Local attractivity for integro-di erential equations with noncompact semigroups

Abstract: In this paper, we are devoted to study the existence and local attractivity of solutions for a class of integro-di erential equations.Under the situation that the nonlinear term satisfy Carathéodory conditions and a noncompactness measure condition, we establish some existence and local attractivity of mild solutions by utilizingMönch xed point theorem, Kuratowskimeasure of noncompactness and resolvent operator theory in the sense of Grimmer.Our investigationswill be situated in the Banach space of real functionswhich are de ned, continuous, and bounded on the right-hand real half axis R+. Moreover an example is given to illustrate our outcomes.


Introduction
Evolution equations are used in many elds of applied mathematics [9,33].In recent years, this type of equation has gained a lot of attention [39].The theory of integro-di erential equations on in nite intervals plays a key role as it addresses certain real world problems; it has gained considerable attention recently and now represents an important nonlinear analysis branch. Broad applications for physics, mechanics, engineering, electric, chemistry, economy and other areas [2,6,35,36] have, in recent years, been found in integrodi erential equations; while numerous research papers and monographs have been written, devoting themselves to functional and integro-di erential equations [3,4,7,21,31,34,37]. Such documents contain speci c qualitative properties like existence, uniqueness, stability and asymptotic behavior of rst-order equations. On the other hand measure of noncompactness is one of the most useful tools in nonlinear and functional analysis, metric xed point theory and the theory of operator equations in Banach spaces which was rst introduced by Kuratowski in [28]. This concept also used to investigate functional equation, ordinary and partial di erential equations, integral and integro-di erential equations. In this context several authors have pre-sented some papers on the existence of solution for nonlinear integral equations which involves the use of measure of noncompactness and many other techniques, for instance see [15,16] and [10,11].
Motivated by the above discussion, in this paper we consider the following integro-di erential equation of the form t ρ t, r, ϑ(r) dr , t ∈ I = [ , +∞), ϑ( ) = ϑ , (1.1) where A is the in nitesimal generator of a strongly semigroup (T(t)) t≥ on a Banach space W with domain D(A).
Here Υ(t) is a closed linear operator on W, with domain D(A) ⊂ D(Υ(t)), which is independent of t. F : I × W × W → W is a Carathéodory function, ρ : ∆ × W → W is a continuous function, ∆ : = {(t, s) ∈ I × I : s ≥ t}, ϑ ∈ W, and (W, . ) is a real Banach space.
Moreover, up to now no work has been reported yet regarding the local attractivity for a class of integro-di erential equations, which motivates this present study. The main contributions of this paper are summarized as follows: • In this work, a general class of integro-di erential equation is considered.
• Using methods of functional analysis, a set of su cient conditions are proposed for ensuring the existence and local attractivity of mild solutions.
• The results are established with the use of the theory of resolvent operator in the sense of Grimmer and Kuratowski measure of noncompactness.We use the fact that the operator-norm continuity of the resolvent operator is equivalent to that of the semigroup generated by the coe cient operator. This property allows us to drop the supposition that the operator semigroup is compact and to show that the operator solution meets the requirements of the Mönch conditions.
• The result in this paper generalizes and improves some of previous ones in this eld.Our paper expands the usefulness of integro-di erential equations, since the literature shows results for existence and attractivity for such equations in the case of semigroup only.
The rest of our paper is as follows.In the next section, we recall some preliminaries about the resolvent operator theory, measure of noncompactness and the xed point theory for condensing maps. Section 3 is devoted to the existence result under a general setting via measure of noncompactness. Section 4 shows the attractivity of the solution to the problem (1.1). The last section gives an example, which illustrates the feasibility of the abstracts results obtained in the paper.

Preliminaries
In this section, we collect some notations, de nitions and supplementary informations that are included in the further considerations of this article.Throughout the paper, W is a Banach space, A and Υ(t) are closed linear operators on W. B represents the Banach space D(A) equipped with the graph norm We recall some knowledge on partial integro-di erential equations and the related resolvent operators. Let us consider the following system for further purposes : (i) R( ) = I (the identity map of W) and R(t) ≤ Ne βt for some constants N > and β ∈ R (ii) For each ϑ ∈ W, R(t)ϑ is strongly continuous for t ≥ .
In what follows, we make the following assumptions.
(H ) The operator A is the in nitesimal generator of a strongly continuous semigroup (T(t)) t≥ on W.
(H ) For all t ≥ , Υ(t) is closed linear operator from D(A) to W and Υ(t) ∈ L (B, W). For any ϑ ∈ W, the map t → Υ(t)ϑ is bounded, di erentiable and the derivative t → Υ ′ (t)ϑ is bounded and uniformly continuous on R + .
The following theorem provides adequate conditions to ensure that the resolvent operator for equation (2.1) exists. In the following, we give some results for the existence of solutions for the following integro-di erential equation.

Theorem 2.2 ([38]). Assume that hypotheses (H ) and (H ) hold. If ϑ is a strict solution of the Eq.(2.2), then the variation of constant formula holds
We recall the following theorem which sets the equivalence between the C -semigroup operator-norm continuity and the integral equations resolvent operator.

2.2) is operator-norm continuous (or continuous in the uniform operator topology) for t > if and only if (T(t)) t≥ is operator-norm continuous for t > .
We remember some characteristics of the measure of noncompactness and the Mönch xed-point theorem to prove the paper's primary outcome.

De nition 2.3 ([12]). Let D be a bounded subset in any Banach space Y. The Kuratowski measure of noncompactness (shortly MNC) is de ned by α(D) = inf{ϵ > : D has a nite cover by sets of diameter ≤ ϵ}.
Let us recall the basic properties of Kuratowski measure of noncompactness.

Lemma 2.4. [12]
Let Y be a Banach space and D , D be bounded subset of Y, then the following properties hold: Denote by ω T (ϑ, ϵ) the modulus of continuity of ϑ on the interval [ , T] i.e Moreover, let us put For more details about properties of ω T (.), one can see [29]. The next results play an important role in demonstrating our key ndings.

Lemma 2.6 ([5]. p. 35). Let A(t), B(t), C(t) and D(t) be real valued nonnegative integrable functions on [ , +∞), for which the inequality
We introduce now the concept of attractivity (stability) of solutions of operator equations in the space BC(I, W). To this end, assume that F is a nonempty subset of the space BC(I, W). Moreover, let Π be an operator de ned on F with values in BC(I, W). Let us consider the operator equation of the form (2.4)

De nition 2.4. We say that solutions of (1.1) are locally attractive if there exists a ball B(ϑ * , δ) in the space
We will say that solutions of (2.4) are uniformly locally attractive when for each ϵ > there exists a T > such that Then F has a xed point in O.

Existence of mild solutions
De nition 3.1. A function ϑ(·) ∈ BC(I, W) is said to be a mild solution of the system (1.1) if The following hypotheses will be further considered in order to present and show the existence outcomes for the issue (1.1).
(C ) The semigroup (T(t)) t≥ is norm continuous for t > .
(C ) There exist constants m ≥ and γ > satisfying (C ) The function F : I × W × W → W is Carathéodory and satis es the following conditions : (ii) There exists an integrable function f : I → R + , such that : for a.e t ∈ I and each Φ i , Ψ i ∈ W , (i = , ) and lim t→+∞ t e −γ(t−r) f(r)dr = .
(C ) The function ρ : ∆ × W → W satis es the following conditions: (i) There exists an integrable function g : I → R + , such that:  Observe that the xed points of the operator Γ are mild solutions of the problem (1.1).

Case1. If t ∈ [ , T], T > , then, we have
We choose T ≥ T, then By hypothesis (C ) − (ii) the above inequality, reduces to This implies that Γ is continuous.
Step 3 Γ(Bq) is equicontinuous. Let t , t ∈ [ , T] with t < t and ϑ ∈ Bq. Then, we have By the continuity of (R(t)) t≥ in the operator-norm topology and the dominated convergence theorem, we conclude that the right hand side of the above inequality tends to zero and independent of ϑ as t → t . Step 4 We show that the Mönch conditions hold.
Since n > , then sup t∈I e −n σ(t) α(K(r)) = . (3.7) We have  Thus we nd that K is relatively compact.
Then from the above inequality, it follows that This implies the uniform local attractivity of solutions of the integro-di erential equation (1.1).

Example
We consider the following class of partial integro-di erential system: Let W = L ( , ). We de ne the operator A induced on W as follows: From [24, p. 173], we know that A is the in nitesimal generator of an analytic C semigroup (T(t)) t≥ on W.
Since the semigroup generated by A is analytic, then it is norm continuous for t > .Thus by Theorem 2.3 the corresponding resolvent operator is operator-norm continuous for t > .
We de ne the operators Υ(t) : Y → X as follows: Υ(t)N = Ξ(t)AN for t ≥ and N ∈ D(A).
We put ϑ(t)(ξ ) = v(t, ξ ) for t ∈ [ , +∞), and de ne We suppose Ξ is a bounded and C function such that Ξ ′ is bounded and uniformly continuous. Accordingly, the assumptions (H ) and (H ) are ful lled. Thus from Theorem 2.1 and Theorem 2.3, the problem (5.1) has a resolvent operator(R(t)) t≥ on W which is norm continuous for t > . We assume moreover that there exists β > a > and Ξ(t) < a e −β t for all t ≥ . Thanks to Lemma 5.