Uniqueness and Existence of Solution for a System of Fractional 𝑞-Difference Equations

Abstract

We prove the existence and uniqueness of solution for a system of fractional differential equations. Our results are based on the nonlinear alternative of Leray-Schauder type and Banach’s fixed-point theorem.

1. Introduction

This paper is mainly concerned with the uniqueness and existence of solution for a system of fractional 𝑞-difference equations given by ^𝐶𝐷^𝛼_𝑞𝑢(𝑡)=𝑓(𝑡,𝑣(𝑡)),1<𝛼≤2,𝑡∈[0,1_𝐶],𝐷^𝛽_𝑞𝑣(𝑡)((𝑡[],𝛼=𝑔𝑡,𝑢)),1<𝛽≤2,𝑡∈0,1₁𝑢(0)−𝛽₁𝐷_𝑞𝑢(0)=𝛾₁𝑢𝜂₁,𝛼₂𝑢(1)+𝛽₂𝐷_𝑞𝑢(1)=𝛾₂𝑢𝜂₂,𝛼₃𝑣(0)−𝛽₃𝐷_𝑞𝑣(0)=𝛾₃𝑣𝜂₃,𝛼₄𝑣(1)+𝛽₄𝐷_𝑞𝑣(1)=𝛾₄𝑣𝜂₄, (1) where ^𝐶𝐷^𝛼_𝑞, ^𝐶𝐷^𝛽_𝑞 is the fractional 𝑞-derivatives of the Caputo type, 1<𝛼, 𝛽≤2, 𝛼_𝑖  (𝑖=1,2,3,4), 𝛽_𝑖  (𝑖=1,2,3,4), 𝛾_𝑖  (𝑖=1,2,3,4), and 𝜂_𝑖  (𝑖=1,2,3,4) are arbitrary real constants, and 𝑓,𝑔∶[0,1]×ℝ→ℝ are given continuous functions.

In the last few years, fractional differential equations (in short FDEs) have been studied extensively. The motivation for those works stems from both the development of the theory of fractional calculus itself and the applications of such constructions in various sciences such as physics, mechanics, chemistry, and engineering. For an extensive collection of such results, we refer the readers to the monographs by Kilbas et al. [1], Miller and Ross [2], Oldham and Spanier [3], Podlubny [4], and Samko et al. [5].

Some basic theory for the initial value problems of fractional differential equations involving Riemann-Liouville differential operator has been discussed by Lakshmikantham and Vatsala ([6–8]), Babakhani and Daftardar-Gejji ([9–11]), Bai [12], and so on. Also, there are some papers which deal with the existence and multiplicity of solutions (or positive solution) for nonlinear FDE of BVPs by using techniques of nonlinear analysis (fixed-point theorems, Leray-Shauder theory, topological degree theory, etc.)—see ([13–18]) and the references therein. The study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications. The reader is referred to the papers ([19–22]) and the references cited therein.

The pioneer work on 𝑞-difference calculus or quantum calculus dates back to Jackson’s papers ([23, 24]), while a systematic treatment of the subject can be found in [25, 26]. For some recent existence results on 𝑞-difference equations, see [27–29] and the references therein.

There has also been a growing interest on the subject of discrete fractional equations on the time scale ℤ. Some interesting results on the topic can be found in a series of papers [30–38]. Fractional 𝑞-difference equations have recently attracted the attention of several researchers. For some earlier work on the topic, we refer to [39, 40], whereas some recent work on the existence theory of fractional 𝑞-difference equations can be found in [41–45]. However, the study of boundary value problems of fractional 𝑞-difference equations is at its infancy and much of the work on the topic is yet to be done.

From the above works, we can see a fact, although the fractional boundary value problems have been investigated by some authors. To the best of our knowledge, there have been few papers which deal with problem (1) for nonlinear fractional differential equation. Motivated by all the works above, in this paper we discuss problem (1). Using nonlinear alternative of Leray-Schauder type, we will give the existence and uniqueness of solution for a system of fractional differential equations with Riemann-Liouville integral boundary conditions of different order (1).

The paper is organized as follows. In Section 2, we give some preliminary results that will be used in the proof of the main results. In Section 3, we establish the uniqueness and existence of a solution for the nonlinear fractional differential equation boundary value problem (1). In last section, we give two examples to illustrate our work.

2. Preliminaries and Lemmas

In this section, we cite some definitions and fundamental results of the 𝑞-calculus as well as of the fractional 𝑞-calculus ([46, 47]). We also give a lemma that will be used in obtaining the main results of the paper.

Let 𝑞∈(0,1) and define [47] [𝑎]_𝑞=𝑞^𝑎−1𝑞−1=𝑎^𝑎−1+⋯+1,𝑎∈ℝ. (2) The 𝑞-analogue of the power (𝑎−𝑏)^𝑛 is (𝑎−𝑏)⁽⁰⁾()=1,𝑎−𝑏^(𝑛)=^𝑛−1_𝑘=0𝑎−𝑏𝑞^𝑘,𝑎,𝑏∈ℝ,𝑛∈ℕ. (3) If 𝛼 is not a positive integer, then (𝑎−𝑏)^(𝛼)=𝑎^𝛼∞_𝑖=01−(𝑏/𝑎)𝑞^𝑖1−(𝑏/𝑎)𝑞^𝛼+𝑖. (4) Note that if 𝑏=0, then 𝑎^(𝛼)=𝑎^𝛼. The 𝑞-gamma function is defined by Γ_𝑞(𝑥)=1−𝑞^(𝑥−1)1−𝑞^𝑥−1,𝑥∈ℝ⧵{0,−1,−2,…},0<𝑞<1 (5) and satisfies Γ_𝑞(𝑥+1)=[𝑥]_𝑞Γ_𝑞(𝑥) (see [47]).

The 𝑞-derivative of a function 𝑓 is here defined by 𝐷_𝑞𝑓(𝑥)=𝑑_𝑞𝑓(𝑥)𝑑_𝑞𝑥=𝑓𝑞𝑥−𝑓(𝑥)𝑞−1𝑥 (6) and 𝑞-derivatives of higher order by 𝐷^𝑛_𝑞𝑓(𝑥)=𝑓(𝑥),if𝐷𝑛=0,_𝑞𝐷_𝑞^𝑛−1𝑓(𝑥),if𝑛∈ℕ. (7) The 𝑞-integral of a function 𝑓 defined in the interval [0,𝑏] is given by ^𝑥₀𝑓(𝑡)𝑑_𝑞𝑡=𝑥1−𝑞^∞_𝑛=0𝑓𝑥𝑞^𝑛𝑞^𝑛,0≤𝑞<1,𝑥∈[0,𝑏]. (8) If 𝑎∈[0,𝑏] and 𝑓 is defined in the interval [0,𝑏], its integral from 𝑎 to 𝑏 is defined by ^𝑏_𝑎𝑓(𝑡)𝑑_𝑞𝑡=^𝑏₀𝑓(𝑡)𝑑_𝑞𝑡−^𝑎₀𝑓(𝑡)𝑑_𝑞𝑡. (9) Similarly, as done for derivatives, an operator 𝐼^𝑛_𝑞 can be defined, namely, by 𝐼⁰_𝑞𝑓(𝑥)(𝑥),𝐼=𝑓_𝑛𝑞𝑓(𝑥)=𝐼_𝑞𝐼_𝑞^𝑛−1𝑓(𝑥),𝑛∈ℕ. (10) The fundamental theorem of calculus applies to these operators 𝐼_𝑞 and 𝐷_𝑞; that is, 𝐷_𝑞𝐼_𝑞𝑓(𝑥)=𝑓(𝑥), (11) and if 𝑓 is continuous at 𝑥=0, then 𝐼_𝑞𝐷_𝑞𝑓(𝑥)=𝑓(𝑥)−𝑓(0). (12) Basic properties of the two operators can be found in the book that is mentioned in [8]. We now point out three formulas that will be used later (_𝑖𝐷_𝑞 denotes the derivative with respect to variable 𝑖) [43]: [𝑎(𝑡−𝑠)]^(𝛼)=𝑎^𝛼(𝑡−𝑠)^(𝛼),_𝑡𝐷_𝑞(𝑡−𝑠)^(𝛼)=[𝛼]_𝑞(𝑡−𝑠)^(𝛼−1),_𝑥𝐷_𝑞^𝑥₀𝑓(𝑥,𝑡)𝑑_𝑞𝑡(𝑥)=^𝑥_0𝑥𝐷_𝑞𝑓(𝑥,𝑡)𝑑_𝑞𝑡+𝑓𝑞𝑥,𝑥. (13)

Remark 1. We note that if 𝛼>0 and 𝑎≤𝑏≤𝑡, then (𝑡−𝑎)^(𝛼)≥(𝑡−𝑏)^(𝛼) [43].

Definition 2 (see [40]). Let 𝛼≥0 and let 𝑓 be a function defined on [0,1]. The fractional 𝑞-integral of the Riemann-Liouville type is (_RL𝐼⁰_𝑞𝑓)(𝑥)=𝑓(𝑥) and _RL𝐼^𝛼_𝑞𝑓(𝑥)=^𝑥₀𝑥−𝑞𝑡^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑡)𝑑_𝑞𝑡,𝛼∈ℝ⁺,𝑥∈[0,1]. (14)

Definition 3 (see [48]). The fractional 𝑞-derivative of the Riemann-Liouville type of order 𝛼≥0 is defined by (_RL𝐷⁰_𝑞𝑓)(𝑥)=𝑓(𝑥) and _RL𝐷^𝛼_𝑞𝑓(𝑥)=𝐷_𝑞^[𝛼]𝐼_𝑞^{[𝛼]−𝛼}𝑓(𝑥),𝛼>0, (15) where [𝛼] is the smallest integer greater than or equal to 𝛼.

Definition 4 (see [48]). The fractional 𝑞-derivative of the Caputo type of order 𝛼≥0 is defined by ^𝐶𝐷^𝛼_𝑞𝑓(𝑥)=𝐼_𝑞^{[𝛼]−𝛼}𝐷_𝑞^[𝛼]𝑓(𝑥),𝛼>0, (16) where [𝛼] is the smallest integer greater than or equal to 𝛼.

Lemma 5. Let 𝛼,𝛽≥0 and let 𝑓 be a function defined on [0,1]. Then the next formulas hold:(1)(𝐼^𝛽_𝑞𝐼^𝛼_𝑞𝑓)(𝑥)=(𝐼_𝑞^𝛼+𝛽𝑓)(𝑥),(2)(𝐷^𝛼_𝑞𝐼^𝛼_𝑞𝑓)(𝑥)=𝑓(𝑥).

Lemma 6 (see [42]). Let 𝛼≥0 and 𝑛∈ℕ. Then the following equality holds: _𝑅𝐿𝐼^𝛼_𝑞𝑅𝐿𝐷^𝑛_𝑞𝑓(𝑥)=_𝑅𝐿𝐷^𝑛_𝑞𝑅𝐿𝐼^𝛼_𝑞𝑓(𝑥)−^𝛼−1_𝑘=0𝑥^{𝛼−𝑛+𝑘}Γ_𝑞()𝐷𝛼+𝑘−𝑛+1^𝑘𝑞𝑓(0). (17)

Lemma 7 (see [48]). Let 𝛼>0 and 𝑛∈ℝ⁺⧵ℕ. Then the following equality holds: 𝐼^𝛼_𝑞^𝐶𝐷^𝛼_𝑞𝑓(𝑥)=𝑓(𝑥)−^[𝛼]−1_𝑘=0𝑥^𝑘Γ_𝑞()𝐷𝑘+1^𝑘𝑞𝑓(0). (18)
For convenience, one introduces the following notations: 𝑏₁=𝛾₁𝛼₂−𝛾₂Δ,𝑏₂=𝛾₁𝛼₂+𝛽₂−𝛾₂𝜂₂Δ,𝑏₃=𝛼₁−𝛾₁Δ,𝑏₄=𝛽₁+𝛾₁𝜂₁Δ,Δ=𝛾₂−𝛼₂𝛽₁+𝛾₁𝜂₁+𝛾₁−𝛼₁𝛼₂+𝛽₂−𝛾₂𝜂₂. (19)
From Lemmas 5 and 7, we can obtain the following lemma.

Lemma 8. Let ℎ∈𝐶[0,1] and Δ≠0; then the unique solution of the linear fractional boundary value problem ^𝐶𝐷^𝛼_𝑞𝑢(𝑡)(𝑡)[],𝛼=ℎ,1<𝛼≤2,𝑡∈0,1₁𝑢(0)−𝛽₁𝐷_𝑞𝑢(0)=𝛾₁𝑢𝜂₁,𝛼₂𝑢(1)+𝛽₂𝐷_𝑞𝑢(1)=𝛾₂𝑢𝜂₂ (20) is given by 𝑢(𝑡)=^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)ℎ(𝑠)𝑑_𝑞𝑠+𝑏₁𝑡+𝑏₂^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)ℎ(𝑠)𝑑_𝑞𝑠+𝑏₃𝑡+𝑏₄−𝛾₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)ℎ(𝑠)𝑑_𝑞𝑠+𝛼₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)ℎ(𝑠)𝑑_𝑞𝑠+𝛽₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()ℎ(𝑠)𝑑𝛼−1^𝑞𝑠. (21)
The following lemma is fundamental in the proofs of our main result.

Lemma 9 (see [49]; nonlinear alternative of Leray-Schauder type). Let 𝐸 be a Banach space with 𝑀⊆𝐸 closed and convex. Assume that 𝑈 is a relatively open subset of 𝐶 with 0∈𝑈 and 𝐹∶𝑈→𝐶 is continuous, compact (i.e., 𝐹(𝑈) is a relatively compact subset of 𝐶) map. Then either(i)𝐹 has a fixed point in 𝑈 or(ii)there exist 𝑢∈𝜕𝑈 and 𝜆∈(0,1) with 𝑢=𝜆𝐹𝑢.

3. Main Results

In this section, we will discuss the uniqueness and existence of solutions for boundary value problem (1).

First of all, we define the Banach space 𝑋={𝑢∣𝑢∈𝐶[0,1]} endowed with the norm ‖𝑢‖=max_𝑡∈[0,1]|𝑢(𝑡)|. For (𝑢,𝑣)∈𝑋×𝑋, let ‖(𝑢,𝑣)‖=max{‖𝑢‖,‖𝑣‖}; then (𝑋×𝑋,‖(⋅,⋅)‖) is a Banach space.

For convenience, we set ▴=𝛾₄−𝛼₄𝛽₃+𝛾₃𝜂₃+𝛾₃−𝛼₃𝛼₄+𝛽₄−𝛾₄𝜂₄, (22) and let ▴≠0. Note 𝔟₁=𝛾₃𝛼₄−𝛾₄▴,𝔟₂=𝛾₃𝛼₄+𝛽₄−𝛾₄𝜂₄▴,𝔟₃=𝛼₃−𝛾₃▴,𝔟₄=𝛽₃+𝛾₃𝜂₃▴. (23)

Employing Lemma 8, system (1) can be expressed as 𝑢(𝑡)=^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝑏₁𝑡+𝑏₂^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝑏₃𝑡+𝑏₄−𝛾₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛼₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛽₂¹₀(1−𝑞𝑠)^(𝛼−2)Γ_𝑞()𝑓((𝑠𝑑𝛼−1𝑠,𝑣))_𝑞𝑠,𝑣(𝑡)=^𝑡₀𝑡−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝑔(𝑠,𝑢(𝑠))𝑑_𝑞𝑠+𝔟₁𝑡+𝔟₂^𝜂3₀𝜂₃−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝑔(𝑠,𝑢(𝑠))𝑑_𝑞𝑠+𝔟₃𝑡+𝔟₄−𝛾₄^𝜂4₀𝜂₄−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝑔(𝑠,𝑢(𝑠))𝑑_𝑞𝑠+𝛼₄¹₀1−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝑔(𝑠,𝑢(𝑠))𝑑_𝑞𝑠+𝛽₄¹₀1−𝑞𝑠^(𝛽−2)Γ_𝑞𝑔((𝑠𝑑𝛽−1𝑠,𝑢))_𝑞𝑠, (24) where 𝑏₁, 𝑏₂, 𝑏₃, 𝑏₄ are given by (21), and 𝔟₁, 𝔟₂, 𝔟₃, 𝔟₄ are given by (22)

From Lemma 8 in Section 2, we can obtain the following lemma.

Lemma 10. Suppose that 𝑓(𝑡,𝑣) and 𝑔(𝑡,𝑢) are continuous; then (𝑢,𝑣)∈𝑋×𝑋 is a solution of BVP (1) if and only if (𝑢,𝑣)∈𝑋×𝑋 is a solution of the integral equations (24).
Let (𝑢,𝑣)∈𝑋×𝑋; define an operator 𝑇∶𝑋×𝑋→𝑋×𝑋 as 𝑇(𝑢,𝑣)(𝑡)=𝑇₁𝑣(𝑡),𝑇₂𝑢(𝑡), (25) where 𝑇₁𝑣(𝑡)=^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝑏₁𝑡+𝑏₂^𝜂1₀(𝜂₁−𝑞𝑠)^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝑏₃𝑡+𝑏₄−𝛾₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛼₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛽₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝑓((𝑠𝑑𝛼−1𝑠,𝑣))_𝑞𝑠,𝑇₂𝑢(𝑡)=^𝑡₀𝑡−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝑔(𝑠,𝑢(𝑠))𝑑_𝑞𝑠+𝔟₁𝑡+𝔟₂^𝜂3₀𝜂₃−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝑔(𝑠,𝑢(𝑠))𝑑_𝑞𝑠+𝔟₃𝑡+𝔟₄−𝛾₄^𝜂4₀𝜂₄−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝑔(𝑠,𝑢(𝑠))𝑑_𝑞𝑠+𝛼₄¹₀1−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝑔(𝑠,𝑢(𝑠))𝑑_𝑞𝑠+𝛽₄¹₀1−𝑞𝑠^(𝛽−2)Γ_𝑞𝑔((𝑠𝑑𝛽−1𝑠,𝑢))_𝑞𝑠; (26)(27) then, by Lemma 10, the fixed point of operator 𝑇 coincides with the solution of system (1).
In the first result, we prove uniqueness of solution of the boundary value problem (1) via Banach’s contraction principle.

Theorem 11. Assume that 𝑓,𝑔∶[0,1]×ℝ→ℝ are continuous functions and the following conditions hold:(H1)there exist two 𝑞-integrable functions 𝐿₁,𝐿₂∶[0,1]→ℝ that satisfy 𝑓(𝑡,𝑢)−𝑓(𝑡,𝑣)≤𝐿₁(𝑡|[]𝑔()()|𝑢−𝑣,∀𝑡∈0,1,𝑢,𝑣∈ℝ;𝑡,𝑢−𝑔𝑡,𝑣)≤𝐿₂(𝑡)|𝑢−𝑣|,∀𝑡∈[0,1],𝑢,𝑣∈ℝ. (28)In addition, assume that 𝜅₁∶=1+𝛼₂𝛿₂𝐼^𝛼_𝑞𝐿₁(1)+𝛾₁𝛿₁𝐼^𝛼_𝑞𝐿₁𝜂₁+𝛾₂𝛿₂𝐼^𝛼_𝑞𝐿₁𝜂₂+𝛽₂𝛿₂𝐼_𝑞^𝛼−1𝐿₁(1)𝜅<1,₂∶=1+𝛼₄𝛿₄𝐼^𝛽_𝑞𝐿₂(1)+𝛾₃𝛿₃𝐼^𝛽_𝑞𝐿₂𝜂₃+𝛾₄𝛿₄𝐼^𝛽_𝑞𝐿₂𝜂₄+𝛽₄𝛿₄𝐼_𝑞^𝛽−1𝐿₂(1)<1, (29) where 𝛿₁𝛼∶=²−𝛾₂+𝛼₂+𝛽₂−𝛾₂𝜂₂|Δ|,𝛿₂𝛼∶=¹−𝛾₁+𝛽₁−𝛾₁𝜂₁|Δ|,𝛿₃𝛼∶=⁴−𝛾₄+𝛼₄+𝛽₄−𝛾₄𝜂₄|▴|,𝛿₄𝛼∶=³−𝛾₃+𝛽₃−𝛾₃𝜂₃|▴|, (30) where Δ and ▴ are given by (19) and (22), respectively. Then system (1) has a unique solution.

Proof. Let us set sup_𝑡∈[0,1]|𝑓(𝑡,0)|=𝑀₁<∞, sup_𝑡∈[0,1]|𝑔(𝑡,0)|=𝑀₂<∞, 𝐴₁=1Γ_𝑞()𝛾𝛼+11+₁𝛿₁𝜂₁^(𝛼−1)+𝛾₂𝛿₂𝜂₂^(𝛼−1)+𝛼₂𝛿₂+𝛽₂𝛿₂Γ_𝑞(𝛼),𝐴₂=1Γ_𝑞𝛾𝛽+11+₃𝛿₃𝜂₃^(𝛽−1)+𝛾₄𝛿₄𝜂₄^(𝛽−1)+𝛼₄𝛿₄+𝛽₄𝛿₄Γ_𝑞𝛽. (31)(32)
Define )},𝐵𝑈={(𝑢,𝑣∈𝑋×𝑋∶‖(𝑢,𝑣)‖≤𝑟_𝑟1=𝑢∈𝐶[0,1]∶|𝑢|≤𝑟₁,𝐵_𝑟2=𝑣∈𝐶[0,1]∶|𝑣|≤𝑟₂. (33)
For 𝑣∈𝐵_𝑟1, we obtain 𝑓(𝑡,𝑣(𝑡))≤𝑓(𝑡,𝑣(𝑡))−𝑓(𝑡,0)+𝑓(𝑡,0)≤𝐿₁(𝑡)|𝑣(𝑡)|+𝑓(𝑡,0)≤𝐿₁(𝑡)𝑟₁+𝑀₁. (34) Then, for 𝑣∈𝐵_𝑟1, 𝑡∈[0,1], we have 𝑇₁𝑣≤sup_[]𝑡∈0,1^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛾₁Δ𝛼₂−𝛾₂𝑡−𝛼₂+𝛽₂−𝛾₂𝜂₂×^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛾₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛼₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛽₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝑓((𝑠𝑑𝛼−1𝑠,𝑣))_𝑞𝑠≤sup_[]𝑡∈0,1^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑟₁+𝑀₁𝑑_𝑞𝑠+𝛾₁Δ𝛼₂−𝛾₂𝑡−𝛼₂+𝛽₂−𝛾₂𝜂₂×^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑟₁+𝑀₁𝑑_𝑞𝑠+𝛾₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑟₁+𝑀₁𝑑_𝑞𝑠+𝛼₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑟₁+𝑀₁𝑑_𝑞𝑠+𝛽₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝐿𝛼−1¹(𝑠)𝑟₁+𝑀₁𝑑_𝑞𝑠≤𝑀₁¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑑_𝑞𝑠+𝛾₁𝛿₁^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑑_𝑞𝑠+𝛾₂𝛿₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑑_𝑞𝑠+𝛾₂𝛿₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑑_𝑞𝑠+𝛾₂𝛿₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝑑𝛼−1^𝑞𝑠+𝑟₁¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₁𝛿₁^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛿₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛿₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛿₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝐿𝛼−1¹(𝑠)𝑑_𝑞𝑠≤𝑀₁1Γ_𝑞()×𝛾𝛼+11+₁𝛿₁𝜂₁^(𝛼−1)+𝛾₂𝛿₂𝜂₂^(𝛼−1)+𝛼₂𝛿₂+𝛽₂𝛿₂Γ_𝑞(𝛼)+𝑟₁¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₁𝛿₁^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛿₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛼₂𝛿₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛽₂𝛿₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝐿𝛼−1¹(𝑠)𝑑_𝑞𝑠. (35) In view of (31), we obtain 𝑇₁𝑣≤𝑀₁𝐴₁+𝑟₁𝜅₁. (36) From the last estimate we deduce that 𝑟₁=𝑀₁𝐴₁/(1−𝜅₁).
By a similar way as done above we have 𝑇₂𝑢≤𝑀₂1Γ_𝑞×𝛾𝛽+11+₃𝛿₃𝜂₃^(𝛽−1)+𝛾₄𝛿₄𝜂₄^(𝛽−1)+𝛼₄𝛿₄+𝛽₄𝛿₄Γ_𝑞𝛽+𝑟₂¹₀1−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝐿₂(𝑠)𝑑_𝑞𝑠+𝛾₃𝛿₃^𝜂3₀𝜂₃−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝐿₂(𝑠)𝑑_𝑞𝑠+𝛾₄𝛿₄^𝜂4₀𝜂₄−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝐿₂(𝑠)𝑑_𝑞𝑠+𝛼₄𝛿₄¹₀1−𝑞𝑠^(𝛽−1)Γ_𝑞𝛽𝐿₂(𝑠)𝑑_𝑞𝑠+𝛽₄𝛿₄¹₀1−𝑞𝑠^(𝛽−2)Γ_𝑞𝐿𝛽−1²(𝑠)𝑑_𝑞𝑠≤𝑀₂𝐴₂+𝑟₂𝜅₂ (37) and 𝑟₂=𝑀₂𝐴₂/(1−𝜅₂).
Therefore, we obtain ‖𝑇(𝑢,𝑣)‖=max𝑇₁𝑣,𝑇₂𝑢=max𝑇₁𝑣,𝑇₂𝑢=𝑟. (38) From the last estimate we can choose 𝑟=max{𝑟₁,𝑟₂}; then, for every (𝑢,𝑣)∈𝑈, we have 𝑇𝑈⊂𝑈.
In order to show that 𝑇 is a contraction, let 𝑢,𝑣,𝑢₁,𝑣₁∈𝑋, and, for any 𝑡∈[0,1], we get 𝑇₁𝑣−𝑇₁𝑣₁≤sup_[]𝑡∈0,1^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))−𝑓𝑠,𝑣₁(𝑠)𝑑_𝑞𝑠+𝛾₁Δ𝛼₂−𝛾₂𝑡−𝛼₂+𝛽₂−𝛾₂𝜂₂×^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))−𝑓𝑠,𝑣₁(𝑠)𝑑_𝑞𝑠+𝛾₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))−𝑓𝑠,𝑣₁(𝑠)𝑑_𝑞𝑠+𝛼₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))−𝑓𝑠,𝑣₁(𝑠)𝑑_𝑞𝑠+𝛽₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝑓((𝑠𝛼−1𝑠,𝑣))−𝑓𝑠,𝑣₁(𝑠)𝑑_𝑞𝑠≤sup_𝑡∈[0,1]^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₁Δ𝛼₂−𝛾₂𝑡−𝛼₂+𝛽₂−𝛾₂𝜂₂×^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛼₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛽₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝐿𝛼−1¹(𝑠)𝑑_𝑞𝑠𝑣−𝑣₁≤sup_[]𝑡∈0,1^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₁𝛿₁^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛿₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛼₂𝛿₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝐿₁(𝑠)𝑑_𝑞𝑠+𝛽₂𝛿₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝐿𝛼−1¹(𝑠)𝑑_𝑞𝑠𝑣−𝑣₁, (39) which, in view of 𝜅₁<1 and (31), implies that 𝑇₁𝑣(𝑡)−𝑇₁𝑣₁(𝑡)≤𝜅₁𝑣−𝑣₁. (40)
Similarly, we have ‖𝑇₂𝑢−𝑇₂𝑢₁‖≤𝜅₂‖𝑢−𝑢₁‖.
Thus, we have 𝑇(𝑢,𝑣)−𝑇𝑢₁,𝑣₁=𝑇₁𝑣−𝑇₁𝑣₁,𝑇₂𝑢−𝑇₂𝑢₁=max𝑇₁𝑣−𝑇₁𝑣₁,𝑇₂𝑢−𝑇₂𝑢₁≤max𝜅₁,𝜅₂max𝑣−𝑣₁,𝑢−𝑢₁=max𝜅₁,𝜅₂𝑣−𝑣₁,𝑢−𝑢₁. (41)
Since 𝜅₁<1, 𝜅₂<1, therefore, the operator 𝑇 is a contraction. Hence, by Banach’s contraction principle, the operator 𝑇 has a unique fixed point, which is the unique solution of the system (1). This completes the proof.

The second result is based on the nonlinear alternative of Leray-Schauder type (Lemma 9).

Theorem 12. Assume that 𝑓,𝑔∶[0,1]×ℝ→ℝ are continuous functions and the following conditions hold:(H2)there exist four functions 𝑝_𝑖(𝑡),𝑞_𝑖(𝑡)∈𝐿¹([0,1],ℝ⁺), 𝑖=1,2, and two nondecreasing functions 𝜑,𝜓∶ℝ⁺→ℝ⁺, such that 𝑓(𝑡,𝑥)≤𝑝₁(𝑡)𝜑(‖𝑥‖)+𝑞₁(𝑡),𝑔𝑡,𝑦≤𝑝₂(𝑡)𝜓𝑦+𝑞₂(𝑡), (42) where (𝑡,𝑥),(𝑡,𝑦)∈[0,1]×ℝ.(H3)There exists a constant 𝑀>0 such that 𝜑(𝑀)𝜔₁+𝜔₂𝜓(𝑀)𝜛<𝑀,₁+𝜛₂<𝑀, (43) where 𝜔₁∶=1+𝛼₂𝛿₂𝐼^𝛼_𝑞𝑝₁(1)+𝛾₁𝛿₁𝐼^𝛼_𝑞𝑝₁𝜂₁+𝛾₂𝛿₂𝐼^𝛼_𝑞𝑝₁𝜂₂+𝛽₂𝛿₂𝐼_𝑞^𝛼−1𝑝₁(1),𝜔₂∶=1+𝛼₂𝛿₂𝐼^𝛼_𝑞𝑞₁(1)+𝛾₁𝛿₁𝐼^𝛼_𝑞𝑞₁𝜂₁+𝛾₂𝛿₂𝐼^𝛼_𝑞𝑞₁𝜂₂+𝛽₂𝛿₂𝐼_𝑞^𝛼−1𝑞₁(1),𝜛₁∶=1+𝛼₄𝛿₄𝐼^𝛽_𝑞𝑝₂(1)+𝛾₃𝛿₃𝐼^𝛽_𝑞𝑝₂𝜂₃+𝛾₄𝛿₄𝐼^𝛽_𝑞𝑝₂𝜂₄+𝛽₄𝛿₄𝐼_𝑞^𝛽−1𝑝₂(1),𝜛₂∶=1+𝛼₄𝛿₄𝐼^𝛽_𝑞𝑞₂(1)+𝛾₃𝛿₃𝐼^𝛽_𝑞𝑞₂𝜂₃+𝛾₄𝛿₄𝐼^𝛽_𝑞𝑞₂𝜂₄+𝛽₄𝛿₄𝐼_𝑞^𝛽−1𝑞₂(1). (44)Then system (1) has at least one solution on [0,1].

Proof. Consider the operator 𝑇∶𝑋×𝑋→𝑋×𝑋 defined by (25). The proof consists of several steps. As a first step, it will be shown that 𝑇 maps bounded sets into bounded sets in 𝑋×𝑋. For a positive number 𝑟, let 𝑈={(𝑢,𝑣)∈𝑋×𝑋∶‖(𝑢,𝑣)‖≤𝑟} be bounded set in 𝑋×𝑋; then, for (𝑢,𝑣)∈𝑈, we have 𝑇₁𝑣≤sup_𝑡∈[0,1]^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛾₁Δ𝛼₂−𝛾₂𝑡−𝛼₂+𝛽₂−𝛾₂𝜂₂×^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛾₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛼₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛽₂Δ𝛼₁−𝛾₁𝑡+𝛽₁+𝛾₁𝜂₁×¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝑓((𝑠𝑑𝛼−1𝑠,𝑣))_𝑞𝑠≤sup_𝑡∈[0,1]^𝑡₀𝑡−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛾₁𝛿₁^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)×𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛿₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)×𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛼₂𝛿₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)×𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛽₂𝛿₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()×𝑝𝛼−1₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠≤𝜑(𝑟)¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝑑_𝑞𝑠+𝛾₁𝛿₁^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛿₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝑑_𝑞𝑠+𝛼₂𝛿₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝑑_𝑞𝑠+𝛽₂𝛿₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝑝𝛼−1¹(𝑠)𝑑_𝑞𝑠+¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑞₁(𝑠)𝑑_𝑞𝑠+𝛾₁𝛿₁^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑞₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛿₂^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑞₁(𝑠)𝑑_𝑞𝑠+𝛼₂𝛿₂¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑞₁(𝑠)𝑑_𝑞𝑠+𝛽₂𝛿₂¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝑞𝛼−1¹(𝑠)𝑑_𝑞𝑠. (45)
As before, it can be shown that 𝑇₁𝑣≤𝜑(𝑟)𝜔₁+𝜔₂. (46) Similarly, we have 𝑇₂𝑢≤𝜓(𝑟)𝜛₁+𝜛₂. (47) Thus, 𝑇 maps bounded sets into bounded sets in 𝑋×𝑋.
Next, we show that 𝑇 maps bounded sets into equicontinuous sets of 𝑋×𝑋. Let 𝑡₁,𝑡₂∈[0,1] with 𝑡₁<𝑡₂ and (𝑢,𝑣)∈𝑈, where 𝑈 is a bounded set of 𝑋×𝑋. Then taking into consideration the inequality (𝑡₂−𝑞𝑠)^(𝛼−1)−(𝑡₁−𝑞𝑠)^(𝛼−1)≤(𝑡₂−𝑡₁), for 0<𝑡₁<𝑡₂, we obtain 𝑇₁𝑣𝑡₂−𝑇₁𝑣𝑡₁≤^𝑡2₀𝑡₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠−^𝑡1₀𝑡₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+1Δ𝛾₁𝛼₂−𝛾₂𝑡₂−𝑡₁×^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛾₂𝛼₁−𝛾₁𝑡₂−𝑡₁×^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛼₂𝛼₁−𝛾₁𝑡₂−𝑡₁×¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑓(𝑠,𝑣(𝑠))𝑑_𝑞𝑠+𝛽₂𝛼₁−𝛾₁𝑡₂−𝑡₁×¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞()𝑓((𝑠𝑑𝛼−1𝑠,𝑣))_𝑞𝑠≤^𝑡1₀𝑡₂−𝑞𝑠^(𝛼−1)−𝑡₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)×𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+^𝑡2_𝑡1𝑡₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+1Δ𝛾₁𝛼₂−𝛾₂𝑡₂−𝑡₁×^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛼₁−𝛾₁𝑡₂−𝑡₁×^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛼₂𝛼₁−𝛾₁𝑡₂−𝑡₁×¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛽₂𝛼₁−𝛾₁𝑡₂−𝑡₁×¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(‖𝑣‖)+𝑞₁(𝑠)𝑑_𝑞𝑠≤^𝑡1₀𝑡₂−𝑡₁Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(𝑟)+𝑞₁(𝑠)𝑑_𝑞𝑠+^𝑡2_𝑡1𝑡₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(𝑟)+𝑞₁(𝑠)𝑑_𝑞𝑠+1Δ𝛾₁𝛼₂−𝛾₂𝑡₂−𝑡₁×^𝜂1₀𝜂₁−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(𝑟)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛾₂𝛼₁−𝛾₁𝑡₂−𝑡₁×^𝜂2₀𝜂₂−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(𝑟)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛼₂𝛼₁−𝛾₁𝑡₂−𝑡₁×¹₀1−𝑞𝑠^(𝛼−1)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(𝑟)+𝑞₁(𝑠)𝑑_𝑞𝑠+𝛽₂𝛼₁−𝛾₁𝑡₂−𝑡₁×¹₀1−𝑞𝑠^(𝛼−2)Γ_𝑞(𝛼)𝑝₁(𝑠)𝜑(𝑟)+𝑞₁(𝑠)𝑑_𝑞𝑠. (48) Clearly, the right-hand side of the above inequality tends to zero independently of 𝑣∈𝑈 as 𝑡₂→𝑡₁. Thus, it follows by the Arzelá-Ascoli theorem that 𝑇₁ is completely continuous. Similarly, 𝑇₂ is completely continuous. Therefore, 𝑇∶𝑋×𝑋→𝑋×𝑋 is completely continuous.
Let us set Ω={(𝑢,𝑣)∈𝑈∶‖(𝑢,𝑣)‖<𝑀}. Note that the operator 𝑇∶Ω→𝑋×𝑋 is continuous and completely continuous. From the choice of Ω, assume that there is (𝑢,𝑣)∈𝜕Ω such that (𝑢,𝑣)=𝜆𝑇(𝑢,𝑣), for some 𝜆∈(0,1). By (H3), we obtain ‖𝑇(<‖𝑇(𝑇‖(𝑢,𝑣)‖=𝜆𝑢,𝑣)‖𝑢,𝑣)‖=max₁𝑣,𝑇₂𝑢<𝑀, (49) which is a contradiction. In consequence, by the nonlinear alternative of Leray-Schauder type (Lemma 9), we deduce that 𝑇 has a fixed point (𝑢,𝑣)∈Ω which is a solution of the system (1). The proof is complete.

In the sequel we present two examples which illustrate Theorems 11 and 12.

4. Examples

Example 1. Consider the following fractional 𝑞-difference nonlocal boundary value problem: ^𝐶𝐷_𝑞^3/2𝑢(𝑡)=𝐿₁2𝑣+tan⁻¹𝑣+sin𝑡,0≤𝑡≤1,_𝐶𝐷_𝑞^7/4𝑣(𝑡)=𝐿₂3𝑢+tan⁻¹𝑢(0)−1𝑢+sin𝑡,0≤𝑡≤1,2𝐷_𝑞𝑢(0)1=𝑢3,14𝑢(1)+34𝐷_𝑞𝑢(1)2=𝑢3,𝑣(0)−12𝐷_𝑞𝑣(0)1=𝑣3,14𝑣(1)+34𝐷_𝑞𝑣(1)2=𝑣3. (50) In this case, 𝛼=3/2, 𝛽=7/4, 𝛼₁=1, 𝛽₁=1/2, 𝛼₂=1/4, 𝛽₂=3/4, 𝛼₃=1, 𝛽₃=1/2, 𝛼₄=1/4, 𝛽₄=3/4, 𝜂₁=1/3, 𝜂₂=2/3, 𝜂₃=1/3, 𝜂₄=2/3, 𝛾₁=1=𝛾₂, 𝛾₃=1=𝛾₄, and 𝐿₁,𝐿₂ are constants to be fixed later on. Moreover, Δ=5/8, ▴=5/8, 𝛿₁=26/15, 𝛿₂=4/3, 𝛿₃=26/15, and 𝛿₄=4/3. Consider 𝑓()=𝐿𝑡,𝑣¹2𝑣+tan⁻¹,𝑔()=𝐿𝑣+sin𝑡𝑡,𝑢²3𝑢+tan⁻¹,[][].𝑢+sin𝑡𝑡∈0,1,𝑢,𝑣∈𝐶0,1 (51)
Clearly, we have𝑓(𝑡,𝑣)−𝑓𝑡,𝑣₁≤𝐿₁𝑣−𝑣₁,𝑔(𝑡,𝑢)−𝑔𝑡,𝑢₁≤𝐿₂𝑢−𝑢₁,𝜅₁𝐿∶=¹Γ_1/2()2√3/21+213+10√2+15√315√32√,𝜅2−1₂𝐿∶=²Γ_1/2()2√7/41+213+10√2+15√315√32√.2−1 (52)
Choose 𝐿₁1∶=Γ_1/2()2√3/21+213+10√2+15√315√32√2−1⁻¹,𝐿₂1∶=Γ_1/2()2√7/41+213+10√2+15√315√32√2−1⁻¹. (53)
Hence all the assumptions of Theorem 11 are satisfied. Therefore, by Theorem 11, the problem (50) has a unique solution.

Example 2. Consider the following fractional boundary value problem:^𝐶𝐷_𝑞^3/2𝑢(𝑡)=14cos𝑡²|𝑣|sin2+𝑒^{−𝑥2(𝑡2+1)}1+𝑡²+1+13,0≤𝑡≤1,_𝐶𝐷_𝑞^3/2𝑣(𝑡)=1((𝑡+|𝑢(𝑡16𝜋sin2𝜋𝑢)))|2(|𝑢(𝑡+11+)|)2,𝑢(0)−10≤𝑡≤1,2𝐷_𝑞𝑢(0)1=𝑢3,14𝑢(1)+34𝐷_𝑞𝑢(1)2=𝑢3,𝑣(0)−12𝐷_𝑞𝑣(0)1=𝑣3,14𝑣(1)+34𝐷_𝑞𝑣(1)2=𝑣3. (54) In this case, 𝛼=3/2, 𝛽=3/2, 𝛼₁=1, 𝛽₁=1/2, 𝛼₂=1/4, 𝛽₂=3/4, 𝛼₃=1, 𝛽₃=1/2, 𝛼₄=1/4, 𝛽₄=3/4, 𝜂₁=1/3, 𝜂₂=2/3, 𝜂₃=1/3, 𝜂₄=2/3, 𝛾₁=1=𝛾₂, 𝛾₃=1=𝛾₄, and 𝐿₁,𝐿₂ are constants to be fixed later on. Moreover, Δ=5/8, ▴=5/8, 𝛿₁=26/15, 𝛿₂=4/3, 𝛿₃=26/15, and 𝛿₄=4/3. Clearly 𝑓(=1𝑡,𝑣)4cos𝑡²|𝑣|sin2+𝑒^{−𝑥2(𝑡2+1)}1+𝑡²+1+13≤18|𝑣|𝑔(=1+1,𝑡,𝑢)((𝑡+|𝑢(𝑡16𝜋sin2𝜋𝑢)))|2(|𝑢(𝑡+11+)|)2≤18|𝑢|+1. (55) Clearly 𝑝₁=1/8, 𝑞₁=1, 𝜑(𝑀)=𝑀, 𝑝₂=1/8, 𝑞₂=1, and 𝜓(𝑀)=𝑀. Consequently, 𝜔₁≈0.567129414, 𝜔₂≈4.536963312, 𝜛₁≈0.567129414, 𝜛₂≈4.536963312, and conditions (43) imply that 𝑀>10.48055997. Thus, all the assumptions of Theorem 12 are satisfied. Therefore, the conclusion of Theorem 12 applies to problems (54).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (11161027, 11262009, and 11226132), the Scientific Research Projects in Colleges and Universities of Gansu Province of China (2013A-043), the Fundamental Research Funds for the Gansu Universities (213061), the Fundamental Research Funds for the Gansu Universities (212084), the Youth Science Foundation of Lanzhou Jiaotong University (2012019), and the National Natural Science Foundation of China (11226132). The authors are thankful to the referees for their careful reading of the paper and insightful comments.