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

Abstract The combined systems of integral equations have become of great importance in various fields of sciences such as electromagnetic and nuclear physics. New classes of the merged type of Urysohn Volterra-Chandrasekhar quadratic integral equations are proposed in this paper. This proposed system involves fractional Urysohn Volterra kernels and also Chandrasekhar kernels. The solvability of a coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type is studied. To realize the existence of a solution of those mixed systems, we use the Perov’s fixed point combined with the Leray-Schauder fixed point approach in generalized Banach algebra spaces.


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 ( ) ∈ C I n , , n , is the space of all n-times continuous and differentiable functions from = [ ] I a 0, to . Let ∈ ( ) ψ C I, 1 be an increasing function. Let  [13] applied the approach of Darbo's fixed point to investigate the following Urysohn-Volterra integral equation: ψ s ψ t  ψ s  p  h s u s s  t  a  , , where ( > ) ∈ + a 0 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 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:   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).

Basic concepts and auxiliary facts
Throughout this paper, + n is the set Let ≼ n be a partial order on n such that: Furthermore, consider 0 n to be the zero of n . Next, we give the concept of generalized metric space. For more details see [14,[27][28][29][30].
3 the following conditions hold: , is called a generalized metric space and ρ is given by Clearly, ρ is a generalized metric on X if and only if ρ i is a metric in the usual sense, for all = … i n 1, 2, , . We indicate that the definitions of sequences, Cauchy sequences, convergence, closed and open subsets and completeness are the same as those for usual metric spaces.

Definition 4. [14]
The generalized Banach algebra X is an algebra which is also a generalized Banach space, such that for all ∈ X Λ , Λ 1 2 , the following holds: 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.
is a completely continuous; Then either:

Existence theory
The investigation of solvability of system (7) is given under the following conditions.
. Clearly ( ∥⋅∥) X, is generalized Banach algebra. Let × → + d X X : 2 be the generalized metric space induced by the norm which is defined as . Note system (7) can be written as ( The operator is the Chandrasekhar integral operator and has the form ( , where the superposition operators , 1 2 are defined as:  as: Clearly, the solution of system (7) is the fixed point of the operator .
Hence, we have Similarly, we get So, we get that  Since ( ) < σ Q 1, is a contractive operator. □ Lemma 8. Suppose that assumptions (C1)-(C6) hold. Then, there exists * K such that, for every solution ( ) ∈ u v X , of the following system: be a solution of (15), then we have For simplicity, suppose that = û u λ and = v v λ . Then, we get

(17)
Similarly, we have Then we have Therefore, we get By adding the last two inequalities, we get . Since < 1, we get which implies Thus, we get