Mild solutions for perturbed evolution equations with infinite state-dependent delay

In this paper, we give sucient conditions to get the existence of mild solutions for two classes of rst order partial and neutral of perturbed evolution equations by using the nonlinear alternative of Avramescu for contractions operators in Fr echet spaces, combined with semigroup theory. The solution here is depending on an innite delay and is giving on the real half-line.


Introduction
In this paper, we give the existence of mild solutions defined on a semi-infinite positive real interval J := [0, +∞) for two classes of first order of semilinear functional and neutral functional perturbed evolution equations with state-dependent delay in a real separable Banach space (E, | • |) when the delay is infinite.
Firstly, we present some preliminary concepts and results in Section 2 and then in Section 3 we study the following semilinear functional perturbed evolution equations with state-dependent delay y (t) = A(t)y(t) + f (t, y ρ(t,yt) ) + h(t, y ρ(t,yt) ), a.e.t ∈ J, where B is an abstract phase space to be specified later, f, h : J ×B → E, ρ : J ×B → IR and 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].Here y t (•) represents the history of the state from time t ≤ 0 up to the present time t.We assume that the histories y t belong to B.
Finally in Section 5, two examples are given to illustrate the abstract theory.
Differential delay equations, or functional differential equations, have been used in modeling scientific phenomena for many years.Often, it has been assumed that the delay is either a fixed constant or is given as an integral in which case is called distributed delay; see for instance the books [20,23,30], and the papers [15,19].
However, complicated situations in which the delay depends on the unknown functions have been proposed in modeling in recent years.These equations are frequently called equations with state-dependent delay.Existence results and among other things were derived recently for functional differential equations when the solution is depending on the delay on a bounded interval for impulsive problems.We refer the reader to the papers by Abada et al. [1], Ait Dads and Ezzinbi [5], Anguraj et al. [6], Hernández et al. [21] and Li et al. [24].
Our main purpose in this paper is to extend some results from the cite literature devoted to state-dependent delay and those considered on a bounded interval for the evolution problems studied in [14].We provide sufficient conditions for the existence of mild solutions on a semiinfinite interval J = [0, +∞) for the two classes of first order semilinear functional and neutral functional perturbed evolution equations with statedependent delay (1) − (2) and ( 3) − (4) with state-dependent delay when the delay is infinite using the nonlinear alternative of Avramescu for contractions maps in Fréchet spaces [7], combined with semigroup theory [4,25].

Preliminaries
We introduce notations, definitions and theorems which are used throughout this paper.
Let C([0, +∞); E) be the space of continuous functions from [0, +∞) into E and B(E) be the space of all bounded linear operators from E into E, with the usual supremum norm  [31]).
Let L 1 ([0, +∞), E) denotes the Banach space of measurable functions y : [0, +∞) → E which are Bochner integrable normed by In this paper, we will employ an axiomatic definition of the phase space B introduced by Hale and Kato in [19] and follow the terminology used in [22].Thus, (B, • B ) will be a seminormed linear space of functions mapping (−∞, 0] into E, and satisfying the following axioms.(A 3 ) The space B is complete.
3. From the equivalence of in the first remark, we can see that for all φ, ψ ∈ B such that φ − ψ B = 0 : We necessarily have that φ(0) = ψ(0). We We have that the spaces BU C, C ∞ and C 0 satisfy conditions (A 1 ) − (A 3 ).However, BC satisfies (A 1 ), (A 3 ) but (A 2 ) is not satisfied.

Example 2.3
The spaces C g , U C 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 Then we have that the spaces C g and C 0 g satisfy conditions (A 3 ).We consider the following condition on the function g.
(g 1 ) For all a > 0, sup They satisfy conditions (A 1 ) and (A 2 ) if (g 1 ) holds.(ii) for each y ∈ B the function f (., y) : J → E is measurable ; (iii) for every positive integer k there exists 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.
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 [3], Engel and Nagel [16] and Pazy [25].
Let X be a Fréchet space with a family of semi-norms { • n } n∈N .We assume that the family of semi-norms { • n } verifies To X we associate a sequence of Banach spaces {(X n , • n )} as follows : For every n ∈ N, we consider the equivalence relation ∼ n defined by : x ∼ n y if and only if x − y n = 0 for 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 {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 The following definition is the appropriate concept of contraction in X.

EJQTDE, 2013
No. 59, p. 5 Definition 2.7 [18] A function f : X → X is said to be a contraction if for each n ∈ N there exists k n ∈ (0, 1) such that The corresponding nonlinear alternative result is as follows Theorem 2.8 (Nonlinear Alternative of Avramescu, [7]).Let X be a Fréchet space and let A, B : X −→ X be two operators satisfying (1) A is a compact operator.
Then one of the following statements holds (Av1) The operator A + B has a fixed point;

Semilinear evolution equations
Before stating and proving the main result, we give first the definition of a mild solution of the semilinear perturbed evolution problem (1) − (2).
Definition 3.1 We say that the function y : R → E is a mild solution of (1) − (2) if y(t) = φ(t) for all t ≤ 0 and y satisfies the following integral equation We always assume that ρ : J × B → R is continuous.Additionally, we introduce the following hypothesis (H φ ) The function t → φ t is continuous from R(ρ − ) into B and there exists a continuous and bounded function L φ : R(ρ − ) → (0, ∞) such that Remark 3.2 The condition (H φ ), is frequently verified by continuous and bounded functions.For more details, see for instance [22].
We will need to introduce the following hypotheses which are assumed thereafter Consider the following space Let us fix τ > 1.For every n ∈ N, we define in B +∞ the semi-norms by and l n is the function from (H3).
Then B +∞ is a Fréchet space with those family of semi-norms • n∈N . where with has a mild solution on (−∞, +∞).
Proof.We transform the problem (1) − (2) into a fixed-point problem.Consider the operator N : B +∞ → B +∞ defined by Clearly, fixed points of the operator N are mild solutions of the problem (1) − (2).
For φ ∈ B, we will define the function x(.) : R → 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 Obviously the operator N having a fixed point is equivalent to F + G having one, so it turns to prove that F + G has a fixed point.First, show that F is continuous and compact.
Step 1 : First, we show the continuity of F .Let (z n ) n∈N be a sequence in B 0 +∞ such that z n → z in B 0 +∞ .By the hypothesis (H1), we have Since f is continuous, by dominated convergence theorem of Lebesgue, we get Step 2 : Show that F transforms any bounded of B 0 +∞ in a bounded set.For each d > 0, there exists a positive constant ξ such that for all z ∈ B d = {z ∈ B 0 +∞ : z n ≤ d} we get F (z) n ≤ ξ.Let z ∈ B d , from assumption (H1) and (H2), we have for each t ∈ [0, n] From (H φ ), Lemma 3.3 and assumption (A 1 ), we have for each Using the nondecreasing character of ψ, we get for each t ∈ [0, n] So there is a positive constant such that Step 3 : F maps bounded sets into equicontinuous sets of B 0 +∞ .We consider B d as in Step 2 and we show that Then by (7) and the nondecreasing character of ψ, we get The right-hand of the above inequality tends to zero as τ 2 − τ 1 −→ 0, since U (t, s) is a strongly continuous operator and the compactness of U (t, s) for t > s, implies the continuity in the uniform operator topology (see [4,25]).As a consequence of Steps 1 to 3 together with the Arzelà-Ascoli theorem it suffices to show that the operator F maps B d into a precompact set in E.
Let t ∈ J be fixed and let ε be a real number satisfying 0 < ε < t.For z ∈ B d , we define every ε,0 < ε < t.Moreover, using the definition of w, we get Step 4 : G is a contraction.Let z, z ∈ B 0 +∞ .By the hypotheses (H1) and (H4), we get for each t ∈ [0, n] and n ∈ N Use the inequality (7), to get Then the operator G is a contraction for all n ∈ N.
Step 5 : For applying Theorem (2.8), we must check (Av2) : i.e. it remains to show that the set Let z ∈ Γ.By (H1) − (H2) and (H4), we have for each t ∈ [0, n] EJQTDE, 2013 No. 59, p. 11 Use Proposition (3.4) and inequality ( 7) We consider the function u(t) := sup The nondecreasing character of ψ gives with the fact that 0 < λ < 1 We consider the function µ defined by From the previous inequality, we have for all t ∈ [0, n] Let us take the right-hand side of the above inequality as v(t).Then, we have From the definition of v, we have Using the nondecreasing character of ψ, we get EJQTDE, 2013 No. 59, p. 12 So, using (6) for each t ∈ [0, n], we get .
Thus, for every t ∈ [0, n], there exists a constant Λ n such that v(t) ≤ Λ n and hence µ(t) ≤ Λ n .Since z n ≤ µ(t), we have z n ≤ Λ n .This shows that the set Γ is bounded.Then the statement (Av2) in Theorem 2.8 does not hold.The nonlinear alternative of Avramescu implies that (Av1) is satisfied, we deduce that the operator F + G has a fixed point z .Then y (t) = z (t) + x(t), t ∈] − ∞, +∞[ is the fixed point of the operator N which is a mild solution of the problem (1) − (2).

Semilinear neutral evolution equations
In this section, we give an existence result for the problem (3) − (4).Firstly we define the concept of the mild solution for that problem.
Proof.Consider the operator N : B +∞ → B +∞ defined by Then, fixed points of the operator N are mild solutions of the problem (3) − (4).
For φ ∈ B, we consider the function x(.) : R → E defined as below by It is obvious that y satisfies (8) if and only if z satisfies z 0 = 0 and Use the inequality (7) to get Therefore Let us fix τ > 0 and assume that then the operator G is a contraction for all n ∈ N.

(A 1 )
If y : (−∞, b) → E, b > 0, is continuous on [0, b] and y 0 ∈ B, then for every t ∈ [0, b) the following conditions hold (i) y t ∈ B; (ii) There exists a positive constant H such that |y(t)| ≤ H y t B ; (iii) There exist two functions K(•), M (•) : R + → R + independent of y with K continuous and M locally bounded such that y t B ≤ K(t) sup{ |y(s)| : 0 ≤ s ≤ t} + M (t) y 0 B .(A 2 ) For the function y in (A 1 ), y t is a B−valued continuous function on [0, b].
Therefore the set Z(t) = {F (z)(t) : z ∈ B d } is totally bounded.So we deduce from Steps 1, 2 and 3 that F is a compact operator.EJQTDE, 2013 No. 59, p. 10
EJQTDE, 2013 No. 59, p. 2 A measurable function y : [0, +∞) → E is Bochner integrable if and only if |y| is Lebesgue integrable.(For the Bochner integral properties, see the classical monograph of Yosida [22]indicate some examples of phase spaces.For other details we refer, for instance to the book by Hino et al.[22]. H2) There exists a function p ∈ L 1 loc (J; R + ) and a continuous nondecreasing function ψ : R + → (0, ∞) and such that |f (t, u)| ≤ p(t) ψ( u B ) for a.e.t ∈ J and each u ∈ B.