Impulsive boundary value problem for a fractional differential equation
- Shuai Yang1Email author and
- Shuqin Zhang2
Received: 14 August 2016
Accepted: 8 November 2016
Published: 17 November 2016
Abstract
In this paper, we discuss a boundary value problem for an impulsive fractional differential equation. By transforming the boundary value problem into an equivalent integral equation, and employing the Banach fixed point theorem and the Schauder fixed point theorem, existence results for the solutions are obtained. For application, we provide some examples to illustrate our main results.
Keywords
MSC
1 Introduction
Fractional differential equations have attracted great attention from many researchers because fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various processes in science and engineering, such as physics, control theory, electrochemistry, biology, viscoelasticity, signal processing, nuclear dynamics, etc. For details, see [1–4] and the references therein. Another important class of differential equations is known as impulsive differential equations. The interest in the study of them is that the impulsive differential system can model the processes which are subject to abrupt changes in their states, refer to [5–9]. Recently, boundary value problems for impulsive fractional differential equations have been attractive to many researchers; see [10–12].
The rest of the paper is organized as follows. In Section 2, we will give some notations, recall some definitions, and introduce some lemmas which are essential to prove our main results. In Section 3, the main results are given, and some examples are presented to demonstrate our main results.
2 Preliminaries
In this section, we introduce notations, definitions, lemmas, and preliminary facts that will be used in the rest of this paper.
Definition 1
[1]
Definition 2
[1]
Lemma 1
[2]
Lemma 2
[2]
We have the following auxiliary lemmas which are useful in the following.
Lemma 3
Proof
Conversely, we assume that \(x(t)\) is a solution of (4). If \(t\in J_{0}\), then, using the fact that \({}^{c}D_{0^{+}}^{\alpha}\) is the left inverse of \(I_{0+}^{\alpha}\), we get \({}^{c}D_{0^{+}}^{\alpha}x(t)=f(t)\). If \(t\in J_{k},k=0,1,\ldots,m\), due to the fact that the Caputo fractional derivative of a constant is equal to zero, we can verify easily that \(x(t)\) satisfies (3), therefore, \(x(t)\) is a solution of (3). The lemma is proved. □
3 Main results
Our first result is based on the Banach fixed point theorem. Before stating and proving the main result, we introduce the following hypotheses.
(H2) \(I_{k},\overline{I}_{k},g\), and h are continuous functions and satisfy the Lipschitz conditions with Lipschitz constants \(L_{2},L_{3},L_{4},L_{5}>0\).
Theorem 1
Proof
The second result is based on the Schauder fixed point theorem. We introduce the following assumptions.
(H4) \(|f(x,t)|\leq l_{1}|x|^{\rho}\), and \(|I_{k}(x)|\leq l_{2}|x|^{\mu }\), \(|\overline{I}_{k}(x)|\leq l_{3}|x|^{\nu}\), \(|h(x)|\leq l_{4}|x|^{\theta}\), \(|g(x)|\leq l_{5}|x|^{\gamma}\), for any \(x\in \mathbb{R},k=1,2,\ldots,m\), where \(l_{i}>0,i=1,2,\ldots,5\), and \(\rho ,\mu,\nu,\theta,\gamma>1\).
Theorem 2
Assume that (H3) or (H4) is satisfied, then the impulsive boundary value problem (1) has at least one solution.
Proof
We will use the Schauder fixed point theorem to prove this result. The proof will be given in several steps.
Step 1. T is a continuous operator.
It is very easy to find that T is continuous since \(f,g,h,I_{k},\overline{I}_{k},k=0,1,\ldots,m\), are continuous functions. We omit the details.
Step 2. T maps bounded sets into uniformly bounded sets in \(PC(J,\mathbb{R})\).
Step 3. T maps bounded sets into equicontinuous sets of \(PC(J,\mathbb{R})\).
As a consequence of Step 1 to 3 together with the Ascoli-Arzela theorem we assert that T is a completely continuous operator. From the Schauder fixed point theorem one deduces that T has at least one fixed point which is a solution of the impulsive boundary value problem (1) and the theorem is proved. □
Remark 1
Remark 2
We state that the coupled system likes (2) can be investigated in the same way as the whole arguments we made above. We omit the details.
In the end of this section, we will give some examples to illustrate our results.
Example 1
Example 2
Example 3
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11371364). The author is very grateful to the anonymous referee for his or her valuable suggestions.
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
References
- Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) View ArticleMATHGoogle Scholar
- Diethelm, K: The Analysis of Fractional Differential Equations. Springer, Berlin (2010) View ArticleMATHGoogle Scholar
- Zhou, Y: Basic Theory of Fractional Differential Equations. World Scientific, London (2014) View ArticleMATHGoogle Scholar
- Ray, SS: Fractional Calculus with Applications for Nuclear Reactor Dynamics. CRC Press, Boca Raton (2015) View ArticleGoogle Scholar
- Bainov, DD, Simeonov, PS: System with Impulsive Effect. Ellis Horwood, Chichester (1989) MATHGoogle Scholar
- Lakshmikantham, V, Bainov, DD, Simeonov, PS: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) View ArticleMATHGoogle Scholar
- Samoilenko, AM, Perestyuk, NA: Impulsive Differential Equations. World Scientific, Singapore (1995) View ArticleMATHGoogle Scholar
- Heidarkhani, S, Salari, A: Nontrivial solutions for impulsive fractional differential systems through variational methods. Comput. Math. Appl. (2016, in press) Google Scholar
- Heidarkhani, S, Salari, A, Caristi, G: Infinitely many solutions for impulsive nonlinear fractional boundary value problems. Adv. Differ. Equ. 2016, 196 (2016) MathSciNetView ArticleGoogle Scholar
- Tian, Y, Bai, Z: Impulsive boundary value problem for differential equations with fractional order. Differ. Equ. Dyn. Syst. 21(3), 253-260 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Shah, K, Khalil, H, Khan, RA: Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos Solitons Fractals 77, 240-246 (2015) MathSciNetView ArticleGoogle Scholar
- Guo, TL, Jiang, W: Impulsive problems for fractional differential equations with boundary value conditions. Comput. Math. Appl. 64, 3281-3291 (2012) MathSciNetView ArticleMATHGoogle Scholar