INITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL DIFFERENTIAL EQUATIONS

In this note we investigate the existence of solutions of initial and boundary value problems defined on a compact interval to some classes of functional differential inclusions. We shall rely on a fixed point theorem for contraction multivalued maps due to Covitz and Nadler and Schafer’s theorem combined with lower semicontinuous multivalued operators with decomposable values.

For any continuous function y defined on the interval [−r, T ] and any t ∈ [0, T ], we denote by y t the element of C([−r, 0], IR n ) defined by Here y t (•) represents the history of the state from time t − r, up to the present time t.
Recently, in [1], the authors studied first and second order initial value problems for equation (1.1), in the case where ρ(t) = 1, by using a fixed point theorem for contraction multivalued maps due to Covitz and Nadler [5] (see also Deimling [6]).Using a fixed point theorem for condensing multivalued maps due to Martelli, the authors have obtained an existence result for the initial value problem (1.1)-(1.2).Here, by using the fixed point theorem for contraction maps and Schaefer's theorem combined with a selestion theorem of Bressan and Colombo for lower semicontinuous multivalued operators with decomposable values, existence results are proposed for problems (1.1)-(1.2) and (1.1)-(1.3).

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts from multivalued analysis which are used throughout this note.
where χ denotes for the characteristic function.
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 : Definition 2.1: Let Y be a separable metric space and let N : Y → P(L 1 ([0, T ], IR n )) 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, T ] × C([−r, 0], IR n ) → P(IR n ) be a multivalued map with nonempty compact values.Assign to F the multivalued operator by letting The operator F is called the Niemytzki operator associated with F. We say F is of lower semi-continuous type (l.s.c.type) if its associated Niemytzki operator F is lower semi-continuous and has nonempty closed and decomposable values.
Next we state a selection theorem due to Bressan and Colombo.Lemma 2.1: [3] Let Y be separable metric space and let N : Y → P(L 1 ([0, T ], IR n )) be a multivalued operator which has property (BC).Then N has a continuous selection, i.e. there exists a continuous function (single-valued) Let (X, d) be a metric space.We use the notations: Then (P b,cl (X), H d ) is a metric space and (P cl (X), H d ) is a generalized metric space.Definition 2.2: A multivalued operator N : X → P cl (X) is called a) γ-Lipschitz if and only if there exists γ > 0 such that For more details on multivalued maps and the proof of known results cited in this section we refer to the books of Deimling [6], Górniewicz [8], Hu and Papageorgiou [9] and Tolstonogov [11].
Our considerations are based on the following fixed point theorem for contraction multivalued operators given by Covitz and Nadler in 1970 [5] (see also Deimling,[6] Theorem 11.1).

Initial Value Problems
Now, we are able to state and prove our main theorems.In this section we shall give two results for the IVP (1.1)-(1.2).Before stating and proving these results, we give the definition of a solution of the IVP (1.1)-(1.2).
We shall show that N satisfies the assumptions of Lemma 2.2.The proof will be given in two steps.
Step 1: Using the fact that F has closed values and from the second part of (H2), we can easily show that

So ỹ ∈ N (y).
Step 2: IR n ) and h 1 ∈ N (y 1 ).Then there exists g 1 (t) ∈ F (t, y 1t ) such that From (H2) it follows that Hence there is w ∈ F (t, y 2t ) such that Consider U : [0, T ] → P(IR n ), given by Since the multivalued operator V (t) = U (t) ∩ F (t, y 2t ) is measurable (see Proposition III.4 in [4]), there exists g 2 (t) a measurable selection for V .So, g 2 (t) ∈ F (t, y 2t ) and Then By the analogous relation, obtained by interchanging the roles of y 1 and y 2 , it follows that So, N is a contraction and thus, by Lemma 2.2, it has a fixed point y, which is solution to (1.1)-(1.2).
By the help of the Schaefer's theorem combined with the selection theorem of Bressan and Colombo for lower semicontinuous maps with decomposable values, we shall present an existence result for the problem (1.1)-(1.2).Before this, let us introduce the following hypotheses which are assumed hereafter: (H4) For each q > 0, there exists a function h q ∈ L 1 ([0, T ], IR + ) such that In the proof of our following theorem, we will need the auxiliary result: Lemma 3.1: [7].Let F : [0, T ] × C([−r, 0], IR n ) → P(IR n ) be a multivalued map with nonempty, compact values.Assume (H3) and (H4) hold.Then F is of l.s.c.type.Theorem 3.2: Suppose, in addition to hypotheses (H3), (H4), the following also holds: We consider the problem ) Clearly from (H4) and the Arzela-Ascoli theorem, the multivalued operator N is continuous and completely continuous.
In order to apply Schaefer's theorem, it remains to show that the set Then λy = N (y) for some λ > 1.Thus, for each t ∈ [0, T ] This implies by (H5) that for each t ∈ [0, T ] we have We consider the function µ defined by , by the previous inequality we have for t ∈ [0, T ] If t * ∈ [−r, 0], then µ(t) = φ and the previous inequality holds.
Let us take the right-hand side of the above inequality as v(t); then we have Using the nondecreasing character of ψ, we get This implies for each t ∈ [0, T ] that This inequality implies that there exists a constant b = b(T, p, ψ) such that v(t) ≤ b, t ∈ [0, T ], and hence µ(t) ≤ b, t ∈ [0, T ].Since for every t ∈ [0, T ], y t ≤ µ(t), we have This shows that E(N ) is bounded.As a consequence of Schaefer's theorem (see [10]), we deduce that N has a fixed point which is a solution of (3.1)-(3.2) and hence from Remark 3.2, a solution to the problem (1.1)-(1.2) .

Boundary Value Problems
In the next theorem we give an existence result for the BVP (1.1), (1.3).
ds and G is the Green's function for the corresponding homogenuous problem which is given by the formula We shall show that N 1 satisfies the assumptions of Lemma 2.1.
Using the same reasoning as in Step 1 of Theorem 3.1, we can show that N 1 (y From (H2) it follows that Hence, there is w ∈ F (t, y 2t ) such that Consider U : [0, T ] → P(IR n ), given by Since the multivalued operator we denote the Banach space of all continuous functions from [0, T ] into IR n with the norm y [0,T ] := sup{|y(t)| : t ∈ [0, T ]}.L 1 ([0, T ], IR n ) denotes the Banach space of measurable functions y : [0, T ] −→ IR n which are Lebesgue integrable normed by 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 d(a, b).