- Research
- Open Access
Multi-point boundary value problems for a coupled system of nonlinear fractional differential equations
- Chengbo Zhai^{1}Email author and
- Mengru Hao^{1}
https://doi.org/10.1186/s13662-015-0487-6
© Zhai and Hao; licensee Springer. 2015
- Received: 25 January 2015
- Accepted: 27 April 2015
- Published: 8 May 2015
Abstract
Keywords
- existence and uniqueness
- solutions
- Riemann-Liouville fractional derivative
- multi-point boundary value problems
- Banach contraction principle
- Krasnoselskii’s fixed point theorem
MSC
- 26A33
- 34B18
- 34B27
1 Introduction
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary non-integer order. The first definition of fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus, was mentioned already in 1695 by Leibniz and L’Hospital. With the help of fractional calculus, we can describe natural phenomena and mathematical models more accurately. The fractional differential equations play an important role in various fields of engineering, physics, economics and biological sciences, etc. (see [1–4] for example). In consequence, the subject of fractional differential equations is gaining much importance and attention. For more details on basic theory of fractional differential equations, one can see the monographs of Diethelm [1], Kilbas et al. [2], Miller and Ross [3], Podlubny [4] and Tarasov [5], and the papers [6–13] and the references therein.
As is known to all, the initial and boundary value problems for nonlinear fractional differential equations arise from the study of models of control, porous media, electrochemistry, viscoelasticity, electromagnetics, etc. Recently, the existence and uniqueness of solutions of initial and boundary value problems for nonlinear fractional equations have been extensively studied (see [14–20]), and some are coupled systems of nonlinear fractional differential equations (see [8, 14, 17, 21, 22]).
Motivated by the above-mentioned works and recent works on coupled systems of fractional differential equations, we consider the existence and uniqueness of solutions of coupled system (1)-(4) by means of the Banach contraction principle and Krasnoselskii’s fixed point theorem. In our paper, we do not suppose that \(a_{1}\), \(a_{2}\), f, g are nonnegative.
With this context in mind, the outline of this paper is as follows. In Section 2 we recall certain results from the theory of continuous fractional calculus. In Section 3 we provide some conditions under which problem (1)-(4) will have a unique solution or at least one solution.
2 Preliminaries
For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems.
Definition 2.1
(see [4])
Lemma 2.2
(see [23])
Let \(\alpha\in\mathbf{R}\). Then \(D^{n}D^{\alpha}_{a^{+}}y(t)=D^{n+\alpha}_{a^{+}}y(t)\), for each \(n\in N_{0}\), where \(y(t)\) is assumed to be sufficiently regular so that both sides of the equality are well defined. Moreover, if \(\beta\in (-\infty,0]\) and \(\gamma\in[0,+\infty)\), then \(D^{\gamma}_{a^{+}}D^{\beta }_{a^{+}}y(t)=D^{\gamma+\beta}_{a^{+}}y(t)\).
Lemma 2.3
(see [23])
The general solution to \(D^{\nu }_{a^{+}}y(t)=0\), where \(n-1<\nu\leq n\) and \(\nu> 0\), is the function \(y(t)=c_{1}t^{\nu-1}+c_{2}t^{\nu-2}+\cdots+c_{n}t^{\nu- n}\), where \(c_{i}\in\mathbf{R}\) for each i.
Lemma 2.4
Proof
3 Main results
This section deals with the existence and uniqueness of solutions to problem (1)-(4).
To establish the main results, we need the following assumptions:
(H_{1}) \(f,g:[0,1]\times\mathbf{R}\times\mathbf{ R}\rightarrow \mathbf{R}\) and \(a_{1},a_{2}:[0,1] \rightarrow\mathbf{R}\);
(H_{2}) f, g, \(a_{1}\), \(a_{2}\) are continuous;
We are ready to state the existence and uniqueness result.
Theorem 3.1
Suppose that conditions (H_{1})-(H_{4}) are satisfied. Then the boundary value problem (1)-(4) has a unique solution.
Proof
Our next result is based on the following well-known fixed point theorem due to Krasnoselskii.
Lemma 3.2
(Krasnoselskii [25])
- (i)
\(Tx+Sy\in K\) whenever \(x,y\in K\);
- (ii)
T is compact and continuous;
- (iii)
S is a contraction mapping.
Now we are ready to state and prove the following existence result.
Theorem 3.3
Suppose that conditions (H_{1})-(H_{3}), (H_{5}), (H_{6}) are satisfied. Then there exists at least one solution of the boundary value problem (1)-(4).
Proof
Thus all the assumptions of Lemma 3.2 are satisfied, so \(S(y_{1},y_{2})=A(y_{1},y_{2})+B(y_{1},y_{2})\) has at least one fixed point. Hence, we obtain that (1)-(4) has at least one solution. □
4 Conclusions
There are few works that deal with multi-point boundary value problems for a coupled system of nonlinear fractional differential equations. In this article, we study multi-point boundary value problems for a coupled system of nonlinear fractional differential equations (1)-(4). By using Green’s function, the Banach contraction principle and Krasnoselskii’s fixed point theorem, we establish some new existence, uniqueness theorems of solutions for multi-point boundary value problems for a coupled system of nonlinear fractional differential equations (1)-(4).
Declarations
Acknowledgements
The research was supported by the Youth Science Foundation of China (11201272) and the Science Foundation of Shanxi Province (2010021002-1).
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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