Existence results for nonlinear coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type

1 Introduction

Integral equations are an important topic in functional analysis (see, [1,2,3], for examples). They are stratified in the characterization of many real life events such as processes encountered in nuclear physics [4], heat conduction [5], electromagnetic [6] and multimedia processing [7].

Among the many integral equations that were established in mathematical analysis and were stratified to many areas of engineering and real life sciences, an efficient and effective role is played by integral equations of fractional kernels [8,9,10,11].

Fractional calculus is an essential and useful branch of mathematical analysis that investigates derivatives and integrals of fractional order. A long time ago, there were many definitions for fractional integral operators, such as Riemann-Liouville, Hadamard, Katugampola and Erdelyi-Kober fractional integral operators. Recently, in 2017, Almeida [12] proposed a new definition of the fractional integral and called this operator ψ-Caputo integral. This new definition is more generalized than Riemann-Liouville, Hadamard, Erdelyi Kober and Caputo operator kinds. Suppose that Cn(I,ℝ),n∈ℕ, is the space of all n-times continuous and differentiable functions from I=[0,a] to ℝ. Let ψ∈C1(I,ℝ) be an increasing function. Let u:I→ℝ be an integrable function. The ψ-Riemann-Liouville fractional integral of order α>0,α∈ℝ of the function u is defined as

where n=[α]+1 and [α] denotes the integral part of α.

In 2018, Darwish et al. [13] applied the approach of Darbo’s fixed point to investigate the following Urysohn-Volterra integral equation:

where (a>0)∈ℝ+ and p∈(0,1). The authors obtained the existence results of the solution of equation (1) under some certain conditions. In the same year, Nieto et al. [14] proposed some new versions of the fixed point theorems in the algebra of generalized Banach spaces. They established the type of Krasnoselskii and Leray-Schauder fixed point for the product and sum of more than or equal two operators.

In 2019, Hashem et al. [15] applied a fixed point approach according to Amar et al. [16] to study the following system:

where (a>0)∈ℝ+ and p,q∈(0,1).

One of the interesting types of integral equations is the so-called Chandrasekhar integral equation. The integral equation of Chandrasekhar’s integral equation is considered by Chandrasekhar [17] to model the process of radiative transfer. From this date onward, this type has attracted a lot of attention from many researchers [18,19,20,21].

In 1998, Banas et al. [18] investigated the solvability of the following quadratic Chandrasekhar integral equation:

where t∈[0,1] and f:[0,1]→ℝ is continuous. Recently, in 2017, Hashem and El-Sayed [19] considered the following two-dimensional quadratic Chandrasekhar integral equations:

They proved that system (4) has at least one continuous solution by applying the Schauder fixed point theorem. In the same year, Hashem [22] investigated a more generalized coupled system of Chandrasekhar integral equations which is given by

The existence of a solution of system (5) was shown via a block operator (2×2)-matrix approach which was proposed by Jeribi et al. [23]. In 2018, Chang and Feng [20] reviewed many results and applications for Chandrasekhar integral equations.

The solvability of quadratic integral equations has been established in many papers; see, for example, [24,25]. More recently, in 2019, Jeribi et al. [21] investigated the solvability of the following coupled system of Chandrasekhar functional integral equations:

In the next year, Jeribi et al. [26] applied a fixed point approach for a 2×2 block operator matrix to study the solvability for infinite system of integral equations.

The merged systems of integral equations have become of great importance now in various fields of science such as electromagnetic and nuclear physics [4,6]. In this paper, we propose the more general coupled system of Urysohn Volterra-Chandrasekhar integral equations, given by

where α and β∈(0,1). It is clear that the proposed system is more general and comprehensive than systems (4), (5) and (6). Also, all functions in the proposed system involve the unknowns u,v, which is an advantage to system (7).

This manuscript is organized as follows: Section 2 is devoted to giving some facts, basic results and definitions which are used in the results. In Section 3, we study the solvability of system (7).

2 Basic concepts and auxiliary facts

Throughout this paper, ℝ+n is the set {Π=(ϖ1,…,ϖn)∈ℝn:ϖi>0∀i=1,2,…,n}. Let Λ=(ϱ1,…,ϱn)∈ℝn. Let ≼n be a partial order on ℝn such that: Π≼nΛ⇔ϖi≤ϱi for all i=1,2,…,n. Therefore, if δ∈ℝ, then Π≼nδ means ϖi≤δ for all i=1,2,…,n. Also, |Π|=(|ϖ1|,…,|ϖn|) and max(Π,Λ)=(max{ϖ1,ϱ1},…,max{ϖn,ϱn}). Furthermore, consider 0n to be the zero of ℝn. Next, we give the concept of generalized metric space. For more details see [14,27,28,29,30].

The following theorem, due to Nieto et al. [14], combines Leray-Schauder with Perov’s fixed point approach in generalized vectorial algebra Banach spaces and is used to prove our results.

3 Existence theory

The investigation of solvability of system (7) is given under the following conditions.

(C1) The functions Vi:I×ℝ2→ℝ,i=1,2 are continuous in t∈I, for any (u,v)∈ℝ2 and continuous in u,v∈ℝ for all t∈I. There exists pij∈ℝ+, i,j=1,2 such that for i=1,2, we have

for all t∈I and (x1,x2,y1,y2)∈ℝ4.

(C2) The functions Fi:I×ℝ2→ℝ,i=1,2 are measurable in t for all u,v∈ℝ and continuous in u,v∈ℝ for almost all t∈I such that

for all (t,u,v)∈I×ℝ2 and

(C3) The functions Hi:I×ℝ2→ℝ,i=1,2 are continuous and

(C4) The mappings pi,qi:I→ℝ,i=1,2 are continuous and there exists Wpi,Wqi, i=1,2 such that

(C5) σ(Q)<1, where Q=(γij)i=1,2,j=1,2 where

(C6) Let p∗=max(ψ(1)−ψ(0))αΓ(α+1)p1j,(ψ(1)−ψ(0))βΓ(β+1)p2j,j=1,2, then the inequality p∗(Wq1+N1K1+Wq2+N2K2)<1 holds.

Let C(I) be the Banach algebra of all continuous real-valued functions on I with the supremum norm ∥u∥∞=supt∈I|u(t)|, for all u∈C(I) and with respect to the pointwise product of functions. Now, let X=C(I)×C(I). Define the generalized norm ∥⋅∥:X→ℝ+2 as

for all (u,v)∈X. Clearly (X,∥⋅∥) is generalized Banach algebra. Let d:X×X→ℝ+2 be the generalized metric space induced by the norm which is defined as

for all (u1,v1),(u2,v2)∈X.

Note system (7) can be written as (u,v)=A(u,v)ℬ(u,v). The operator A(x,y)=(A1(u,v),A2(u,v)), where

The operator ℬ is the Chandrasekhar integral operator and has the form ℬ(x,y)=(ℬ1(u,v),ℬ2(u,v)), where the superposition operators ℬ1,ℬ2 are defined as:

and

Define T:C(I)×C(I)→X as:

Clearly, the solution of system (7) is the fixed point of the operator T.