Existence results for some integro-di erential equations with state-dependent nonlocal conditions in Fréchet Spaces

Abstract: In this work, we present existence of mild solutions for partial integro-di erential equations with state-dependent nonlocal local conditions. We assume that the linear part has a resolvent operator in the sense given by Grimmer. The existence of mild solutions is proved bymeans of Kuratowski’s measure of noncompactness and a generalized Darbo xed point theorem in Fréchet space. Finally, an example is given for demonstration.

In the past few decades, interest in a variety of problems such as delay di erential equations (DDEs), retarded di erential equations (RDEs), neutral delay di erential equations (NDDEs) has expanded. The main areas of application are biological science, economics, materials science, medicine, public health and robotics ( [13,16,17]).
The theory of integro-di erential equations has become an active area of investigation due to their applications in the elds such as engineering, mechanics, physics, chemistry, biology, economics, ecology and so on. One can see [22] and references therein. In various works the problem of existence of solutions of the Cauchy problem for integro-di erential equations has been studied; we refer the reader to books [1,15,18] and to papers [2,3]. In addition, the nonlinear integro-di erential equations with resolvent operator serve as an abstract formulation of partial integro-di erential equations that arise in many physical phenomena.
On the other hand, nonlocal conditions are known to make a much better description of real models than classical initial ones. Byszewski's work [7] provides the rst result as well as the physical signi cance for nonlocal issues. It then generated increased interest in many nonlocal issues regarding di erential equations. Some basics outcomes on nonlocal issues are obtained see [8,9,22] and the references therein for additional commentary and citations. Hernandez and O'Regan's work [15] proposes the concept of state-dependent nonlocal conditions which generalizes many nonlocal conditions. Recently, with this class of conditions, Hernandez studied the existence and uniqueness of solution for a general class of abstract di erential equations with state dependent delay (see [14]) and in [4], Benchohra and al discussed the existence of mild solutions for non-linear fractional integrodi erential equations. Motivated by the previously mentionned works, in this paper we will extend some such results of mild solutions for the following abstract integro-di erential system with state-dependent delays : In our current paper, we will investigate the existence of solutions for the previously mentioned integrodi erential system since this problem still has not been considered in the literature. The main contributions of this paper are summarized as follows : 1. The study of integro-di erential equations via measure of noncompactness in the form (1) is an untreated topic in the literature and this is an additional motivation for writing this paper.
2. We establish some su cient conditions for the nonlocal existence by means of Darbo xed-point Theorem via the noncompactness measure in Fréchet space.
3. The results are established with the use of the theory of resolvent operator in the sense of Grimmer. The structure of this work is as follows: Sect. 2, is preliminaries on some basic de nitions, lemmas and notations. Sect. 3 is focused upon existence of mild solution of Eq. (1). In Sect. 4, we provide a concrete example to illustrate the e ciency of our results. The last section is devoted to our conclusions.

Notations and preliminaries
In this section, we give basic concepts, De nitions and Lemmas which will be used in the sequel, to obtain the main results.
By L(X) we denote the Banach space of bounded linear operators from X into X, with norm Let L (J, X) denote the Banach space of measurable functions ϑ : J → X which are Bochner integrable, endowed with the norm Let C(J, X) be the Banach space of continuous functions from J into X, furnished with the norm Finally, we introduce the space C(R + ) which is the Fréchet space of all continuous functions ϑ : R + −→ X, equipped with the family seminorms We recall some basic results about the resolvent operators for the following linear homogeneous equation where A and Υ(t) are closed linear operators on X. In the following Y represents the Banach space D(A) equipped with the graph norm ϑ Y := Aϑ + ϑ for ϑ ∈ Y. C(R + , Y), L(Y, X) stand for the space of all continuous functions from R + into X and the set of all bounded linear operators from Y into X, respectively.

De nition 2.1 ([12]). A resolvent operator for Eq. (2) is a bounded linear operator valued function R(t) ∈ L(X)
for t ≥ , having the following properties : (i) R( ) = I and R(t) ≤ Ne βt for some constants N and β .
In order to get the existence of the resolvent operators, we assume the following assumptions : (H )The operator A is the in nitesimal generator of a strongly continuous semigroup (T(t)) t≥ on X.
(H )For all t ≥ , Υ(t) is a closed linear operator from D(A) to X. For any ϑ ∈ X, 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 the system (2) exists.

Theorem 2.1 ([12]). Assume that (H )-(H ) hold. Then there exists a unique resolvent operator to the Cauchy problem (2).
In what follows, we give some results for the existence of solutions for the following integro-di erential equation.

Theorem 2.2 ([12]). Assume that hypotheses (H ) and (H ) hold. If ϑ is a strict solution of the Eq.(3), then the variation of constant formula holds
We recall the following de nition of the notion of a sequence of measures of noncompactness [10,11].  [20,21]).

Remark 2.1. Notice that if the set D is equicontinuous, then ω n (D) = .
In the sequel, we need the useful following results for the computation of α(·) De nition 2.4. A nonempty subset D ⊂ F is said to be bounded if for n ∈ N, there exists Mn > such that y n ≤ Mn , for each y ∈ D.

Lemma 2.4 ([6]). If D is a bounded subset of a Banach space X, then for each ϵ >
there is a sequence

De nition 3.1. A function ϑ ∈ C([−r, +∞); X) is called the mild solution of the system
In this work, we will work under the following assumptions : (A ) There exists a constant M > such that R(t) L(X) ≤ M for every t ∈ R + . (A ) (i) The function t → F(t, ϑ) is measurable on R + for each ϑ ∈ C, and the function ϑ → F(t, ϑ) is continuous on C for a.e. t ∈ R + . (ii) There exists a function f ∈ L loc (R + , R + ) and a continuous nondecreasing function Ξ : R + → R + such that F(t, ϑ) ≤ f(t)Ξ ( ϑ ∞) for a.e t ∈ R + and each ϑ ∈ C.
(iii) For each bounded set D ⊂ C and for each t ∈ [ , n], n ∈ N, we have where α is a measure of noncompactness on the Banach space X.
Then the nonlocal state dependent problem (1) has at least one mild solution.
By De nition 3.1, it is easy to see that the mild solution of nonlocal problem (1) is equivalent to the xed point of the operator Θ de ned by (6). We de ne the ball Bq n = B( , qn) = {ϑ ∈ C([−r , +∞), X) : ϑ n ≤ qn}.
Firstly, we claim that Θ(Bq n ) ⊂ Bq n . In fact, for any n ∈ N, and each ϑ ∈ Bq n and t ∈ [ , n], by (A ) − (A ), we have Thus Θ(ϑ) ≤ qn. This proves that Θ transforms the ball Bq n into Bq n . We complete the proof in the following steps.
Step 1 : Θ : Bq n −→ Bq n is continuous. Let {ϑ k } k∈N be a sequence such that ϑ k → ϑ in Bq n . Then for each t ∈ [ , n], we have Since ϑ k → ϑ as k → ∞, the Lebesgue dominated convergence theorem implies that Thus Θ is continuous.

Thus αn((ΘK)(t)) ≤ M δn + fn αn(K).
From Steps 1 to 3, together with Theorem 2.6, we can conclude that Θ has at least one xed point in Bq n which is a mild solution of problem (1).

An example
In this section, an example is provided to illustrate the obtained theory. We consider the following integro-di erential equation with state-dependent nonlocal initial conditions: where en(z) = π / sin(nz), ≤ z ≤ π, n = , , · · · is the orthonormal set of eigenvectors of A. It is known that A is the in nitesimal generator of a analytic semigroup T(t) t≥ in X, which is given by Then with these settings system (10) can be written in the abstract form Furthermore, we can check that the assumptions of Theorem 3.1 hold. Consequently, the problem (10) has at least one mild solution on [−r; +∞).

Conclusion
In this paper, we consider a class of integro-di erential system with state-dependent delay and nonlocal conditions in a Frechet space. More precisely, using the generalized Darbo xed point theorem associated with the theory of measures of noncompactness and the theory of the resolvent operator, we established a new set of conditions which guarantees the existence of a mild solution for the considered system. In further works, we intend to extend the obtained results for stochastic integro-di erential systems.