Global Existence and Stability for Second Order Functional Evolution Equations with Infinite Delay

In this article, the authors give sufficient conditions for existence and at-tractivity of mild solutions for second order semi-linear functional evolution equation in Banach spaces using Schauder's fixed point theorem. An example is provided to illustrate the result.


Introduction
In this paper, we consider the existence and attractivity of mild solutions of the second order evolution equation y (t) − A(t)y(t) = f (t, y t ), t ∈ J := [0, ∞), where (E, | • |) a real Banach space, {A(t)} 0≤t<+∞ is a family of linear closed operators from E into E that generate an evolution system of operators {U (t, s)} (t,s)∈J×J for 0 ≤ s ≤ t < +∞, f : J × B → E is a Carathéodory function, B is an abstract phase space to be specified later, ỹ ∈ E, and φ ∈ B.
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 up to the present time t.We assume that the histories y t belong to B.
Differential equations on infinite intervals frequently occur in mathematical modeling of various applied problems.For example, in the study of unsteady flow of a gas through a semi-infinite porous medium [3,19], the analysis of the mass transfer on a rotating disk in a non-Newtonian fluid [4], heat transfer in the radial flow between parallel circular disks [22], investigation of the temperature distribution in the problem of phase change of solids with temperature dependent thermal conductivity [22], as well as numerous problems arising in the study of circular membranes [2,9,10], plasma physics [4], nonlinear mechanics, and non-Newtonian fluid flows [2].
This paper is organized as follows.In Section 2, we recall some definitions and facts about evolution systems.In Section 3, we prove the existence of mild solutions to the problem (1.1)-(1.2).In Section 4, we show the attractivity of mild solutions, and in the last section, an example is given to show the applicability of our results.
To our knowledge, no papers devoted to the global existence and the attractivity of mild solutions of problem (1.1)-(1.2) have appeared in the literature.The present work attempts to fill that gap.

Preliminaries
Let E be a Banach space with the norm | • | and let BC(J, E) be the Banach space of all bounded and continuous functions y mapping J into E with the usual supremum norm Let X be the space defined by X = y : R → E | y | J ∈ BC(J, E) and y 0 ∈ B , where by y | J we mean the restriction of y to J.
In this paper, we will use an axiomatic definition of the phase space B introduced by Hale and Kato in [13] and follow the terminology used in [18].Thus, (B, • B ) will be a seminormed linear space of functions mapping (−∞, 0] into E, and satisfying the following axioms. (ii) there exists a positive constant H such that |y(t)| ≤ H y t B ; (iii) there exist functions K, M : R + → R + independent of y with K continuous and M locally bounded such that: The space B is complete.
Remark 2.1.In the sequel, we assume that K and M are bounded on J and For additional details we refer the reader, for example, to the book by Hino et al. [18].
In what follows, let {A(t), t ≥ 0} be a family of closed linear operators on the Banach space E with domain D(A(t)) that is dense in E and independent of t.The existence of solutions to the problem (1.1)-(1.2) is related to the existence of an evolution operator U (t, s) for the homogeneous problem This concept of evolution operator has been developed by Kozak [21].
s ≤ t}, is called an evolution operator of the equation (2.1) if the following conditions hold.
(D 1 ) For any x ∈ E the map (t, s) −→ U (t, s)x is continuously differentiable and: (a) for any t ∈ J, U (t, t) = 0; (b) for all (t, s) ∈ ∆ and for any x and ∂ 3 ∂s 2 ∂t U (t, s)x exist, and: (Corduneanu [7]).Let C ⊂ BC(R + , E) be a set satisfying the following conditions: (ii) the functions belonging to C are equicontinuous on any compact interval of R + ; (iii) the set C(t) := {y(t) : y ∈ C} is relatively compact on any compact interval of R + ; Then C is relatively compact in BC(R + , E).
Our final lemma is the well known Schauder fixed point theorem [11].
Lemma 2.4.Let C be a nonempty closed convex bounded subset of a Banach space E. Then any continuous compact mapping T : C → C has a fixed point.

Main result
We begin with the definition of a mild solution to our problem.Definition 3.1.A function y ∈ X is called a mild solution to the problem (1.1)-(1.2),if y is continuous and To prove our results we introduce the following conditions.
(H 1 ) There exists a constant M ≥ 1 and ω > 0 such that Proof.It is clear that the fixed points of the operator T : X → X defined by are mild solutions of problem (1.1)-(1.2).For φ ∈ B, let x : (−∞, +∞) → E be the function defined by It is clear that y satisfies (3.2) if and only if z satisfies z 0 = 0 and for all t ∈ J z(t) = t 0 U (t, s) f (t, x s + z s )ds. (3.3) In the sequel, we always take X 0 to be the Banach space X 0 = {z ∈ X : z 0 = 0} endowed with the norm Now, we can consider the operator L : X 0 → X 0 given by The problem (1.1) having a solution is equivalent to L having a fixed point.To prove that problem (1.1) does in fact have a solution, we begin with the following estimation.
For any z ∈ X 0 and t ∈ J, we have Now, we will show that the operator L satisfied the conditions of Schauder's fixed point theorem.
Step 1. L is continuous.Let (z k ) k∈N be a sequence in X 0 such that z k → z in X 0 ; then for any t ∈ J, we obtain Hence, from the continuity of the function f and the Lebesgue dominated convergence theorem, we obtain So L is continuous.
Let D η be a bounded subset of X 0 .To show that L(D η ) is relatively compact we will use Lemma 2.3.
L(D η ) is equicontinuous.Let s, t ∈ [0, b] with t > s and z ∈ D η .Then, we have From inequality (3.4), we obtain The right-hand side of the above inequality tends to zero as t − s → 0, which implies that L(D η ) is equicontinuous.
Let t ∈ J be a fixed and let 0 < ε < t ≤ b.For z ∈ D η , we define Since U (t, s) is a compact operator, and the set

Attractivity of solutions
In this section we study the local attractivity of solutions the problem (1.1)-(1.2).Definition 4.1 ([8]).Solutions of (1.1) are locally attractive if there exists a closed ball B(z * , σ) in the space X 0 for some z * ∈ X such that, for any solutions z and z of (1.1)-(1.2) belonging to B(z * , σ), we have lim Under the assumptions of Section 3, let z * be a solution to (1.1)-(1.2) and B(z * , σ) the closed ball in X 0 where σ satisfies Then, for z ∈ B(z * , σ), from (H Consequently, the solutions of problem (1.1)-(1.2) are locally attractive.
is the image of bounded set in E by U (t, s), we see that Λ ε is precompact in E. Furthermore, for z ∈ D η , we haveThe right-hand side tends to zero as ε → 0, so L ε (z) converge uniformly to L(z), which implies that D η (t) is precompact in E.L is equiconvergent.Let z ∈ D; then from conditions (H 1 )-(H 3 ) and (3.4), we have which implies that L is equiconvergent.Therefore, by Lemma 2.3, L(D η ) is relatively compact.Hence, by Lemma 2.4, the operator L has at least one fixed point which in turn is a mild solution of problem (1.1)-(1.2).