Existence and stability of solutions for a system of quadratic integral equations in Banach algebras

The aim of this paper is to prove the existence and stability of solutions of a system of quadratic integral equations in the Banach algebra of continuous and bounded functions on unbounded rectangle. The main tool used in our considerations is the multiple fixed point theorem which is a consequence of Darbo’s fixed point theorem and the technique associated with measures of noncompactness. We also present an illustrative example.


Introduction
The theory of integral operators and integral equations is an important part of nonlinear analysis. It is caused by the fact that this theory is frequently applicable in other branches of mathematics and in mathematical physics, engineering, economics, biology as well in describing problems connected with real world (cf. [19,25]). It is well-known that a lot of problems investigated in the theories of radiative transfer and neutron transport, and in the kinetic theory of gases and also several real world problems can be described with the help of various quadratic integral equations. Especially the quadratic integral equations of Chandrasekhar type can be very often encountered in several applications see [12,18]. In the last years there appeared many papers devoted to the quadratic integral equations and several types of integral operators are investigated, we refer to the papers of Benchohra and al. [3], Banaś and al. [9,14,15,16,20,21] and references therein.
On the other hand, the classical theory of integral operators and equations can be generalized with the help of the Stieltjes integral having kernels dependent

[204]
Said Baghdad on one or two variables. Such an approach was presented and developed in many research papers see the results of Abbas and Benchohra [1,2,7], Banaś and al. [8,15,17].
The goal of this paper is to investigate a finite system of nonlinear quadratic integral equations of Hadamard-Volterra-Stieltjes type applying the concept of multiple fixed point of condensing operators and we will use the technique associated with measures of noncompactness. There are some papers in this direction, Aghajani and al. [4,5], Dhage [20] and references therein.
It is worthwhile mentioning that in applications, the most useful measures of noncompactness are those defined in an axiomatic way [6,11]. Such a direction of investigations has been initiated in the papers [10,13], where the authors introduced the so-called condition (m) related to the operation of multiplication in an algebra and playing a crucial role in the use of the technique of measures of noncompactness in Banach algebras setting. Moreover, if we apply an approach to the measure of noncompactness concept associated with a suitable axiomatic definition, then we create the possibility to characterize solutions of investigated operator equations see [9,12,13,14,16,21]. Consider the following system of quadratic fractional integral equations

Preliminaries
Assume that (X, · ) is an infinite dimensional Banach space with zero element θ. Denote by B(x, r) the closed ball with center at x and radius r. We write B r to denote the ball B(θ, r). If U is a subset of X then the symbols U , Conv U stand for the closure and closed convex hull of U , respectively. Apart from this the symbol diam U will denote the diameter of a set U while U denotes the norm of U ; that is, U = sup{ u : u ∈ U }. Moreover, we denote by M X the family of all nonempty and bounded subsets of X and by N X its subfamily consisting of all relatively compact sets.
Here, we give some basic facts concerning measures of noncompactness, which is defined axiomatically in terms of some natural conditions.
The family ker ψ described in (A1) is said to be the kernel of the measure of noncompactness ψ.

Remark 2.2
Observe that the intersection set A ∞ from (A6) is a member of the family ker ψ. In fact, since ψ(A ∞ ) ≤ ψ(A n ) for any n, we infer that ψ(A ∞ ) = 0.
In the sequel we will assume that the space X has the structure of a Banach algebra. In such a case we write uv in order to denote the product of elements u, v ∈ X. Similarly, we will write U V to denote the product of subsets U , V of X; that is, One says that the measure of noncompactness µ defined on a Banach algebra X satisfies condition (m) if for arbitrary sets U, V ∈ M X the following inequality is satisfied Remark 2.4 It turns out that the above defined condition (m) is very convenient in considerations connected with the use of the technique of measures of noncompactness in Banach algebras. Apart from this the majority of measures of noncompactness satisfy this condition. [206]

Said Baghdad
We recall the well known fixed point theorem of Darbo.
Theorem 2.5 ( [5,6,9]) Let Ω be a nonempty, bounded, closed and convex subset of a Banach space X and let F : Ω → Ω be a continuous mapping. Assume that there exists a constant k ∈ [0, 1) such that ψ(F A) ≤ kψ(A) for any nonempty subset A of Ω. Then F has a fixed point in the set Ω.
Remark 2.6 Let us denote by Fix F the set of all fixed points of the operator F which belongs to Ω. It can be shown that the set Fix F belongs to the family ker ψ.
. . , X n be Banach spaces and µ 1 , µ 2 , . . . , µ n be measures of noncompactness in X 1 , X 2 , . . . , X n , respectively. Assume that the function ξ : R n + → R + is convex with ξ(a 1 , a 2 , . . . , a n ) = 0 if and only if a i = 0 for i = 1, 2, . . . , n. Then defines a measure of noncompactness in the product space In what follows we will work in the Banach space BC consisting of all real functions defined, continuous and bounded on J. This space is furnished with the standard norm The space BC has the structure of a Banach algebra, i.e. if w, v ∈ BC, we write wv to denote the product of elements u and v, and (wv)(t, s) = w(t, s)v(t, s).

Remark 2.10
It is clear that the product space BC × · · · × BC n turns out to be a Banach space if equipped with the norm Existence and stability of solutions for a system of quadratic integral equations

[207]
In order to define a measure of noncompactness in the space BC, fix a nonempty bounded subset Y of BC and T > 1. For u ∈ Y and 1 , 2 > 0 let us denote by ω T (u, 1 , 2 ) the modulus of continuity of the function u on the rectangle Finally, consider the function ψ defined on the family M X by the formula It can be shown that the function ψ is a measure of noncompactness in the space BC. The kernel ker ψ consists of nonempty and bounded sets Y such that functions from Y are locally equicontinuous on J and the thickness of the bundle formed by functions from Y tends to zero at infinity. This property will permit us to characterize solutions of the system (1) considered in the next section. Let L 1 (J, R) be the Banach space of functions u : J → R that are Lebesgue integrable with the norm Definition 2.12 Let r 1 , r 2 ≥ 0, σ = (1, 1) and r = (r 1 , r 2 ). For u ∈ L 1 (J, R), define the Hadamard partial fractional integral of order r by the expression where Γ(·) is the Euler gamma function.
For more details about Hadamard fractional integral see [22]. Let a nondegenerate rectangle I = [a, b] × [c, d] ⊂ R 2 be given. We consider a real function p : I → R defined on I. For a given sub-rectangle (a 1 , d 1 ) + p(a 1 , c 1 ). [208]

Said Baghdad
Let us de define where the supremum is taken over all finite systems of nonoverlapping rectangles I i ⊂ I (i.e. for the interiors I • i of the rectangles I i we assume that I If is fixed, then we denote the usual variation of the function p(s, γ) (resp. p(α, t)) in the interval     It is clear that the Riemann integral is a special case of the Stieltjes integral, obtained by setting g(x) = x. The Stieltjes integral exists under several conditions, we will restrict ourselves to only one theorem in this direction.
We consider the Hadamard-Stieltjes fractional integral of a function u : J → R of order r = (r 1 , r 2 ) of the form for any Q 1 , Q 2 , . . . , Q n of D, then the operator F has a multiple fixed point in D, i.e. there exist x * 1 , . . . , x * n ∈ D such that Proof. see [4].
Let us assume that Ω is a nonempty subset of the space BC and F is an operator on Ω n with values in BC n . Consider the following equation Definition 2.19 A solution u * = (u 1 (x, y), . . . , u n (x, y)) of equation (2) is said to be globally attractive if for each solution v * = (v 1 (x, y), . . . , v n (x, y)) of (2) we have In the case when this limit is uniform, i.e. when for each > 0 there exists T > 0 such that sup we will say that solutions of (2) are uniformly globally attractive. [210]

Said Baghdad
Remark 2.20 This concept of stability in the case of equation (F u)(x, y) = u(x, y) can be found in [14,21].

Main results
We will give an existence and stability results for the system (1) applying Theorem 2.18 under the following assumptions: (H 1 ) The function g is continuous and there exist continuous and bounded functions p i : J → R + , i = 1, . . . , n such that (H 2 ) The operator G : BC → BC is a bounded linear operator with spectral radius r σ (G) < 1.
for all (x, y) ∈ J and u i ∈ R.
(ii) The operator G is a bounded linear operator, i.e. the operator G is linear and there exists a constant l > 0 such that for each u ∈ BC, Gu ≤ l u .

[213]
Since lim Then the operator F is continuous on B r × · · · × B r .
is finite. Now, we consider the operator T defined on BC × · · · × BC by For an arbitrarily function u = (u 1 , . . . , u n ) of the space BC × · · · × BC and a fixed (x, y) in J, using our assumptions, we get Clearly the operator T transforms the set B r × · · · × B r into the ball B r and F (B r × · · · × B r ) ⊆ ϕ + G(B r ) · T (B r × · · · × B r ). [214]

Said Baghdad
Next, we take a nonempty Y ⊂ B r , using Definition 2.3 and applying Theorem 2.11, we obtain Finally, in view of Theorem 2.18 we deduce that F has at least one multiple fixed point in B r which is a solution of the system (1). Moreover, taking into account the fact that the set Fix F ∈ ker ψ and the characterization of sets belonging to ker ψ (Remark 2.6) we conclude that all solutions of the system (1) are globally attractive in the sense of Definition 2.19.
Existence and stability of solutions for a system of quadratic integral equations
Consequently from Theorem 3.2 the system (3) has at least solution in BC and solutions of the system (3) are globally attractive.