Monotone iterative method for nonlinear fractional q-difference equations with integral boundary conditions
- Yanfeng Li^{1} and
- Wengui Yang^{1}Email author
https://doi.org/10.1186/s13662-015-0630-4
© Li and Yang 2015
Received: 29 April 2015
Accepted: 1 September 2015
Published: 17 September 2015
Abstract
This paper investigates the existence of positive solutions for a class of nonlinear fractional q-difference equations with integral boundary conditions. By applying monotone iterative method and some inequalities associated with the Green’s function, the existence results of positive solutions and two iterative schemes approximating the solutions are established. An explicit example is given to illustrate the main result.
Keywords
MSC
1 Introduction
The monotone iterative method is an interesting and effective technique for investigating the existence of solutions/positive solutions for nonlinear boundary value problems. This method has been paid more and more attention due to the advantage that the first term of the iterative sequences may be taken to be a constant function or a simple function; see [1–8] and the references therein. For instance, by means of the monotone iterative technique and the method of lower and upper solutions, Xu and Liu [9] studied the maximal and minimal solutions for a coupled system of fractional differential-integral equations with two-point boundary conditions. In [10], by means of monotone iterative technique, Zhang et al. investigated the existence and uniqueness of the positive solution for a fractional differential equation with derivatives. By applying the monotone iteration method, Zhang et al. [11] obtained the positive extremal solutions and iterative schemes for approximating the solution of fractional differential equations with nonlinear terms depending on the lower-order derivatives on a half-line.
Since 2010, fractional q-difference equations have gained considerable popularity and importance due to the fact that they can describe the natural phenomena and the mathematical model more accurately. For some recent contributions on the topic, see [12–18] and the references incited therein. For example, under different conditions, Graef and Kong [19, 20] investigated the existence of positive solutions for boundary value problems with fractional q-derivatives in terms of different ranges of λ, respectively. By applying the nonlinear alternative of Leray-Schauder type and Krasnoselskii fixed point theorems, the author [21] established sufficient conditions for the existence of positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions. By applying some standard fixed point theorems, Agarwal et al. [22] and Ahmad et al. [23] showed some existence results for sequential q-fractional integrodifferential equations with q-antiperiodic boundary conditions and nonlocal four-point boundary conditions, respectively. In [24], relying on the contraction mapping principle and a fixed point theorem due to O’Regan, Ahmad et al. were concerned with new boundary value problems of nonlinear q-fractional differential equations with nonlocal and sub-strip type boundary conditions. In [25], Yang et al. obtained the existence and uniqueness of positive solutions for a class of nonlinear q-fractional boundary value problems and established the iterative schemes for approximating the solutions.
Motivated by the results mentioned above and the effectiveness and feasibility of monotone iterative method, we consider the existence of positive solutions for fractional q-difference boundary value problem (1.1). In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. The main theorems are formulated and proved in Section 3. At last, an explicit example is given to illustrate the main result in Section 4.
2 Preliminaries
For the convenience of the reader, we present some necessary definitions and lemmas of fractional q-calculus theory. These details can be found in the recent literature; see [26] and references therein.
Definition 2.1
([26])
More generally, if \(\alpha\in\mathbb{R}\), then \((1-q)^{(\alpha)}=\prod_{n=0}^{\infty}((1-q^{n+1})/(1-q^{1+\alpha +n}) )\).
Definition 2.2
([26])
Lemma 2.3
([26])
- (1)
\((I_{q}^{\beta}I_{q}^{\alpha}f)(x)=I_{q}^{\alpha+\beta}f(x)\),
- (2)
\((D_{q}^{\alpha}I_{q}^{\alpha}f)(x)=f(x)\).
Lemma 2.4
([12])
- (H1)
\(g:[0,1]\rightarrow[0,\infty)\) is continuous and \(\sigma=\mu \int_{0}^{1}s^{\alpha-1}g(s)\, d_{q}s<1\), \(\theta=\mu\int_{0}^{1}s^{\alpha}g(s)\, d_{q}s\).
- (H2)
\(h:[0,1]\rightarrow[0,\infty)\) is continuous and \(0<\int_{0}^{1}(1-qs)^{(\alpha-1)} h(s)\, d_{q}s<\infty\).
Now we derive the corresponding Green’s function for boundary value problem (1.1), and obtain some properties of the Green’s function.
Lemma 2.5
Proof
Lemma 2.6
([21])
Lemma 2.7
Proof
3 Main results
Lemma 3.1
Assume that (H1) and (H2) hold. \(\mathscr{T}\) is a completely continuous operator and \(\mathscr{T}(\mathscr {K})\subseteq\mathscr{K}\).
Proof
In view of (2.12) we conclude that \(\mathscr {T}(\mathscr{K})\subseteq\mathscr{K}\). Applying the Arzela-Ascoli theorem and standard arguments, we conclude that \(\mathscr{T}\) is a completely continuous operator. The proof is completed. □
Theorem 3.2
Proof
We will divide our proof into four steps.
Step 2. The iterative sequence \(\{v_{k}\}\) is increasing, and there exists \(v^{\ast}\in\mathscr{K}_{a}\) such that \(\lim_{k\rightarrow\infty}\|v_{k}-v^{\ast}\|=0\), and \(v^{\ast}\) is a positive solution of problem (1.1).
Step 3. The iterative sequence \(\{w_{k}\}\) is decreasing, and there exists \(w^{\ast}\in\mathscr{K}_{a}\) such that \(\lim_{k\rightarrow\infty}\|w_{k}-w^{\ast}\|=0\), and \(w^{\ast}\) is a positive solution of problem (1.1).
Corollary 3.3
Remark 3.4
The iterative schemes in Theorem 3.2 start off with the zero function and a known simple function which is helpful for computational purpose, respectively.
Remark 3.5
Of course, \(w^{\ast}=v^{\ast}\) may happen and then problem (1.1) has only one solution in \(\mathscr{K}_{a}\). For example, in the case the Lipschitz condition is satisfied by the functions involved, the solutions \(v^{\ast}\) and \(w^{\ast}\) coincide, and then problem (1.1) will have a unique solution in \(\mathscr{K}_{a}\).
4 An example
Example 4.1
Declarations
Acknowledgements
The authors sincerely thank the editor and reviewers for their valuable suggestions and useful comments to improve the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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