Multivalued evolution equations with infinite delay in Fréchet spaces, Electron

In this paper, sufficient conditions are given to investigatethe existence of mild solutions on a semi-infinite interval for two classes of first order semilinear functional and neutral functional differential evolution inclusions with infinite delay using a recent nonlinear alternative for contractive multivalued maps in � � ; � �


Introduction
In this paper, we consider the existence of mild solutions defined on a semi-infinite positive real interval J := [0, +∞), for two classes of first order semilinear functional and neutral functional differential evolution inclusions with infinite delay in a real Banach space (E, | • |).Firstly, in Section 3, we study the following evolution inclusion of the form y (t) ∈ A(t)y(t) + F (t, y t ), a.e.t ∈ J (1) where F : J ×B → P(E) is a multivalued map with nonempty compact values, P(E) is the family of all subsets of E, φ ∈ B are given functions and {A(t)} 0≤t<+∞ is a family of linear closed (not necessarily bounded) operators from E into E that generate an evolution system of operators {U(t, s)} (t,s)∈J×J for 0 ≤ s ≤ t < +∞.For any continuous function y and any t ≥ 0, we denote by y t the element of B defined by y t (θ) = y(t + θ) for θ ∈ (−∞, 0].We assume that the histories y t belongs to some abstract phase space B, to be specified later. In Section 4, we consider the following neutral evolution inclusion of the form d dt [y(t) − g(t, y t )] ∈ A(t)y(t) + F (t, y t ), a.e.t ∈ J 1 Corresponding author EJQTDE, 2008 No. 33, p. 1 where A(•), F and φ are as in problem (1) − (2) and g : J × B → E is a given function.
Finally in Section 5, two examples are provided illustrating the abstract theory.
Functional differential and partial differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years.A good guide to the literature for functional differential equations is the books by Hale [25] and Hale and Verduyn Lunel [27], Kolmanovskii and Myshkis [36], Kuang [37] and Wu [41] and the references therein.
During the last decades, several authors considered the problem of existence of mild solutions for semilinear evolution equations with finite delay.Some results can be found in the books by Ahmed [1,2], Heikkila and Lakshmikantham [28] and Pazy [38] and Wu [41] and the references therein.When the delay is infinite, the notion of the phase space B plays an important role in the study of both qualitative and quantitative theory.A usual choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato in [26], see also Corduneanu and Lakshmikantham [17], Hino et al. [31], Kappel and Schappacher [33], and Schumacher [39].
An extensive theory is developed for inclusion (1) with A(t) = A. We refer the reader to the books by Heikkila and Lakshmikantham [28], Kamenski et al [34] and the pioneer Hino and Murakami paper [30].By means of fixed point arguments, Benchohra and his collaborators have studied many classes of first and second order functional differential inclusions with local and nonlocal conditions in [7,8,10,11,13,14,15,23] on a bounded interval.Extension to the semiinfinite interval is given by Benchohra and Ntouyas in [9,12] and by Henderson and Ouahab [29] with finite delay.
When A is depending on the time, Arara et al [3] considered control multivalued problem on a bounded interval [0, b] and very recently Baghli and Benchohra [4,5] provided uniqueness results for some classes of partial and neutral functional differential evolution equations on the interval J = [0, +∞) when the delay is finite.The perturbed problem with infinite delay is studied in [6].Our main purpose in this paper is to look for the multivalued version of these problems.
Sufficient conditions are established to get existence results of mild solutions which are fixed points of the appropriate operators of the semilinear functional and the neutral functional differential evolution problems by applying the nonlinear alternative of Leray-Schauder type due to Frigon [21] for contractive multivalued maps in Fréchet spaces, combined with the semigroup theory [1,2,38].

Preliminaries
We introduce here notations, definitions and preliminary facts from multivalued analysis which are used throughout this paper.
Let C([0, +∞); E) be the space of continuous functions from [0, +∞) into E and EJQTDE, 2008 No. 33, p. 2 B(E) be the space of all bounded linear operators from E into E, with the norm A measurable function y : [0, +∞) → E is Bochner integrable if and only if |y| is Lebesgue integrable.(For the Bochner integral properties, see Yosida [42] for instance).
In this paper, we will employ an axiomatic definition of the phase space B introduced by Hale and Kato in [26] and follow the terminology used in [31].Thus, (B, • B ) will be a seminormed linear space of functions mapping (−∞, 0] into E, and satisfying the following axioms : 2. Since • B is a seminorm, two elements φ, ψ ∈ B can verify φ−ψ B = 0 without necessarily φ(θ) = ψ(θ) for all θ ≤ 0.

Example 2.3
The spaces C g , UC g , C ∞ g and C 0 g .Let g be a positive continuous function on (−∞, 0].We define : is bounded on (−∞, 0] ; = 0 , endowed with the uniform norm We consider the following condition on the function g. (g 1 ) For all a > 0, sup Then we have that the spaces C g and C 0 g satisfy conditions (A 3 ).They satisfy conditions (A 1 ) and (A 2 ) if (g 1 ) holds.

Example 2.4
The space C γ .For any real constant γ, we define the functional space C γ by Then in the space C γ the axioms (A 1 ) − (A 3 ) are satisfied.

EJQTDE, 2008 No. 33, p. 4
In what follows, we assume that {A(t), t ≥ 0} is a family of closed densely defined linear unbounded operators on the Banach space E and with domain D(A(t)) independent of t.Definition 2.5 We say that a family {A(t)} t≥0 generates a unique linear evolution system {U(t, s)} (t,s)∈∆ for ∆ := {(t, s) ∈ J × J : 0 ≤ s ≤ t < +∞} satisfying the following properties : 1. U(t, t) = I where I is the identity operator in E, 3. U(t, s) ∈ B(E) the space of bounded linear operators on E, where for every (t, s) ∈ ∆ and for each y ∈ E, the mapping (t, s) → U(t, s)y is continuous.
More details on evolution systems and their properties could be found on the books of Ahmed [1], Engel and Nagel [19] and Pazy [38].
Let X be a Fréchet space with a family of semi-norms To X we associate a sequence of Banach spaces {(X n , • n )} as follows : For every n ∈ IN, we consider the equivalence relation ∼ n defined by : x ∼ n y if and only if x − y n = 0 for all x, y ∈ X.We denote X n = (X| ∼n , • n ) the quotient space, the completion of X n with respect to • n .To every Y ⊂ X, we associate a sequence the {Y n } of subsets Y n ⊂ X n as follows : For every x ∈ X, we denote [x] n the equivalence class of x of subset X n and we defined respectively, the closure, the interior and the boundary of Y n with respect to • in X n .We assume that the family of semi-norms { • n } verifies : Let (X, d) be a metric space.We use the following notations : ).Then (P b,cl (X), H d ) is a metric space and (P cl (X), H d ) is a generalized (complete) metric space (see [35]).EJQTDE, 2008 No. 33, p. 5 Definition 2.6 A multivalued map G : J → P cl (X) is said to be measurable if for each x ∈ E, the function Y : J → X defined by is measurable where d is the metric induced by the normed Banach space X.Definition 2.7 A function F : J × B −→ P(X) is said to be an L 1 loc -Carathéodory multivalued map if it satisfies : (iii) for every positive constant k there exists Finally, we say that G has a fixed point if there exists x ∈ X such that x ∈ G(x).
For each y ∈ B +∞ let the set S F,y known as the set of selectors from F defined by For more details on multivalued maps we refer to the books of Deimling [18], Górniewicz [24], Hu and Papageorgiou [32] and Tolstonogov [40].
ii) For every x ∈ X and every ∈ (0, ∞) n , there exists y ∈ F (x) such that x − y n ≤ x − F (x) n + n for every n ∈ N Theorem 2.9 (Nonlinear Alternative of Frigon, [21,22]).Let X be a Fréchet space and U an open neighborhood of the origin in X and let N : U → P(X) be an admissible multivalued contraction.Assume that N is bounded.Then one of the following statements holds : (C1) N has a fixed point; (C2) There exists λ ∈ [0, 1) and x ∈ ∂U such that x ∈ λ N (x).EJQTDE, 2008 No. 33, p. 6

Semilinear Evolution Inclusions
The main result of this section concerns the semilinear evolution problem (1) − (2).Before stating and proving this one, we give first the definition of the mild solution.
Definition 3.1 We say that the function y(•) : (−∞, +∞) → E is a mild solution of the evolution system (1) − (2) if y(t) = φ(t) for all t ∈ (−∞, 0] and the restriction of y(•) to the interval J is continuous and there exists f (•) ∈ L 1 (J; E) : f (t) ∈ F (t, y t ) a.e. in J such that y satisfies the following integral equation : We will need to introduce the following hypothesis which are assumed hereafter : (H1) There exists a constant M ≥ 1 such that : (H2) The multifunction F : J × B −→ P(E) is L 1 loc -Carathéodory with compact and convex values for each u ∈ B and there exist a function p ∈ L 1 loc (J; R + ) and a continuous nondecreasing function ψ : J → (0, ∞) such that: F (t, u) P(E) ≤ p(t) ψ( u B ) for a.e.t ∈ J and each u ∈ B.
(H3) For all R > 0, there exists l R ∈ L 1 loc (J; R + ) such that : for each t ∈ J and for all u, v ∈ B with u B ≤ R and v B ≤ R and For every n ∈ IN, we define in B +∞ the family of semi-norms by : Clearly, the fixed points of the operator N are mild solutions of the problem (1)−( 2).We remark also that, for each y ∈ B +∞ , the set S F,y is nonempty since, by (H2), F has a measurable selection (see [16], Theorem III.6).
For φ ∈ B, we will define the function x(.) : (−∞, +∞) → E by Then x 0 = φ.For each function z ∈ B +∞ , set It is obvious that y satisfies (5) if and only if z satisfies z 0 = 0 and where f (t) ∈ F (t, z t + x t ) a.e.t ∈ J.
+∞ , the multivalued operator F : B 0 +∞ → P(B 0 +∞ ) by : Obviously the operator inclusion N has a fixed point is equivalent to the operator inclusion F has one, so it turns to prove that F has a fixed point.
Let z ∈ B 0 +∞ be a possible fixed point of the operator F .Given n ∈ N, then z should be solution of the inclusion z ∈ λ F (z) for some λ ∈ (0, 1) and there exists EJQTDE, 2008 No. 33, p. 8 Assumption (A 1 ) gives Using the nondecreasing character of ψ, we get Then We consider the function µ defined by By the previous inequality, we have Let us take the right-hand side of the above inequality as v(t).Then, we have EJQTDE, 2008 No. 33, p. 9 From the definition of v, we have Using the nondecreasing character of ψ, we get This implies that for each t ∈ [0, n] and using the condition ( 6), we get Thus, for every t ∈ [0, n], there exists a constant Λ n such that v(t) ≤ N n and hence Clearly, U is an open subset of B 0 +∞ .We shall show that F : U → P(B 0 +∞ ) is a contraction and an admissible operator.First, we prove that F is a contraction ; Let z, z ∈ B 0 +∞ and h ∈ F (z). Then there exists f (t) ∈ F (t, z t + x t ) such that for each t ∈ [0, n] From (H3) it follows that Consider U : [0, n] → P(E), given by Since the multivalued operator V(t) = U (t) ∩ F (t, z t + x t ) is measurable (in [16], see Proposition III.4), there exists a function f (t), which is a measurable selection for V. So, f (t) ∈ F (t, z t + x t ) and using (A 1 ), we obtain for each t ∈ [0, n] Then we have Therefore, By an analogous relation, obtained by interchanging the roles of z and z, it follows that So, F is a contraction for all n ∈ N.
Now we shall show that F is an admissible operator.Let z ∈ B 0 +∞ .Set, for every n ∈ N, the space and let us consider the multivalued operator F : B 0 n → P cl (B 0 n ) defined by : where f ∈ S n F,y = {v ∈ L 1 ([0, n], E) : v(t) ∈ F (t, y t ) for a.e.t ∈ [0, n]}.From (H1) − (H3) and since F is a multivalued map with compact values, we can prove that for every z ∈ B 0 n , F (z) ∈ P cl (B 0 n ) and there exists z ∈ B 0 n such that z ∈ F (z ).Let h ∈ B 0 n , y ∈ U and > 0. Assume that z ∈ F (z), then we have

EJQTDE, 2008 No. 33, p. 11
Since h is arbitrary, we may suppose that h ∈ B(z , So, F is an admissible operator contraction.From the choice of U there is no z ∈ ∂U such that z = λ F (z) for some λ ∈ (0, 1).Then the statement (C2) in Theorem 2.9 does not hold.We deduce that the operator F has a fixed point z .Then y (t) = z (t) + x(t), t ∈ (−∞, +∞) is a fixed point of the operator N, which is a mild solution of the evolution inclusion problem (1) − (2).

Semilinear Neutral Evolution Inclusions
In this section, we give existence results for the neutral functional differential evolution problem with infinite delay (3) − (4).Firstly we define the mild solution.
Definition 4.1 We say that the function y(•) : (−∞, +∞) → E is a mild solution of the neutral evolution system (3) − (4) if y(t) = φ(t) for all t ∈ (−∞, 0], the restriction of y(•) to the interval J is continuous and there exists f (•) ∈ L 1 (J; E) : f (t) ∈ F (t, y t ) a.e. in J such that y satisfies the following integral equation We consider the hypotheses (H1) − (H3) and we will need the following assumptions : (H4) There exists a constant M 0 > 0 such that : (H5) There exists a constant 0 < L < 1 M 0 K n such that : ) for all t ∈ J and φ ∈ B.
For every n ∈ IN, let us take here l n (t) = M K n [L + l n (t)] for the family of seminorm { • n } n∈IN defined in Section 3. In what follows we fix τ > 1 and assume Theorem 4.2 Suppose that hypotheses (H1) − (H6) are satisfied and moreover with Then the neutral evolution problem (3) − (4) has a mild solution.
Consider the multivalued operator N : B +∞ → P(B +∞ ) defined by : Clearly, the fixed points of the operator N are mild solutions of the problem (3)−(4).We remark also that, for each y ∈ B +∞ , the set S F,y is nonempty since, by (H2), F has a measurable selection (see [16], Theorem III.6).
Obviously the operator inclusion N has a fixed point is equivalent to the operator inclusion F has one, so it turns to prove that F has a fixed point.
Let z ∈ B 0 +∞ be a possible fixed point of the operator F. Given n ∈ N, then z should be solution of the inclusion z ∈ λ F(z) for some λ ∈ (0, 1) and there exists EJQTDE, 2008 No. 33, p. 14 Using the inequality (8) and the nondecreasing character of ψ, we obtain We consider the function µ defined by Let us take the right-hand side of the above inequality as v(t).Then, we have From the definition of v, we have v(0) = δ n and Using the nondecreasing character of ψ, we get EJQTDE, 2008 No. 33, p. 15 This implies that for each t ∈ [0, n] and using the condition (10), we get .
We can show as in Section 3 that F is an admissible operator and we shall prove now that F : U → P(B 0 +∞ ) is a contraction.
Let z, z ∈ B 0 +∞ and h ∈ F(z).Then there exists f (t) ∈ F (t, z t + x t ) such that for each t ∈ [0, n] h(t) = g(t, z t + x t ) − U(t, 0)g(0, φ) + t 0 U(t, s)A(s)g(s, z s + x s )ds + t 0 U(t, s)f (s)ds From (H3) it follows that H d (F (t, z t + x t ), F (t, z t + x t )) ≤ l n (t) z t − z t B .
Consider U : [0, n] → P(E), given by Since the multivalued operator V(t) = U (t) ∩ F (t, z t + x t ) is measurable (in [16], see Proposition III.4), there exists a function f (t), which is a measurable selection for V. So, f (t) ∈ F (t, z t + x t ) and using (A 1 ), we obtain for each t ∈ [0, n]  Therefore, given by H d (A, B) = max sup 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