Next Article in Journal
On Global Convergence of Third-Order Chebyshev-Type Method under General Continuity Conditions
Next Article in Special Issue
A New Approach to Fractional Kinetic Evolutions
Previous Article in Journal
Maximum Likelihood Estimation for Mixed Fractional Vasicek Processes
Previous Article in Special Issue
Elementary Fractal Geometry. 2. Carpets Involving Irrational Rotations
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

A Langevin-Type q-Variant System of Nonlinear Fractional Integro-Difference Equations with Nonlocal Boundary Conditions

1
Department of Mathematics, Texas A& M University, Kingsville, TX 78363-8202, USA
2
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
*
Author to whom correspondence should be addressed.
Fractal Fract. 2022, 6(1), 45; https://doi.org/10.3390/fractalfract6010045
Submission received: 26 November 2021 / Revised: 27 December 2021 / Accepted: 7 January 2022 / Published: 14 January 2022
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)

Abstract

:
We introduce a new class of boundary value problems consisting of a q-variant system of Langevin-type nonlinear coupled fractional integro-difference equations and nonlocal multipoint boundary conditions. We make use of standard fixed-point theorems to derive the existence and uniqueness results for the given problem. Illustrative examples for the obtained results are also presented.

1. Introduction

The Langevin equation provides a decent approach to describe the evolution of fluctuating physical phenomena. Examples include anomalous diffusion [1], time evolution of the velocity of the Brownian motion [2,3], diffusion with inertial effects [4], gait variability [5], harmonization of a many-body problem [6], financial aspects [7], etc. However, the failure of the ordinary Langevin equation for correct description of the dynamical systems in complex media led to its several generalizations. One such example is that of the Langevin equation, involving fractional-order derivative operators, which provides a more flexible model for fractal processes. For some recent results on Langevin equation, see ([8,9,10,11,12]) and the references therein.
The topic of q-difference equations has evolved into an important area of research, as such equations are always completely controllable and appear in the q-optimal control problem [13]. Furthermore, the variational q-calculus is regarded as a generalization of the continuous variational calculus due to the presence of an extra parameter q whose nature may be physical or economical. The variational calculus on the q-uniform lattice is concerned with the study of the q-Euler equation and its applications to commutation equations, and isoperimetric and Lagrange problems. In other words, the q-Euler–Lagrange equation is solved for finding the extremum of the functional involved instead of solving the Euler–Lagrange equation [14]. There do exist q-variants of certain significant concepts, such as q-analogues of fractional operators, q-Laplace transform, q-Taylor’s formula, etc.
Fractional-order operators are found to be of great utility in improving the mathematical modeling of several real-world problems. The variational principles based on fractional derivative operators lead to the class of fractional Euler–Lagrange equations [15]. In addition, one can find some interesting results on optimal control theories for fractional differential systems in the articles [16,17,18,19,20,21].
The popularity of fractional calculus in the recent years led to the birth of the fractional analogue of q-difference equations (fractional q-difference equations), for instance, see [22,23]. One can find interesting results on nonlinear boundary value problems involving fractional q-derivative and q-integral operators, and different kinds of boundary conditions in the articles [24,25,26,27,28,29,30,31,32,33,34,35,36,37]. In a recent work [38], the authors studied the existence of solutions for a nonlinear fractional q-integro-difference equation equipped with q-integral boundary conditions. However, it is observed that there are a few results for coupled systems of fractional q-integro-difference equations [39]. More recently, a coupled system of nonlinear fractional q-integro-difference equations with q-integral coupled boundary conditions was studied in [40].
The objective of the present work is to enrich the literature on boundary value problems of coupled systems of fractional q-integro-difference equations. Keeping in mind the importance of the fractional Langevin equation, we introduce and study a new problem consisting of a coupled system of Langevin-type nonlinear fractional q-integro-difference equations complemented with nonlocal multipoint boundary conditions. The proposed problem is interesting in the sense that it enhances the literature on fractional q-variant of Langevin equations with mixed nonlinearities in terms of the parameter q . On the other hand, the consideration of multipoint non-separated boundary conditions involving the values of the unknown functions together with their q-derivatives at the end points as well as the interior nonlocal positions of given domain extends the scope of the present work to a more general situation (also see Section 5). For the motivation of nonlocal boundary conditions, we recall that nonlocal multipoint boundary conditions appear in feedback controls problems, optimal boundary control of (finite) string vibrations arising from interior arbitrary positions, etc. For more details, see [41,42,43,44]. In precise terms, we investigate the following boundary value problem:
c D q p 1 ( c D q p 2 + λ 1 ) x ( t ) = α 1 f 1 ( t , x ( t ) , y ( t ) ) + β 1 I q ξ 1 g 1 ( t , x ( t ) , y ( t ) ) , 0 t 1 , c D q r 1 ( c D q r 2 + λ 2 ) y ( t ) = α 2 f 2 ( t , x ( t ) , y ( t ) ) + β 2 I q ξ 2 g 2 ( t , x ( t ) , y ( t ) ) , 0 t 1 ,
μ 1 x ( 0 ) μ 2 t ( 1 p 2 ) D q x ( t ) | t = 0 = j = 1 n a j y ( η j ) , μ 3 y ( 0 ) μ 4 t ( 1 r 2 ) D q y ( t ) | t = 0 = j = 1 n b j x ( η j ) , σ 1 x ( 1 ) + σ 2 D q x ( 1 ) = j = 1 n k j D q y ( η j ) , σ 3 y ( 1 ) + σ 4 D q y ( 1 ) = j = 1 n m j D q x ( η j ) ,
where c D q p i and c D q r i denote the fractional q-derivative operators of the Caputo type, 0 < p i , r i 1 , 0 < q < 1 , I q ξ i ( . ) denotes Riemann–Liouville integral of order ξ i > 0 , f i , g i are given continuous functions, λ i 0 , α i , β i , i = 1 , 2 , and a j , b j , k j , m j , j = 1 , , n are real constants and μ 1 , μ 2 , μ 3 , μ 4 , σ 1 , σ 2 , σ 3 , σ 4 R , η j ( 0 , 1 ) , j = 1 , , n .
Here, one can notice that the right-hand sides of the fractional q-Langevin equations in the system (1) involve the usual as well as q-integral-type nonlinearities. These equations correspond to different combinations of nonlinearities, such as ordinary nonlinearities, f 1 ( t , x ( t ) , y ( t ) ) and f 2 ( t , x ( t ) , y ( t ) ) for β 1 = 0 = β 2 , purely q-integral-type nonlinearities, I q ξ 1 g 1 ( t , x ( t ) , y ( t ) ) and I q ξ 2 g 2 ( t , x ( t ) , y ( t ) ) for α 1 = 0 = α 2 , and so on.
The paper is organized as follows. In Section 2, we recall some general concepts and results on q-calculus and fractional calculus. We then solve a linear variant of the given problem that provides a platform to define the solution for the problem at hand. Section 3 is devoted to the main existence results, which are established with the aid of some classical fixed-point theorems. The paper concludes with an illustrative example.

2. Preliminaries on Fractional q-Calculus

Here, we recall some basic definitions and known results on fractional q-calculus.
Definition 1.
Let β 0 , 0 < q < 1 , and f be a function defined on [ 0 , 1 ] . The fractional q-integral of the Riemann–Liouville type is ( I q 0 f ) ( t ) = f ( t ) and
( I q β f ) ( t ) = 0 t ( t q s ) ( β 1 ) Γ q ( β ) f ( s ) d q ( s ) , β > 0 , t [ 0 , 1 ] ,
where
Γ q ( β ) = ( 1 q ) ( β 1 ) ( 1 q ) β 1 , 0 < q < 1
and satisfies the relation:
Γ q ( β + 1 ) = [ β ] q Γ q ( β ) , with
[ β ] q = q β 1 q 1 , ( 1 q ) ( 0 ) = 1 , ( 1 q ) ( n ) = k = 0 n 1 ( 1 q k + 1 ) , n N .
More generally, if α R , then
( 1 q ) ( α ) = i = 0 ( 1 q i + 1 ) ( 1 q 1 + α + i ) .
For 0 < q < 1 , we define the q-derivative of a real valued function f as
D q f ( t ) = f ( t ) f ( q t ) ( 1 q ) t , t 0 , D q f ( 0 ) = lim n f ( s q n ) f ( 0 ) s q n , s 0 .
For more details, see [22].
Definition 2
([45]). The fractional q-derivative of the Riemann–Liouville type of order β 0 is defined by ( D q 0 f ) ( t ) = f ( t ) and
( D q β f ) ( t ) = ( D q [ β ] I q [ β ] β f ) ( t ) , β > 0 ,
where [ β ] is the smallest integer greater than or equal to β .
Definition 3
([45]). The fractional q-derivative of the Caputo type of order β 0 is defined by
( c D q β f ) ( t ) = ( I q [ β ] β D q [ β ] f ) ( t ) , β > 0 ,
where [ β ] is the smallest integer greater than or equal to β .
Definition 4.
(q-Beta function) For any x , y > 0 ,
B q ( x , y ) = 0 1 t ( x 1 ) ( 1 q t ) ( y 1 ) d q t
is called the q-beta function.
Recall that
B q ( x , y ) = Γ q ( x ) Γ q ( y ) Γ q ( x + y ) .
Lemma 1
([45]). Let β , γ 0 and let f be a function defined on [ 0 , 1 ] . Then
(i) ( I q γ I q β f ) ( t ) = ( I q β + γ f ) ( t ) ,
(ii) ( D q β I q β f ) ( t ) = f ( t ) .
Lemma 2
([45]). Let β > 0 . Then the following equality holds:
( I q β c D q β f ) ( t ) = f ( t ) k = 0 [ β ] 1 t k Γ q ( k + 1 ) ( D q k f ) ( 0 ) .
Lemma 3
([25]). Let β 0 and n N . Then the following equality holds:
( I q β D q n f ) ( t ) = D q n I q β f ( t ) k = 0 [ β ] 1 t β n + k Γ q ( β n + k ) ( D q k f ) ( 0 ) .
Lemma 4
([46]). For β R + , λ ( 1 , ) , the following is valid
I q β ( x a ) ( λ ) = Γ q ( λ + 1 ) Γ q ( β + λ + 1 ) ( x a ) ( β + λ ) , 0 < a < x < b .
In particular, for λ = 0 , a = 0 , using q-integration by parts, we have
( I q β 1 ) ( x ) = 1 Γ q ( β ) 0 x ( x q t ) ( β 1 ) d q t = 1 Γ q ( β ) 0 x D q ( ( x t ) ( β ) ) [ β ] q d q t = 1 Γ q ( β + 1 ) 0 x D q ( ( x t ) ( β ) ) d q t = 1 Γ q ( β + 1 ) x ( β ) .
In order to define the solution for the problem (1) and (2), we need the following lemma.
Lemma 5.
Let Λ 0 and h 1 , h 2 C ( [ 0 , 1 ] , R ) . Then the unique solution of the following linear system of equations:
c D q p 1 ( c D q p 2 + λ 1 ) x ( t ) = h 1 ( t ) , 0 t 1 , c D q r 1 ( c D q r 2 + λ 2 ) y ( t ) = h 2 ( t ) , 0 t 1 ,
subject to the boundary conditions (2) is given by
x ( t ) = 0 t ( t q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) h 1 ( u ) d q u λ 1 0 t ( t q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u δ 1 ρ 1 t p 2 + ρ 5 Λ 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) h 2 ( u ) d q u λ 2 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u δ 2 ρ 2 t p 2 + ρ 6 Λ 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) h 1 ( u ) d q u λ 1 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u δ 3 ρ 3 t p 2 + ρ 7 Λ 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) h 2 ( u ) d q u λ 2 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u + σ 1 ρ 3 t p 2 + ρ 7 Λ 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) h 1 ( u ) d q u λ 1 0 1 ( 1 q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u + σ 2 ρ 3 t p 2 + ρ 7 Λ 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) h 1 ( u ) d q u λ 1 0 1 ( 1 q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u δ 4 ρ 4 t p 2 + ρ 8 Λ 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) h 1 ( u ) d q u λ 1 0 η j ( η j q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u + σ 3 ρ 4 t p 2 + ρ 8 Λ 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) h 2 ( u ) d q u λ 2 0 1 ( 1 q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u + σ 4 ρ 4 t p 2 + ρ 8 Λ 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) h 2 ( u ) d q u λ 2 0 1 ( 1 q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u ,
and
y ( t ) = 0 t ( t q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) h 2 ( u ) d q u λ 2 0 t ( t q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u δ 1 ρ 9 t r 2 + ρ 13 Λ 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) h 2 ( u ) d q u λ 2 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u δ 2 ρ 10 t r 2 + ρ 14 Λ 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) h 1 ( u ) d q u λ 1 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u δ 3 ρ 11 t r 2 + ρ 15 Λ 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) h 2 ( u ) d q u λ 2 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u + σ 1 ρ 11 t r 2 + ρ 15 Λ 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) h 1 ( u ) d q u λ 1 0 1 ( 1 q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u + σ 2 ρ 11 t r 2 + ρ 15 Λ 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) h 1 ( u ) d q u λ 1 0 1 ( 1 q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u δ 4 ρ 12 t r 2 + ρ 16 Λ 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) h 1 ( u ) d q u λ 1 0 η j ( η j q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u + σ 3 ρ 12 t r 2 + ρ 16 Λ 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) h 2 ( u ) d q u λ 2 0 1 ( 1 q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u + σ 4 ρ 12 t r 2 + ρ 16 Λ 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) h 2 ( u ) d q u λ 2 0 1 ( 1 q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u ,
where
Λ = ( δ 1 δ 2 μ 1 μ 3 ) ( σ 1 + σ 2 [ p 2 ] q ) ( σ 3 + σ 4 [ r 2 ] q ) σ 1 ( δ 1 δ 6 + μ 2 μ 3 [ p 2 ] q ) ( σ 3 + σ 4 [ r 2 ] q ) + ( μ 1 μ 3 δ 7 δ 8 μ 2 δ 2 δ 7 σ 3 μ 2 μ 4 σ 1 σ 3 δ 1 δ 2 δ 7 δ 8 μ 4 δ 1 δ 8 σ 1 ) [ p 2 ] q [ r 2 ] q σ 3 ( δ 2 δ 5 + μ 1 μ 4 [ r 2 ] q ) ( σ 1 + σ 2 [ p 2 ] q ) μ 3 δ 5 δ 8 σ 1 [ p 2 ] q μ 1 δ 6 δ 7 σ 3 [ r 2 ] q + δ 5 δ 6 σ 1 σ 3 ,
δ 1 = j = 1 n a j , δ 2 = j = 1 n b j , δ 3 = j = 1 n k j , δ 4 = j = 1 n m j , δ 5 = j = 1 n a j η j r 2 , δ 6 = j = 1 n b j η j p 2 , δ 7 = j = 1 n k j η j r 2 1 , δ 8 = j = 1 n m j η j p 2 1 , ρ 1 = μ 3 σ 1 ( σ 3 + σ 4 [ r 2 ] q ) ( δ 2 δ 7 σ 3 + μ 4 σ 1 σ 3 ) [ r 2 ] q , ρ 2 = δ 1 σ 1 ( σ 3 + σ 4 [ r 2 ] q ) μ 1 δ 7 σ 3 [ r 2 ] q + δ 5 σ 1 σ 3 , ρ 3 = ( μ 1 μ 3 δ 1 δ 2 ) ( σ 3 + σ 4 [ r 2 ] q ) + μ 1 μ 4 σ 3 [ r 2 ] q + δ 2 δ 5 σ 3 , ρ 4 = ( μ 1 μ 3 δ 7 δ 1 δ 2 δ 7 μ 4 δ 1 σ 1 ) [ r 2 ] q μ 3 δ 5 σ 1 , ρ 5 = μ 3 ( σ 1 + σ 2 [ p 2 ] q ) ( σ 3 + σ 4 [ r 2 ] q ) + μ 4 σ 3 [ r 2 ] q ( σ 1 + σ 2 [ p 2 ] q ) μ 3 δ 7 δ 8 [ p 2 ] q [ r 2 ] q + δ 6 δ 7 σ 3 [ r 2 ] q , ρ 6 = δ 1 ( σ 1 + σ 2 [ p 2 ] q ) ( σ 3 + σ 4 [ r 2 ] q ) δ 5 σ 3 ( σ 1 + σ 2 [ p 2 ] q ) ( δ 1 δ 7 δ 8 + μ 2 δ 7 σ 3 ) [ p 2 ] q [ r 2 ] q , ρ 7 = ( δ 1 δ 6 + μ 2 μ 3 [ p 2 ] q ) ( σ 3 + σ 4 [ r 2 ] q ) + ( μ 4 δ 1 δ 8 + μ 2 μ 4 σ 3 ) [ p 2 ] q [ r 2 ] q + μ 3 δ 5 δ 8 [ p 2 ] q δ 5 δ 6 σ 3 , ρ 8 = ( μ 3 δ 5 + μ 4 δ 1 [ r 2 ] q ) ( σ 1 + σ 2 [ p 2 ] q ) + μ 2 μ 3 δ 7 [ p 2 ] q [ r 2 ] q + δ 1 δ 6 δ 7 [ r 2 ] q , ρ 9 = δ 2 σ 3 ( σ 1 + σ 2 [ p 2 ] q ) μ 3 δ 8 σ 1 [ p 2 ] q + δ 6 σ 1 σ 3 , ρ 10 = μ 1 σ 3 ( σ 1 + σ 2 [ p 2 ] q ) ( μ 2 σ 1 σ 3 + δ 1 δ 8 σ 1 ) [ p 2 ] q , ρ 11 = ( μ 1 μ 3 δ 8 δ 1 δ 2 δ 8 μ 2 δ 2 σ 3 ) [ p 2 ] q μ 1 δ 6 σ 3 , ρ 12 = ( μ 1 μ 3 δ 1 δ 2 ) ( σ 1 + σ 2 [ p 2 ] q ) + μ 2 μ 3 σ 1 [ p 2 ] q + δ 1 δ 6 σ 1 , ρ 13 = δ 2 ( σ 1 + σ 2 [ p 2 ] q ) ( σ 3 + σ 4 [ r 2 ] q ) δ 6 σ 1 ( σ 3 + σ 4 [ r 2 ] q ) ( δ 2 δ 7 δ 8 + μ 4 δ 8 σ 1 ) [ p 2 ] q [ r 2 ] q , ρ 14 = μ 1 ( σ 1 + σ 2 [ p 2 ] q ) ( σ 3 + σ 4 [ r 2 ] q ) + μ 2 σ 1 [ p 2 ] q ( σ 3 + σ 4 [ r 2 ] q ) μ 1 δ 7 δ 8 [ p 2 ] q [ r 2 ] q + δ 5 δ 8 σ 1 [ p 2 ] q , ρ 15 = ( μ 1 δ 6 + μ 2 δ 2 [ p 2 ] q ) ( σ 3 + σ 4 [ r 2 ] q ) + μ 1 μ 4 δ 8 [ p 2 ] q [ r 2 ] q + δ 2 δ 5 δ 8 [ p 2 ] q , ρ 16 = ( δ 2 δ 5 + μ 1 μ 4 [ r 2 ] q ) ( σ 1 + σ 2 [ p 2 ] q ) + ( μ 2 δ 2 δ 7 + μ 2 μ 4 σ 1 ) [ p 2 ] q [ r 2 ] q + μ 1 δ 6 δ 7 [ r 2 ] q δ 5 δ 6 σ 1 .
Proof. 
Applying the q-integral operators I q p 1 and I q r 1 , respectively, on the first and second equations of (4), we obtain
( c D q p 2 + λ 1 ) x ( t ) = I q p 1 h 1 ( t ) c 0 , ( c D q r 2 + λ 2 ) y ( t ) = I q r 1 h 2 ( t ) d 0 ,
where c 0 and d 0 are arbitrary real constants. Now, applying the q-integral operators I q p 2 and I q r 2 , respectively, to both sides of the above equations, we obtain
x ( t ) = 0 t ( t q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) h 1 ( u ) d q u λ 1 0 t ( t q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u c 0 t p 2 Γ q ( p 2 + 1 ) c 1 ,
y ( t ) = 0 t ( t q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) h 2 ( u ) d q u λ 2 0 t ( t q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u d 0 t r 2 Γ q ( r 2 + 1 ) d 1 ,
where c 1 , d 1 R are arbitrary constants. By using the conditions (2), we obtain a system of equations in the unknown constants c 0 , c 1 , d 0 and d 1 given by
μ 2 [ p 2 ] q Γ q ( p 2 + 1 ) c 0 μ 1 c 1 + δ 5 Γ q ( r 2 + 1 ) d 0 + δ 1 d 1 = F 1 , δ 6 Γ q ( p 2 + 1 ) c 0 + δ 2 c 1 + μ 4 [ r 2 ] q Γ q ( r 2 + 1 ) d 0 μ 3 d 1 = F 2 , ( σ 1 + σ 2 [ p 2 ] q ) Γ q ( p 2 + 1 ) c 0 σ 1 c 1 + δ 7 [ r 2 ] q Γ q ( r 2 + 1 ) d 0 = F 3 , δ 8 [ p 2 ] q Γ q ( p 2 + 1 ) c 0 ( σ 3 + σ 4 [ r 2 ] q ) Γ q ( r 2 + 1 ) d 0 σ 3 d 1 = F 4 ,
where δ 1 , δ 2 , δ 5 , δ 6 , δ 7 , δ 8 are given in (8), and
F 1 = δ 1 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) h 2 ( u ) d q u λ 2 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u , F 2 = δ 2 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) h 1 ( u ) d q u λ 1 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u , F 3 = δ 3 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) h 2 ( u ) d q u λ 2 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u σ 1 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) h 1 ( u ) d q u λ 1 0 1 ( 1 q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u σ 2 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) h 1 ( u ) d q u λ 1 0 1 ( 1 q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u , F 4 = δ 4 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) h 1 ( u ) d q u λ 1 0 η j ( η j q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u σ 3 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) h 2 ( u ) d q u λ 2 0 1 ( 1 q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u σ 4 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) h 2 ( u ) d q u λ 2 0 1 ( 1 q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u .
Solving the system (11) for c 0 , c 1 , d 0 and d 1 , we find that
c 0 = Γ q ( p 2 + 1 ) Λ ρ 1 F 1 + ρ 2 F 2 + ρ 3 F 3 + ρ 4 F 4 , c 1 = 1 Λ ρ 5 F 1 + ρ 6 F 2 + ρ 7 F 3 + ρ 8 F 4 , d 0 = Γ q ( r 2 + 1 ) Λ ρ 9 F 1 + ρ 10 F 2 + ρ 11 F 3 + ρ 12 F 4 , d 1 = 1 Λ ρ 13 F 1 + ρ 14 F 2 + ρ 15 F 3 + ρ 16 F 4 ,
where Λ is given by (7). Substituting the values of c 0 , c 1 , d 0 and d 1 in (9) and (10) yields the solution (5) and (6). By direct computation, one can obtain the converse of the lemma. This completes the proof.  □
Let C = { x | x C ( [ 0 , 1 ] , R ) } be the space equipped with the norm x = sup t [ 0 , 1 ] | x ( t ) | . Obviously, ( C , . ) is a Banach space. Then, the product space ( C × C , . ) is also a Banach space with the norm ( x , y ) = x + y for ( x , y ) C × C .
In view of Lemma 5, we define an operator G : C × C C × C by
G ( x , y ) ( t ) = G 1 ( x , y ) ( t ) G 2 ( x , y ) ( t ) ,
where
G 1 ( x , y ) ( t ) = α 1 0 t ( t q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 t ( t q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 t ( t q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u δ 1 ρ 1 t p 2 + ρ 5 Λ ( α 2 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u ) δ 2 ρ 2 t p 2 + ρ 6 Λ ( α 1 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u ) δ 3 ρ 3 t p 2 + ρ 7 Λ ( α 2 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u ) + σ 1 ρ 3 t p 2 + ρ 7 Λ ( α 1 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 1 ( 1 q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u ) + σ 2 ρ 3 t p 2 + ρ 7 Λ ( α 1 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 1 ( 1 q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u ) δ 4 ρ 4 t p 2 + ρ 8 Λ ( α 1 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 η j ( η j q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u ) + σ 3 ρ 4 t p 2 + ρ 8 Λ ( α 2 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 1 ( 1 q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u ) + σ 4 ρ 4 t p 2 + ρ 8 Λ ( α 2 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 1 ( 1 q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u ) ,
G 2 ( x , y ) ( t ) = α 2 0 t ( t q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 t ( t q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 t ( t q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u δ 1 ρ 9 t r 2 + ρ 13 Λ ( α 2 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u ) δ 2 ρ 10 t r 2 + ρ 14 Λ ( α 1 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u ) δ 3 ρ 11 t r 2 + ρ 15 Λ ( α 2 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u ) + σ 1 ρ 11 t r 2 + ρ 15 Λ ( α 1 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 1 ( 1 q u ) ( p 2 1 ) Γ q ( p 2 ) x ( u ) d q u ) + σ 2 ρ 11 t r 2 + ρ 15 Λ ( α 1 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 1 ( 1 q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u ) δ 4 ρ 12 t r 2 + ρ 16 Λ ( α 1 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) f 1 ( u , x ( u ) , y ( u ) ) d q u + β 1 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) g 1 ( u , x ( u ) , y ( u ) ) d q u λ 1 0 η j ( η j q u ) ( p 2 2 ) Γ q ( p 2 1 ) x ( u ) d q u ) + σ 3 ρ 12 t r 2 + ρ 16 Λ ( α 2 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 1 ( 1 q u ) ( r 2 1 ) Γ q ( r 2 ) y ( u ) d q u ) + σ 4 ρ 12 t r 2 + ρ 16 Λ ( α 2 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) f 2 ( u , x ( u ) , y ( u ) ) d q u + β 2 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) g 2 ( u , x ( u ) , y ( u ) ) d q u λ 2 0 1 ( 1 q u ) ( r 2 2 ) Γ q ( r 2 1 ) y ( u ) d q u ) .

3. Existence and Uniqueness Results

In the sequel, we set the notation
Ψ 1 = | α 1 | Γ q ( p 1 + p 2 + 1 ) + | α 1 | ( γ 1 + γ 2 ) | Λ | , Ψ 2 = | α 2 | ( γ 3 + γ 4 ) | Λ | , Ψ 3 = | α 1 | ( γ 13 + γ 14 ) | Λ | , Ψ 4 = | α 2 | Γ q ( r 1 + r 2 + 1 ) + | α 2 | ( γ 15 + γ 16 ) | Λ | , Φ 1 = | β 1 | Γ q ( p 1 + p 2 + ξ 1 + 1 ) + | β 1 | ( γ 5 + γ 6 ) | Λ | , Φ 2 = | β 2 | ( γ 7 + γ 8 ) | Λ | , Φ 3 = | β 1 | ( γ 17 + γ 18 ) | Λ | , Φ 4 = | β 2 | Γ q ( r 1 + r 2 + ξ 2 + 1 ) + | β 2 | ( γ 19 + γ 20 ) | Λ | , Θ 1 = | λ 1 | Γ q ( p 2 + 1 ) + | λ 1 | ( γ 9 + γ 10 ) | Λ | + | λ 2 | ( γ 11 + γ 12 ) | Λ | , Θ 2 = | λ 1 | ( γ 21 + γ 22 ) | Λ | + | λ 2 | Γ q ( r 2 + 1 ) + | λ 2 | ( γ 23 + γ 24 ) | Λ | ,
γ 1 = | δ 2 | | ρ 2 + ρ 6 | η j p 1 + p 2 + | σ 1 | | ρ 3 + ρ 7 | Γ q ( p 1 + p 2 + 1 ) , γ 2 = | δ 4 | | ρ 4 + ρ 8 | η j p 1 + p 2 1 + | σ 2 | | ρ 3 + ρ 7 | Γ q ( p 1 + p 2 ) , γ 3 = | δ 1 | | ρ 1 + ρ 5 | η j r 1 + r 2 + | σ 3 | | ρ 4 + ρ 8 | Γ q ( r 1 + r 2 + 1 ) , γ 4 = | δ 3 | | ρ 3 + ρ 7 | η j r 1 + r 2 1 + | σ 4 | | ρ 4 + ρ 8 | Γ q ( r 1 + r 2 ) , γ 5 = | δ 2 | | ρ 2 + ρ 6 | η j p 1 + p 2 + ξ 1 + | σ 1 | | ρ 3 + ρ 7 | Γ q ( p 1 + p 2 + ξ 1 + 1 ) , γ 6 = | δ 4 | | ρ 4 + ρ 8 | η j p 1 + p 2 + ξ 1 1 + | σ 2 | | ρ 3 + ρ 7 | Γ q ( p 1 + p 2 + ξ 1 ) , γ 7 = | δ 1 | | ρ 1 + ρ 5 | η j r 1 + r 2 + ξ 2 + | σ 3 | | ρ 4 + ρ 8 | Γ q ( r 1 + r 2 + ξ 2 + 1 ) , γ 8 = | δ 3 | | ρ 3 + ρ 7 | η j r 1 + r 2 + ξ 2 1 + | σ 4 | | ρ 4 + ρ 8 | Γ q ( r 1 + r 2 + ξ 2 ) , γ 9 = | δ 2 | | ρ 2 + ρ 6 | η j p 2 + | σ 1 | | ρ 3 + ρ 7 | Γ q ( p 2 + 1 ) , γ 10 = | δ 4 | | ρ 4 + ρ 8 | η j p 2 1 + | σ 2 | | ρ 3 + ρ 7 | Γ q ( p 2 ) , γ 11 = | δ 1 | | ρ 1 + ρ 5 | η j r 2 + | σ 3 | | ρ 4 + ρ 8 | Γ q ( r 2 + 1 ) , γ 12 = | δ 3 | | ρ 3 + ρ 7 | η j r 2 1 + | σ 4 | | ρ 4 + ρ 8 | Γ q ( r 2 ) , γ 13 = | δ 2 | | ρ 10 + ρ 14 | η j p 1 + p 2 + | σ 1 | | ρ 11 + ρ 15 | Γ q ( p 1 + p 2 + 1 ) , γ 14 = | δ 4 | | ρ 12 + ρ 16 | η j p 1 + p 2 1 + | σ 2 | | ρ 11 + ρ 15 | Γ q ( p 1 + p 2 ) , γ 15 = | δ 1 | | ρ 9 + ρ 13 | η j r 1 + r 2 + | σ 3 | | ρ 12 + ρ 16 | Γ q ( r 1 + r 2 + 1 ) , γ 16 = | δ 3 | | ρ 11 + ρ 15 | η j r 1 + r 2 1 + | σ 4 | | ρ 12 + ρ 16 | Γ q ( r 1 + r 2 ) , γ 17 = | δ 2 | | ρ 10 + ρ 14 | η j p 1 + p 2 + ξ 1 + | σ 1 | | ρ 11 + ρ 15 | Γ q ( p 1 + p 2 + ξ 1 + 1 ) , γ 18 = | δ 4 | | ρ 12 + ρ 16 | η j p 1 + p 2 + ξ 1 1 + | σ 2 | | ρ 11 + ρ 15 | Γ q ( p 1 + p 2 + ξ 1 ) , γ 19 = | δ 1 | | ρ 9 + ρ 13 | η j r 1 + r 2 + ξ 2 + | σ 3 | | ρ 12 + ρ 16 | Γ q ( r 1 + r 2 + ξ 2 + 1 ) , γ 20 = | δ 3 | | ρ 11 + ρ 15 | η j r 1 + r 2 + ξ 2 1 + | σ 4 | | ρ 12 + ρ 16 | Γ q ( r 1 + r 2 + ξ 2 ) , γ 21 = | δ 2 | | ρ 10 + ρ 14 | η j p 2 + | σ 1 | | ρ 11 + ρ 15 | Γ q ( p 2 + 1 ) , γ 22 = | δ 4 | | ρ 12 + ρ 16 | η j p 2 1 + | σ 2 | | ρ 11 + ρ 15 | Γ q ( p 2 ) , γ 23 = | δ 1 | | ρ 9 + ρ 13 | η j r 2 + | σ 3 | | ρ 12 + ρ 16 | Γ q ( r 2 + 1 ) , γ 24 = | δ 3 | | ρ 11 + ρ 15 | η j r 2 1 + | σ 4 | | ρ 12 + ρ 16 | Γ q ( r 2 ) .
In the following theorem, we prove the existence of a unique solution to the system (1) and (2) by applying the Banach contraction mapping principle [47].
Theorem 1.
Let f 1 , f 2 : [ 0 , 1 ] × R × R R , g 1 , g 2 : [ 0 , 1 ] × R × R R be continuous functions satisfying the following conditions:
(A1)
There exist positive constants ι 1 , ι 2 such that for each t [ 0 , 1 ] and x i , y i R , i = 1 , 2 ,
| f 1 ( t , x 1 , y 1 ) f 1 ( t , x 2 , y 2 ) | ι 1 ( | x 1 x 2 | + | y 1 y 2 | ) ,
| f 2 ( t , x 1 , y 1 ) f 2 ( t , x 2 , y 2 ) | ι 2 ( | x 1 x 2 | + | y 1 y 2 | ) .
(A2)
There exist positive constants κ 1 , κ 2 such that for each t [ 0 , 1 ] and x i , y i R , i = 1 , 2 ,
| g 1 ( t , x 1 , y 1 ) g 1 ( t , x 2 , y 2 ) | κ 1 ( | x 1 x 2 | + | y 1 y 2 | ) ,
| g 2 ( t , x 1 , y 1 ) g 2 ( t , x 2 , y 2 ) | κ 2 ( | x 1 x 2 | + | y 1 y 2 | ) .
Then the system (1) and (2) has a unique solution on [ 0 , 1 ] , provided that
Y = ( Ψ 1 + Ψ 3 ) ι 1 + ( Ψ 2 + Ψ 4 ) ι 2 + ( Φ 1 + Φ 3 ) κ 1 + ( Φ 2 + Φ 4 ) κ 2 + Θ 1 + Θ 2 < 1 .
where Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 , Φ 1 , Φ 2 , Φ 3 , Φ 4 , Θ 1 , Θ 2 are given in (13).
Proof. 
Let N 1 , N 2 , M 1 , M 2 be finite numbers such that
N 1 = sup t [ 0 , 1 ] | f 1 ( t , 0 , 0 ) | , N 2 = sup t [ 0 , 1 ] | f 2 ( t , 0 , 0 ) | , M 1 = sup t [ 0 , 1 ] | g 1 ( t , 0 , 0 ) | , M 2 = sup t [ 0 , 1 ] | g 2 ( t , 0 , 0 ) | .
Now we show that G B ϱ B ϱ , where B ϱ = { ( x , y ) G × G : ( x , y ) ϱ } with
ϱ ( Ψ 1 + Ψ 3 ) N 1 + ( Ψ 2 + Ψ 4 ) N 2 + ( Φ 1 + Φ 3 ) M 1 + ( Φ 2 + Φ 4 ) M 2 1 Y ,
where Y is given in (15). For any ( x , y ) B ϱ , t [ 0 , 1 ] , using ( A 1 ) , we have
| f 1 ( t , x ( t ) , y ( t ) ) | | f 1 ( t , x ( t ) , y ( t ) ) f 1 ( t , 0 , 0 ) | + | f 1 ( t , 0 , 0 ) | ι 1 ( | x ( t ) | + | y ( t ) | ) + | f 1 ( t , 0 , 0 ) | ι 1 ( x + y ) + N 1 ι 1 ϱ + N 1 .
Similarly, we can find that
| f 2 ( t , x ( t ) , y ( t ) ) | ι 2 ϱ + N 2 , | g 1 ( t , x ( t ) , y ( t ) ) | κ 1 ϱ + M 1 , | g 2 ( t , x ( t ) , y ( t ) ) | κ 2 ϱ + M 2 .
Then we have
G 1 ( x , y ) sup t [ 0 , 1 ] { | α 1 | 0 t ( t q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 t ( t q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 t ( t q u ) ( p 2 1 ) Γ q ( p 2 ) | x ( u ) | d q u + | δ 1 | | ρ 1 t p 2 + ρ 5 | | Λ | ( | α 2 | 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) | y ( u ) | d q u ) + | δ 2 | | ρ 2 t p 2 + ρ 6 | | Λ | ( | α 1 | 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) | x ( u ) | d q u ) + | δ 3 | | ρ 3 t p 2 + ρ 7 | | Λ | ( | α 2 | 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) | y ( u ) | d q u ) + | σ 1 | | ρ 3 t p 2 + ρ 7 | | Λ | ( | α 1 | 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 1 ( 1 q u ) ( p 2 1 ) Γ q ( p 2 ) | x ( u ) | d q u ) + | σ 2 | | ρ 3 t p 2 + ρ 7 | | Λ | ( | α 1 | 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 1 ( 1 q u ) ( p 2 2 ) Γ q ( p 2 1 ) | x ( u ) | d q u ) + | δ 4 | | ρ 4 t p 2 + ρ 8 | | Λ | ( | α 1 | 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 η j ( η j q u ) ( p 2 2 ) Γ q ( p 2 1 ) | x ( u ) | d q u ) + | σ 3 | | ρ 4 t p 2 + ρ 8 | | Λ | ( | α 2 | 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 1 ( 1 q u ) ( r 2 1 ) Γ q ( r 2 ) | y ( u ) | d q u ) + | σ 4 | | ρ 4 t p 2 + ρ 8 | | Λ | ( | α 2 | 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 1 ( 1 q u ) ( r 2 2 ) Γ q ( r 2 1 ) | y ( u ) | d q u ) } ( ι 1 r + N 1 ) sup t [ 0 , 1 ] { | α 1 | 0 t ( t q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) d q u + | α 1 | | δ 2 | | ρ 2 t p 2 + ρ 6 | | Λ | × 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) d q u + | α 1 | | σ 1 | | ρ 3 t p 2 + ρ 7 | | Λ | 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) d q u + | α 1 | | σ 2 | | ρ 3 t p 2 + ρ 7 | | Λ | 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) d q u + | α 1 | | δ 4 | | ρ 4 t p 2 + ρ 8 | | Λ | 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) d q u } + ( κ 1 r + M 1 ) sup t [ 0 , 1 ] { | β 1 | 0 t ( t q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) d q u + | β 1 | | δ 2 | | ρ 2 t p 2 + ρ 6 | | Λ | × 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) d q u + | β 1 | | σ 1 | | ρ 3 t p 2 + ρ 7 | | Λ | 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) d q u + | β 1 | | σ 2 | | ρ 3 t p 2 + ρ 7 | | Λ | 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) d q u + | β 1 | | δ 4 | | ρ 4 t p 2 + ρ 8 | | Λ | × 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) d q u } + ( ι 2 r + N 2 ) sup t [ 0 , 1 ] { | α 2 | | δ 1 | | ρ 1 t p 2 + ρ 5 | | Λ | × 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) d q u + | α 2 | | δ 3 | | ρ 3 t p 2 + ρ 7 | | Λ | 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) d q u + | α 2 | | σ 3 | | ρ 4 t p 2 + ρ 8 | | Λ | 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) d q u + | α 2 | | σ 4 | | ρ 4 t p 2 + ρ 8 | | Λ | 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) d q u } + ( κ 2 r + M 2 ) sup t [ 0 , 1 ] { | β 2 | | δ 1 | | ρ 1 t p 2 + ρ 5 | | Λ | 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) d q u + | β 2 | | δ 3 | | ρ 3 t p 2 + ρ 7 | | Λ | × 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) d q u + | β 2 | | σ 3 | | ρ 4 t p 2 + ρ 8 | | Λ | 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) d q u + | β 2 | | σ 4 | | ρ 4 t p 2 + ρ 8 | | Λ | 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) d q u } + ϱ sup t [ 0 , 1 ] { | λ 1 | 0 t ( t q u ) ( p 2 1 ) Γ q ( p 2 ) d q u + | λ 2 | | δ 1 | | ρ 1 t p 2 + ρ 5 | | Λ | 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) d q u + | λ 1 | | δ 2 | | ρ 2 t p 2 + ρ 6 | | Λ | 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) d q u + | λ 2 | | δ 3 | | ρ 3 t p 2 + ρ 7 | | Λ | 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) d q u + | λ 1 | | σ 1 | | ρ 3 t p 2 + ρ 7 | | Λ | 0 1 ( 1 q u ) ( p 2 1 ) Γ q ( p 2 ) d q u + | λ 1 | | σ 2 | | ρ 3 t p 2 + ρ 7 | | Λ | 0 1 ( 1 q u ) ( p 2 2 ) Γ q ( p 2 1 ) d q u + | λ 1 | | δ 4 | | ρ 4 t p 2 + ρ 8 | | Λ | 0 η j ( η j q u ) ( p 2 2 ) Γ q ( p 2 1 ) d q u + | λ 2 | | σ 3 | | ρ 4 t p 2 + ρ 8 | | Λ | 0 1 ( 1 q u ) ( r 2 1 ) Γ q ( r 2 ) d q u + | λ 2 | | σ 4 | | ρ 4 t p 2 + ρ 8 | | Λ | 0 1 ( 1 q u ) ( r 2 2 ) Γ q ( r 2 1 ) d q u } ( ι 1 ϱ + N 1 ) Ψ 1 + ( ι 2 ϱ + N 2 ) Ψ 2 + ( κ 1 ϱ + M 1 ) Φ 1 + ( κ 2 ϱ + M 2 ) Φ 2 + ϱ Θ 1 ,
where Ψ 1 , Ψ 2 , Φ 1 , Φ 2 , Θ 1 are given in (13).
Furthermore, we obtain
G 2 ( x , y ) sup t [ 0 , 1 ] { | α 2 | 0 t ( t q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 t ( t q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 t ( t q u ) ( r 2 1 ) Γ q ( r 2 ) | y ( u ) | d q u + | δ 1 | | ρ 9 t r 2 + ρ 13 | | Λ | ( | α 2 | 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) | y ( u ) | d q u ) + | δ 2 | | ρ 10 t r 2 + ρ 14 | | Λ | ( | α 1 | 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) | x ( u ) | d q u ) + | δ 3 | | ρ 11 t r 2 + ρ 15 | | Λ | ( | α 2 | 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) | y ( u ) | d q u ) + | σ 1 | | ρ 11 t r 2 + ρ 15 | | Λ | ( | α 1 | 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 1 ( 1 q u ) ( p 2 1 ) Γ q ( p 2 ) | x ( u ) | d q u ) + | σ 2 | | ρ 11 t r 2 + ρ 15 | | Λ | ( | α 1 | 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 1 ( 1 q u ) ( p 2 2 ) Γ q ( p 2 1 ) | x ( u ) | d q u ) + | δ 4 | | ρ 12 t r 2 + ρ 16 | | Λ | ( | α 1 | 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) | f 1 ( u , x ( u ) , y ( u ) ) | d q u + | β 1 | 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) | g 1 ( u , x ( u ) , y ( u ) ) | d q u + | λ 1 | 0 η j ( η j q u ) ( p 2 2 ) Γ q ( p 2 1 ) | x ( u ) | d q u ) + | σ 3 | | ρ 12 t r 2 + ρ 16 | | Λ | ( | α 2 | 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 1 ( 1 q u ) ( r 2 1 ) Γ q ( r 2 ) | y ( u ) | d q u ) + | σ 4 | | ρ 12 t r 2 + ρ 16 | | Λ | ( | α 2 | 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) | f 2 ( u , x ( u ) , y ( u ) ) | d q u + | β 2 | 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) | g 2 ( u , x ( u ) , y ( u ) ) | d q u + | λ 2 | 0 1 ( 1 q u ) ( r 2 2 ) Γ q ( r 2 1 ) | y ( u ) | d q u ) } ( ι 1 r + N 1 ) sup t [ 0 , 1 ] { | α 1 | | δ 2 | | ρ 10 t r 2 + ρ 14 | | Λ | 0 η j ( η j q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) d q u + | α 1 | | σ 1 | | ρ 11 t r 2 + ρ 15 | | Λ | 0 1 ( 1 q u ) ( p 1 + p 2 1 ) Γ q ( p 1 + p 2 ) d q u + | α 1 | | σ 2 | | ρ 11 t r 2 + ρ 15 | | Λ | 0 1 ( 1 q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) d q u + | α 1 | | δ 4 | | ρ 12 t r 2 + ρ 16 | | Λ | 0 η j ( η j q u ) ( p 1 + p 2 2 ) Γ q ( p 1 + p 2 1 ) d q u } + ( κ 1 r + M 1 ) sup t [ 0 , 1 ] { | β 1 | | δ 2 | | ρ 10 t r 2 + ρ 14 | | Λ | × 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) d q u + | β 1 | | σ 1 | | ρ 11 t r 2 + ρ 15 | | Λ | 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 1 ) Γ q ( p 1 + p 2 + ξ 1 ) d q u + | β 1 | | σ 2 | | ρ 11 t r 2 + ρ 15 | | Λ | 0 1 ( 1 q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) d q u + | β 1 | | δ 4 | | ρ 12 t r 2 + ρ 16 | | Λ | × 0 η j ( η j q u ) ( p 1 + p 2 + ξ 1 2 ) Γ q ( p 1 + p 2 + ξ 1 1 ) d q u } + ( ι 2 r + N 2 ) sup t [ 0 , 1 ] { | α 2 | 0 t ( t q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) d q u + | α 2 | | δ 1 | | ρ 9 t r 2 + ρ 13 | | Λ | 0 η j ( η j q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) d q u + | α 2 | | δ 3 | | ρ 11 t r 2 + ρ 15 | | Λ | 0 η j ( η j q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) d q u + | α 2 | | σ 3 | | ρ 12 t r 2 + ρ 16 | | Λ | 0 1 ( 1 q u ) ( r 1 + r 2 1 ) Γ q ( r 1 + r 2 ) d q u + | α 2 | | σ 4 | | ρ 12 t r 2 + ρ 16 | | Λ | 0 1 ( 1 q u ) ( r 1 + r 2 2 ) Γ q ( r 1 + r 2 1 ) d q u } + ( κ 2 r + M 2 ) sup t [ 0 , 1 ] { | β 2 | 0 t ( t q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) d q u + | β 2 | | δ 1 | | ρ 9 t r 2 + ρ 13 | | Λ | × 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) d q u + | β 2 | | δ 3 | | ρ 11 t r 2 + ρ 15 | | Λ | 0 η j ( η j q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) d q u + | β 2 | | σ 3 | | ρ 12 t r 2 + ρ 16 | | Λ | 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 1 ) Γ q ( r 1 + r 2 + ξ 2 ) d q u + | β 2 | | σ 4 | | ρ 12 t r 2 + ρ 16 | | Λ | × 0 1 ( 1 q u ) ( r 1 + r 2 + ξ 2 2 ) Γ q ( r 1 + r 2 + ξ 2 1 ) d q u } + ϱ sup t [ 0 , 1 ] { | λ 2 | 0 t ( t q u ) ( r 2 1 ) Γ q ( r 2 ) d q u + | λ 2 | | δ 1 | | ρ 9 t r 2 + ρ 13 | | Λ | 0 η j ( η j q u ) ( r 2 1 ) Γ q ( r 2 ) d q u + | λ 1 | | δ 2 | | ρ 10 t r 2 + ρ 14 | | Λ | 0 η j ( η j q u ) ( p 2 1 ) Γ q ( p 2 ) d q u + | λ 2 | | δ 3 | | ρ 11 t r 2 + ρ 15 | | Λ | 0 η j ( η j q u ) ( r 2 2 ) Γ q ( r 2 1 ) d q u + | λ 1 | | σ 1 | | ρ 11 t r 2 + ρ 15 | | Λ | 0 1 ( 1 q u ) ( p 2