The Iterative Positive Solution for a System of Fractional q-Difference Equations with Four-Point Boundary Conditions

Abstract

In this work, we investigate the following system of fractional q-difference equations with four-point boundary problems: where and are the fractional Riemann–Liouville q-derivative of order α and β, respectively, , , , , and . By virtue of monotone iterative approach, the iterative positive solutions are obtained. An example to illustrate our result is given.

1. Introduction

In [1, 2], Jackson studied the q-difference calculus firstly; since then, many authors have investigated this subject duo to applications of the q-difference calculus in quantum mechanics, particle physics, hypergeometric series, and complex analysis [3, 4]. The extension of q-difference calculus is the fractional q-difference calculus, which was originally investigated by Al-Salam [5] and Agarwal [6]. In the past decade, in many works concerning nonlinear fractional q-difference boundary value problem, the results of the existence and the uniqueness of solutions have been given. In [7], Ferreira considered the existence of positive solutions to the nonlinear fractional q-difference equation:

In [8], Ferreira studied the existence of positive solutions to the nonlinear fractional q-difference equation:

By using a fixed-point theorem in partially ordered sets, Garzi and Agheli [9] studied the existence and uniqueness of a positive and nondecreasing solution to the fractional q-difference equation:where and .

In [10], Guo and Kang obtained the existence and uniqueness of a positive solution for the fractional q-difference equation of the formby virtue of fixed-point theorems for the mixed monotone operator. Here, and .

Recently, by using the monotone iterative approach, in [11], Wang investigated the iterative positive solutions of the following fractional q-difference equations with three-point boundary conditions:

It should be noted that the existence of positive solutions of problem (5) had been studied by Li et al. [12] by means of a fixed-point theorem in cones. The novel idea of [11] is to find the positive solution.

Motivated by the above mentioned works, in this paper, we consider the following system of fractional q-difference equations with four-point boundary conditions:where and are the fractional Riemann–Liouville q-derivative of order α and β, respectively, , , , , and .

By using the monotone iterative approach, in this paper, we will construct two convergent monotone iterative schemes for seeking one coupled positive solution and obtain the coupled positive solution of problem (6). To the best of our knowledge, there is no paper to study the iterative coupled positive solutions for the coupled system of fractional q-difference boundary value problems. It is noted that we may investigate the approximate solutions of problem (6) by numerical approximation algorithms, which will be presented as another paper. For the latest development of numerical approximation algorithms of some boundary value problems, see [13–17] and the references therein.

2. Preliminaries

Let , the q-derivative of a function f is defined byand q-derivatives of higher order by

The q-integral of a function f defined in the interval is given by

Similar to the derivatives, the operator is given by

Define

The q-analogue of the power function with is

Moreover, if , then

Remark 1. If , then . If and , then .
The q-gamma function [18] is defined byand satisfies .

Definition 1. We say is a solution of system (6), if satisfies the first and second equations of (6) and boundary conditions of (6).

Definition 2. (see [19]). Let and f be a function defined on . The fractional q-integral of the Riemann–Liouville type is

Definition 3. (see [19]). The fractional q-derivative of the Riemann–Liouville type is defined bywhere n is the smallest integer greater than or equal to α.

Lemma 1 (see [19]). Let and f be a function defined on . Then, the following formulas hold:(1)(2)

Lemma 2 (see [13]). Let and n be a positive integer. Then, the following equality holds:

By Lemmas 1 and 2, Guo and Kang in [10] obtained the following lemma.

Lemma 3. For any , the boundary value problemhas a unique solution:whereis the Green function of BVP (18).

Similarly, we have the following.

Lemma 4. For any , the boundary value problemhas a unique solution:whereis the Green function of BVP (21).

Lemma 5. (see [10]). For and defined as in Lemmas 3 and 4, respectively, we have(i) and are two continuous functions(ii) where (iii) where

3. Main Result

In this paper, we will employ the Banach space , equipped with norm for each . Define two cones and in as follows:

Now, we define the operators by

From Lemmas 3 and 4, BVP (6) can be transformed into the following system of integral equations:

By (26), we know that is a solution of (6) if and only if and .

In order to facilitate our investigation, we make the following assumptions: (H1) is nondecreasing with respect to , and there exists a positive constant , such that (H2) is nondecreasing with respect to u, and there exists a positive constant , such that (H3)  (H4)

Remark 2. The conditions (H1) and (H2) imply that, for , we have and .

Theorem 1. Assume that conditions (H1)–(H4) hold and there exist two positive constants and such thatthen the fractional q-difference system (6) has one positive solution , where and . Moreover, for each , there exist constants , such thatwhich can be obtained by monotone iterative schemes and generated byi.e., and as .

Proof. For any , we know that there exist two constants and with such thatFrom Lemma 5 and condition (H1), we obtainwhere are two positive constants satisfyingThus, maps into . For each , there exist two constants and with such thatSimilarly, by Lemma 5 and condition (H2), we can get thatwhere are two positive constants satisfyingwhich implies that maps into . On the other hand, the proof of completely continuous and are as the same as in [12], and we omit it here.
Let . In the following, we will prove and . In fact, for any and , by conditions (29) and (30), we obtainwhich implies that and . So, and .
Taking and , then , , , and . Thus, there exist constants such thatLet and be two positive numbers satisfying , , andSetObviously, by and , and . By (H1) and (H2), we haveFrom conditions (H1) and (H2), we know that and are two nondecreasing operators. Thus, by induction, we can obtainBy the compactness of the operators and , we have that and are two sequentially compact sets. Therefore, there exist and , such that converges to and converges to as , respectively. Since the operators and are continuous, and and we obtain and as , which implies that system (6) has a positive solution , and , , and , where are constants and , which can be achieved by the monotone scheme:with initial values and defined as in (42).
In the following, we give an example to illustrate the existence of positive solutions of BVP (6).

Example 1. Consider the following system of fractional q-difference with boundary conditions:where , , , , , , , andObviously, and are nondecreasing with respect to and u, respectively, andChoosing and , we haveSo, conditions (H1) and (H2) hold. Moreover, we can show thatwhich implies that (H3) and (H4) hold. Moreover, we know that there exist two positive constants and such that (29) and (30) hold, respectively. Thus, it follows from Theorem 1 that boundary value problem of fractional q-difference system (47) has one iterative positive solution which can be obtained with the aid of monotone iterative sequences.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of China (11571136).