fixed point y which is a solution to problem (1.2).
Remark 4.4.The reasoning used above can be applied to obtain the existence results for the following first-and second-order semilinear neutral functional integrodifferential inclusions of Volterra type:

. 1 )
By C([−r,b],E) we denote the Banach space of all continuous functions from [−r,b] into E with the norm A,b) = inf a∈A d(a,b) and d(a,B) = inf b∈B d(a,b).
where * = b 0 l(s)ds.Then the IVP (1.1) has at least one mild solution on [−r,b].Proof.Transform problem (1.1) into a fixed-point problem.Consider the multivalued operator

t 0 C 0 S
(t − s) f s, y s ds + t (t − s)g(s)ds.