Weak Solutions for a Coupled System of Partial Pettis Hadamard Fractional Integral Equations

In this paper we investigate the existence of weak solutions under the Pettis integrability assumption for a coupled system of partial integral equations via Hadamard’s fractional integral, by applying the technique of measure of weak noncompactness and Mönch’s fixed point theorem.


Introduction
In this paper N and R denote the sets of positive integers, respectively the set of real numbers, while N 0 := N ∪ {0} and R + 0 := [0, ∞).The fractional calculus represents a powerful tool in applied mathematics to study many problems from different fields of science and engineering, with many break-through results found in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology and bioengineering [25,40].There has been a significant development in fractional differential and integral equations in recent years; see the monographs of Abbas et al. [1,2], Kilbas et al. [26], Miller and Ross [28], and the papers of Abbas et al. [3], Darwish et al. [16,17,18,19,20,21], Vityuk et al. [41,42], and the references therein.
In [14], Butzer et al. investigated properties of the Hadamard fractional integral and derivative.In [15], they obtained the Mellin transform of the Hadamard fractional integral and differential operators, and in [36], Pooseh et al. obtained expansion formulas of the Hadamard operators in terms of integer order derivatives.Many other interesting properties of those operators and others are summarized in [37], and the references therein.
The measure of weak noncompactness was introduced by De Blasi [22].The strong measure of noncompactness was developed first by Banas and Goebel [6] and subsequently developed and used in many papers; see for example, Akhmerov et al. [4], Alvàrez [5], Benchohra et al. [10,12], Guo et al. [23], Mönch et al. [30,31], Szufla [38], and the references therein.Recently in [7,8] Benchohra et al. used the measure of weak noncompactness for some classes of fractional differential equations and inclusions, while in [9], a class of hyperbolic differential equations involving the Caputo fractional derivative was considered.Some applications of the measure of weak noncompactness to ordinary differential and integral equations in Banach spaces are reported in [11,27,33,39] and the references therein.Some recent results on coupled systems of operator equations in b-metric spaces are given in [34].
This paper deals with the existence of weak solutions to the following coupled system of Hadamard partial fractional integral equations of the form, for (x, y) ∈ J, where is the Euler gamma function and E is a real (or complex) Banach space with norm • E and dual E * , such that E is the dual of a weakly compactly generated Banach space X.
The present paper initiates the use of the measure of weak noncompactness and Mönch's fixed point theorem to the coupled system (1.1).

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.Let C := C(J, E) be the Banach space of continuous functions u : J → E with the norm It is clear that the product space C := C × C is a Banach space with the norm Denote by L ∞ (J, E), the Banach space of essentially bounded measurable functions u : J → E equipped with the norm u L ∞ = inf{c > 0 : u(x, y) E ≤ c, a.e.(x, y) ∈ J}.
Definition 2.3.[35] The function u : J → E is said to be Pettis integrable on J if and only if there is an element u j ∈ E corresponding to each j ⊂ J such that φ(u j ) = j φ(u(s, t))dtds for all φ ∈ E * , where the integral on the right hand side is assumed to exist in the sense of Lebesgue, (by definition, u j = j u(s, t)dtds).
Let P (J, E) be the space of all E-valued Pettis integrable functions on J, and L 1 (J, R), be the Banach space of Lebesgue integrable functions u : J → R. Define the class P 1 (J, E) by The space P 1 (J, E) is normed by where λ stands for the Lebesgue measure on J.
Proposition 2.4.[35] If u ∈ P 1 (J, E) and h is a measurable and essentially bounded E−valued function, For all that follows, the sign " " denotes the Pettis integral.
Let us recall the definitions of Pettis integral and Hadamard integral of fractional order.
The De Blasi measure of weak noncompactness is the map The De Blasi measure of weak noncompactness satisfies the following properties: The next result follows directly from the Hahn-Banach theorem.
For a given set V of functions v : J → E let us denote by For our purposes, we will need the following fixed point theorem: Theorem 2.11.[32] Let Q be a nonempty, closed, convex and equicontinuous subset of a metrizable locally convex vector space C(J, E) holds for every subset V ⊂ Q, then the operator T has a fixed point.

Existence Results
Let us start by defining what we mean by a solution of the integral equation (1.1).
Definition 3.1.A pair (u, v) ∈ C is said to be a solution of (1.1) if (u, v) satisfies equation (1.1) on J.
Further, we present conditions for the existence of a solution of equation (1.1).
Proof.Define the operators N i : C → C; i = 1, 2 by and Consider the continuous operator N : C → C defined by First notice that, the hypothesis (H 2 ) implies that From (H 3 ) we have that for all (x, y) ∈ J, the functions ln x s Let R, R i > 0; i = 1, 2 be such that , and consider the set Clearly, the subset Q is closed, convex and equicontinuous.We shall show that the operator N satisfies all the assumptions of Theorem 2.11.The proof will be given in several steps.

An Example
Clearly, the functions f and g are continuous.
For each u, v ∈ E and (x, y) ∈