AN EXISTENCE RESULT FOR NEUTRAL FUNCTIONAL DIFFERENTIAL INCLUSIONS IN A BANACH SPACE L.GUEDDA AND

In this paper we prove the existence of mild solutions for semilinear neutral functional differential inclusions with un- bounded linear part generating a noncompact semigroup in a Ba- nach space. This work generalizes the result given in (4).


Introduction
Semilinear neutral functional differential inclusion has been the object of many studies by many researchers in the recent years.We only mention the works of some authors ( [1], [2], [6]).The method which consists in defining an integral multioperator for which fixed points set coincides with the solutions set of differential inclusion has been often applied to existence problems.
Our aim in this paper is to give an existence result for initial value problems for first order semilinear neutral functional differential inclusions in a separable Banach space E of the form: x(t) = ϕ(t), t ∈ [−r, 0], (1.2) where A : D(A) ⊂ E → E is the infinitesimal generator of an uniformly bounded analytic semigroup of linear operators {S(t)} t≥0 on a separable Banach space E; the multimap F : The function x t (.) represents the history of the state from time t − r, up the present time t.
In [8] using topological methods of multivalued analysis, existence results for semilinear inclusions with unbounded linear part generating a noncompact semigroup in Banach space were given.In this paper, using the method of fractional power of closed operators theory and by giving a special measure of noncompactness, we extend this line of attack to the problem (1.1)-(1.2).More precisely in section 3 we give the measure of noncompactness for which the integral multioperator is condensing, this will allow us to give an existence result for the problem (1.1)-(1.2),and by using the properties of fixed points set of condensing operators we deduce that the mild solutions set is compact.

Preliminaries
Along this work, E will be a separable Banach space provided with norm ., A : D(A) ⊂ E → E is the infinitesimal generator of an uniformly bounded analytic semigroup of linear operators {S(t)} t≥0 in E. We will assume that 0 ∈ ρ(A) and that S(t) ≤ M for all t ∈ [0, T ].Under these conditions it is possible to define the fractional power (−A) α , 0 < α ≤ 1, as closed linear operator on its domain D(−A) α .Furthermore, D(−A) α is dense in E and the function x α = (−A) α x defines a norm in D(−A) α .If X α is the space D(−A) α endowed with the norm .α , then X α is a Banach space and there exists c α > 0 such that (−A) α S(t) For additional details respect of fractional power of a linear operator and semigroup theory, we refer the reader to [11] and [16] .. Let X be a Banach space and (Y, ≥) a partially ordered set.A function Ψ : A measure of noncompactness Ψ is called: EJQTDE, 2008 No. 9, p. 2 with the natural ordering, and Ψ(Ω) < +∞ for every bounded set Ω ∈ P (X).
If Y is a cone in a Banach space we will say that the measure of noncompactness Ψ is regular if Ψ(Ω) = 0 is equivalent to the relative compactness of Ω.
One of most important examples of measure of noncompactness possessing all these properties, is the Hausdorff measure of noncompactness Let K(X) denotes the class of compact subsets of X, Kv(X) denotes the class of compact convex subsets of X, and (Q, d) a metric space.
A multimap G :

y).
A multimap G : X → P (Q) is said to be quasicompact if its restriction to every compact subset A ⊂ X is compact.
In the sequel, C([−r, T ], E) denotes the space of continuous functions from [−r, T ] to E endowed with the supremum norm.For any In section 3 we establish an existence result to the problem (1.1)-(1.2) using the following well known results.(See [8]).
Lemma 3. Let E be a separable Banach space and J an operator which satisfies the following conditions: J 1 ) There exists D > 0 such that Then, EJQTDE, 2008 No. 9, p. 4 [3]).

Lemma 4. ([8]
).If G is a convex closed subset of a Banach space E, and Γ : G → Kv(G) is closed Θ condensing, where Θ is nonsingular measure of noncompactness defined on subsets of G, then F ixΓ = ∅.

Lemma 5. ([8]
).Let Z be a closed subset of a Banach space E and F : Z → K(E) a closed multimap, which is α-condensing on every bounded subset of Z, where α is a monotone measure of noncompactness.If the fixed points set F ixF is bounded, then it is compact.

Existence Result
Let us define what we mean by a mild solution of the problem (1.1)- EJQTDE, 2008 No. 9, p. 5 and To establish our result we consider the following conditions: where, for s ∈ [−r, 0], Ω(s) = {u(s); u ∈ Ω} .Assume also that H) there exist constants d 1 , d 2 , ω, θ ∈ R + and 0 < α < 1, such that h is X α -valued, and (i) for every u ∈ C([−r, 0], E), and t ∈ [0, T ] We note that from assumptions (F 1) − (F 3) it follows that the superposition multioperator is correctly defined (see [8]) and is weakly closed in the following sense: [8]).Since the family {S(t)} t≥0 is an analytic semigroup [16], the operator function s → AS(t − s) is continuous in the uniform operator topology on [0, t) which from the estimate and the Bochner's theorem implies that AS(t − s)h(s, x s ) is integrable on [0, t).Now we shall prove our main result.
where f ∈ sel F (x), and the operator Υ : The proof will be given in four steps.
Step 1.The multivalued operator Γ is closed.
The multivalued operator Γ can be written in the form Γ = where the operators Γ i , i = 1, 2, 3 are defined as follows: the multivalued operator and the operator Γ an arbitrary sequence such that, for n ≥ 1 f n (t) ∈ F (t, x n t ), a.e.t ∈ [0, T ], and Since {S(t)} t≥0 is a strongly continuous semigroup (see [3]), the operator Υ satisfies the properties (J 1 ) and (J 2 ) of Lemma 3, by using hypothesis (F 3) we have that sequence {f n } ∞ n=1 is integrably bounded.Hypothesis (F 4) implies that E).Moreover, by using the fact that the operator sel F is closed, we have f 0 ∈ sel F (x 0 ).EJQTDE, 2008 No. 9, p. 8 Consequently in the space C([−r, T ], E), with f 0 ∈ sel F (x 0 ).On the other hand, using (H) − (iii), for t ∈ [0, T ] we get Using hypothesis (H)−(ii) and the estimate in the family No. 9, p. 9 and f 0 ∈ sel F (x 0 ).Thus z 0 ∈ Γ(x 0 ) and hence Γ is closed.Now in the space C([−r, T ], E) we consider the measure of noncompactness Θ defined in the following way: for every bounded subset Ω ⊂ C([−r, T ], E) where and mod c Ω is the module of equicontinuity of the set Ω ⊂ C([−r, T ], E) given by: and L > 0 is chosen so that where M is the constant from the estimation in the family {S(t)} t≥0 , the constants d 1 , d 2 from (H) − (i), the constant ω from (H) − (ii), the function β from the hypothesis (F 3) and the function κ from the hypothesis (F 4).From the Arzelá-Ascoli theorem, the measure Θ give a nonsingular and regular measure of noncompactness in C([−r, T ], E).
Let Ω ⊂ C([−r, T ], E) be a bounded subset such that where the inequality is taking in the sense of the order in R 3 induced by the positive cone R 3 + .We will show that (3.4) implies that Ω is relatively compact in C([−r, T ], E).From the inequality (3.4) follows immediately that (3.5) χ(Ω([−r, 0])) = 0.
Indeed, we have Remark that from (3.5) it follows that sup We give now an upper estimate for χ({f (s), f ∈ sel F (Ω)}, for s ∈ [0, t], t ≤ T .By using (3.6) and the assumption (F 4) we have Since the measure χ is monotone, by using (H 1 ) − (iii) and Lemma 1, we obtain for s ∈ By multiplying both sides with e −Lt , we have (3.8)sup The multifunction G : s → AS(t − s)h(s, Ω s ), s ∈ [0, t) is integrable and integrably bounded.Indeed for any x ∈ Ω we have: EJQTDE, 2008 No. 9, p. 12 Using the assumption (H) − (ii) and Lemma 1, we get By lemma 2, we get for every s ∈ [0, t] By multiplying both sides with e −Lt and bearing in mind the definition of q 2 , we get EJQTDE, 2008 No. 9, p. 13 From the inequalities (3.7)-(3.9),remark 2 and the fact that ω (−A) α < 1, we get Using the inequality (3.4), the last inequality implies that (3.10) Ψ(Ω) = 0.
EJQTDE, 2008No.9, p. 15 where x ∈ Ω, is equicontinuous on C([−r, T ], E).Let 0 ≤ t ≤ t ≤ T , and x ∈ Ω.We have Ω s )ds is relatively compact for every t ∈ [0, T ], the first term on the right hand side converge to zero when t → t uniformly on x ∈ Ω.As consequence we get