On the existence of mild solutions for neutral functional differential inclusions in Banach space

A theorem on existence of mild solutions for partial neutral functional differential inclusions with unbounded linear part generating a noncompact semigroup in Banach space is established. Semilinear neutral functional differential inclusion has been the object of many stud- ies by many researchers in the recent years. The method which consists in defining an integral multioperator for wich fixed points set coincides whith the solutions set of differential inclusion has been often applied to existence problems. In the case of inclu- sions on infinite dimensional spaces its direct application is complicated by the fact that the integral multioperator is noncompact except if one impose a severe compactness assumption. In this paper using the method of condensing integral multioperators and fractional power of closed operators theory, we study the existence of mild solutions for initial value problems for first order semilinear neutral functional differential inclusions in a separable Banach space E for the form: d dt (x(t) h(t,xt)) 2 Ax(t) + F(t,xt),a.e. t 2 (0,T) (1.1) x(t) = '(t), t 2 ( r,0), (1.2) where A : D(A) � E ! E is the infinitesimal generator of an uniformly bounded analytic semigroup of linear operators, {e At }t�0 on a separable Banach space E; the multimap F : (0,T) × C(( r,0),E) ! P(E) and h : (0,T) × C(( r,0),E) ! E, are given functions, 0 < r < 1,' 2 C(( r,0),E), where P(E) denotes the class of all nonempty subsets of E, and C(( r,0),E) denotes the space of continuous functions from ( r,0) to E. For any continuous function x defined on ( r,T) and any t 2 (0,T), we denote by xt the element of C(( r,0),E) defined by


Introduction
Semilinear neutral functional differential inclusion has been the object of many studies by many researchers in the recent years.The method which consists in defining an integral multioperator for wich fixed points set coincides whith the solutions set of differential inclusion has been often applied to existence problems.In the case of inclusions on infinite dimensional spaces its direct application is complicated by the fact that the integral multioperator is noncompact except if one impose a severe compactness assumption.
For any continuous function x defined on [−r, T ] and any t ∈ [0, T ], we denote by x t the element of C([−r, 0] , E) defined by For any u ∈ C([−r, 0] , E) the norm of u is defined by u = sup{ u(θ) : θ ∈ [−r, 0]}.
The function x t (.) represents the history of the state from time t − r, up the present time t.
Our work was motivated by the paper of E. Hernandez [3].Using the theory of condensing operators, one can clarify certain conditions given in [3] in the form of estimates.Let us mention that existence results for semilinear differential inclusion with χ-regularity condition for the multivalued nonlinearity, where χ is the Hausdorf measure of noncompactness, were obtained in the works of N.S.Papageorgiou [9,10], and existence results for impulsive neutral functional differential inclusion by S.K. Ntouyas [7].See also [2,5,6].A general existence theorem was given by V.V. Obukhovskii [8] for a semilinear functional differential inclusions with an analytic semigroup and upper Carathéodory type nonlinearity.In case where the linear part generates a strongly continuous semigroup, and the multivalued nonlinearity satisfies simple and general conditions of boundedness and χ-regularity, existence results were obtained in the paper of J.F. Couchouron and M.I.Kamenskii [1].In this paper we use the results given in [1] and in the book of M. Kamenskii et al. [4] to study the multivalued part of our integral multioperators.

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, {e At } t≥0 , on a separable Banach space E. We will assume that 0 ∈ ρ(A) and that e At ≤ 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 e At ≤ cα t α , for t > 0. Also the inclusion X α → X β for 0 < β ≤ α ≤ 1 is continuous.
For additional details respect of semigroup theory, we refer the reader to Pazy [11].Let Y + be the positive cone of an ordered Banach space (Y + , ≤).A function Ψ defined on the set of all bounded subsets of the Banach space X with values in Y + is called a measure of noncompactness on X if Ψ(Ω) = Ψ( − coΩ) for all bounded subsets Ω ⊂ X, where − coΩ denote the closed convex hull of Ω.The measure Ψ is called nonsingular if for every a ∈ X, Ω ∈ P (X), Ψ({a} ∪ Ω) = Ψ(Ω), monotone, if Ω 0, Ω 1 ∈ P (X) and Ω 0 ⊆ Ω 1 implly Ψ(Ω 0 ) ≤ Ψ(Ω 1 ).One of most important example of measure of noncompactness, is the Hausdorf measure of noncompactness defined on each bounded set Ω of X by: χ(Ω) = inf{ε > 0; Ω has a finite ε-net in X} 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.EJQTDE, 2007 No. 2, p. 2 A multimap G : Z → K(X) is called Ψ−condensing if for every bounded set Ω ⊂ E, that is not relatively compact we have Ψ(G(Ω)) Ψ(Ω), where Z ⊂ X.
A multivalued G : X → K(Q) is u.s.c.at a point x ∈ X, if for every ε > 0 there exists neighborhood V (x) such that G(x ) ⊂ W ε (G(x)), for every x ∈ V (x).Here by W ε (A) we denote the ε-neighborhood of a set A, i.e.W ε (A) = {y ∈ Y : d(y, A) < ε}, where d(y, A) = inf x∈A d(x, y).
A multimap G : For all this definitions see for example [4].
In the following C([−r, T ], E) is the space of continuous functions from [−r, T ] to E endowed with the supremum norm.For any x ∈ C([−r, T ], E), In section 3 we establish some existence results to the problem (1.1)-(1.2) using the following well known results.(See [4]).
Lemma 2.1.Let E be a separable Banach space and G : [0, T ] → P (E) an integrable, integrably bounded multifunction such that Lemma 2.2.Let E be a separable Banach space and S an operator which satisfies the following conditions: An example of this operator is the operator S : where x 0 ∈ E, and A is an unbounded linear operator generating a C 0 −semigroup in E (see [1]).
Lemma 2.4.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 Results
Let us define what we mean by a mild solution of the problem (1.1)-(1.2). and To establish our results we consider the following conditions: Suppose that the multimap We note that from assumptions (F 1) − (F 3) it follows that the superposition multioperator is correctly defined (see [4]) and is weakly closed in the following sense: if the sequences [4]).Also from the assumption (H) − (ii), the function (−A) α h is continuous.Since the family e At t≥0 is an analytic semigroup [11], the operator function s → Ae A(t−s) is continuous in the uniform operator topology on [0, t) which from the estimate and the Bochner's theorem implies that Ae A(t−s) h(s, x s ) is integrable on [0, t).
where f ∈ sel F (x), and the operator S : The proof will be given in four steps.
Step 1.The multivalued operator Γ is closed.
The operator Γ can be written in the form Γ = 3 1 Γ i where the operators Γ i , i = 1, 2, 3 are defined as follows: the multivalued operator Γ and the operator Γ The operator S satisfies the properties S1 and S2 of the Lemma 2.2, since e At is a strongly continuous operator (see [1]).Hypothesis (F 3) implies that {f n } ∞ n=1 is integrably bounded, hypothesis (F 4) implies that χ({f E).So we can assume without loss of generality, that f n → w f 0 .Lemma 2.2 implies that Sf n → Sf 0 in C([0, T ], E) and by using the fact that the operator sel F is closed, we get f 0 ∈ sel F (x 0 ).Consequently On the other hand, we have the inequalities: For t ∈ [−r, 0] we have: Using hypothesis (H) − (ii) and the estimate in the family Ae At t>0 we have: ) and 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]) 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 sup where M is the constant from the estimation in the family of e At t≥0 , the constants d 1 , d 2 from (H) − (i), 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]).
We shall give now an upper estimate for mod c ΓΩ.We have shown that for any t ∈ [0, T ].From the conditions (F 3) and (F 4) follows that the sequence {f ∈ sel F (x), x ∈ Ω} is semicompact in L 1 ([0, T ], E), and hence the set [1]).Therefore, the set E). Consequently: (3.10) mod c Γ 1 Ω = 0. Now we will show that the set where x ∈ Ω, is equicontinuous on C([−r, T ], E]).Let 0 ≤ t ≤ t ≤ T,and x ∈ Ω.We have Since χ t 0 Ae A(t−s) h(s, Ω s )ds = 0, for all t ∈ [0, T ] the first term on the right hand side converge to zero when t → t uniformly on x ∈ Ω.As consequence we have (3.11)mod c Γ 3 Ω = 0.
From the condition (H1) − (ii) follows immediately that (3.12) Indeed for −r ≤ t ≤ s ≤ 0, and x ∈ Ω, we have For 0 ≤ t ≤ s ≤ T, and x ∈ Ω, we have Since mod c ΓΩ ≤ where L is a positive constant.Consider the ball where r is a constant chosen so that where x 0 = ϕ(0) − h(0, ϕ).Note that the last inequality implies Step 3. The multioperator Γ maps the ball B r (0) into itself.
Let x ∈ B r (0) and y ∈ Γ(x) with where f ∈ sel F (x).

t 0 e
A(t−s) f (s) ds + +e −Lt