A METHOD OF UPPER AND LOWER SOLUTIONS FOR FUNCTIONAL DIFFERENTIAL INCLUSIONS MOUFFAK BENCHOHRA

In this paper, a fixed point theorem for condensing maps combined with upper and lower solutions are used to investigate the existence of solutions for first order functional differential inclusions. Initial Value Problem, Convex Multivalued Map, Functional


Introduction
This paper is concerned with the existence of solutions for the initial multivalued problem: where is a nonempty compact and convex valued For any continuous function defined on the interval and any , we denote by the element of defined by ) ) ) Here represents the history of the state from time , up to the present time .
The method of upper and lower solutions has been successfully applied to study the existence of multiple solutions for initial and boundary value problems of first order functional differential equations.This method has been used only in the context of single-valued, functional differential equations.We refer to the papers of Haddock and Nkashama [6], Hristova and Bainov [10], Liz and Nieto [13], and Nieto, Jiang and Jurang [15].For other results on functional differential equations using other methods, we refer to the books of Erbe, Qingai and Zhang [5], Hale [7], Henderson [9], and the survey paper of Ntouyas [16].Notice that very recently this method has been used for initial and boundary value problems for differential inclusions in the papers of Benchohra and Boucherif [2], Benchohra and Ntouyas [3],and Halidias and Papageorgiou [8].
In this paper we establish an existence result for the problem (1)-(2).Our approach is based on the existence of upper and lower solutions and a fixed point theorem for condensing maps developed by Martelli [14].

Preliminaries
We will briefly recall some basic definitions and facts from multivalued analysis that we will use in the sequel.
is the space of all absolutely continuous functions .EGÐN ß Ñ CÀ N Ä ' ' is the Banach space of all continuous functions normed by denotes the set of all nonempty compact and convex subsets of GGÐ\Ñ \.
An upper semi-continuous map is said to be condensing [14] if for any KÀ \ Ä # \ bounded subset with , we have , where denotes the We remark that a compact map is the easiest example of a condensing map.For more details on multivalued maps, see the books of Deimling [4] and Hu and Papageorgiou [11].
The multivalued map is said to be measurable if, for every , the , there exists such that Ð333Ñ 5  !− P ÐN ß Ñ : ' The following concept of lower and upper solutions for (1)-( 2) was introduced by Halidias and Papageorgiou in [8] for second order multivalued boundary value problems.It will be the basic tool in the approach that follows.
A function is said to be a lower solution of ( 1)-( 2) if there exists Definition 2.3: " " For the multivalued map and for each we define by Our main result is based on the following: [12] Lemma 2.4: Let be a Banach space and a real compact interval.Let \ N J À N ‚ \ Ä GGÐ\Ñ P W Á g be an -Caratheodory multivalued map with and let be a " " J > linear continuous mapping from to , then the operator , is a closed graph operator in .GÐN ß \Ñ ‚ GÐN ß \Ñ Let be an u.s.c. and condensing map.If the set Lemma 2.5: [14] KÀ \ Ä GGÐ\Ñ QÀ oe Ö@ − \À @ − KÐ@Ñ  "× -for some is bounded, then has a fixed point.K

Main Result
We are now in a position to state and prove our result for the problem (1)-( 2).Theorem 3.1: Suppose is an -Caratheordory multivalued J À N ‚ GÐN ß Ñ Ä GGÐ Ñ P !" ' ' map which satisfies the condition there exist and in lower and upper solutions, respectively, for the Transform the problem into a fixed point problem.Consider the following modified problem

C oe ß Ð%Ñ
! 9 where is the truncation operator defined by 4) is a fixed point of the operator defined by a.e. on and a.e. on W oeÖ@−P ÐNß ÑÀ@Ð>Ñ−JÐ>ßÐ CÑ Ñ >−N×ß For each the set is nonempty.Indeed, by there exists where We shall show that is a completely continuous multivalued map, u.s.c. with convex K closed values.The proof will be given in several steps.
is convex for each .
Step 1: KÐCÑ C − G ÐN ß Ñ !" ' Indeed, if belong to , then there exist and such that

Let
. Then for each we have Thus, for each we get sends bounded sets in into equicontinuous sets.

> !
We have From Lemma 2.4, it follows that is a closed graph operator.Also, from the de-> ‰ W µ " J finition of we have > Now, we are going to show that the set Step 5: Let then for some .Thus there exists such that From the definition of there exists such that 7 Thus we obtain " This shows that is bounded.Hence, Lemma 2.5 applies and has a fixed point which Q K is a solution to problem (3)-( 4).