Open Access

Existence of Solutions to Anti-Periodic Boundary Value Problem for Nonlinear Fractional Differential Equations with Impulses

Advances in Difference Equations20112011:915689

https://doi.org/10.1155/2011/915689

Received: 20 October 2010

Accepted: 20 January 2011

Published: 8 February 2011

Abstract

This paper discusses the existence of solutions to antiperiodic boundary value problem for nonlinear impulsive fractional differential equations. By using Banach fixed point theorem, Schaefer fixed point theorem, and nonlinear alternative of Leray-Schauder type theorem, some existence results of solutions are obtained. An example is given to illustrate the main result.

1. Introduction

In this paper, we consider an antiperiodic boundary value problem for nonlinear fractional differential equations with impulses
(1.1)

where is a positive constant, , denotes the Caputo fractional derivative of order , , , and satisfy that , , , and represent the right and left limits of at .

Fractional differential equations have proved to be an excellent tool in the mathematic modeling of many systems and processes in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, electromagnetic, porous media, and so forth. In consequence, the subject of fractional differential equations is gaining much importance and attention (see [16] and the references therein).

The theory of impulsive differential equations has found its extensive applications in realistic mathematic modeling of a wide variety of practical situations and has emerged as an important area of investigation in recent years. For the general theory of impulsive differential equations, we refer the reader to [7, 8]. Recently, many authors are devoted to the study of boundary value problems for impulsive differential equations of integer order, see [912].

Very recently, there are only a few papers about the nonlinear impulsive differential equations and delayed differential equations of fractional order.

Agarwal et al. in [13] have established some sufficient conditions for the existence of solutions for a class of initial value problems for impulsive fractional differential equations involving the Caputo farctional derivative. Ahmad et al. in [14] have discussed some existence results for the two-point boundary value problem involving nonlinear impulsive hybrid differential equation of fractional order by means of contraction mapping principle and Krasnoselskii's fixed point theorem. By the similar way, they have also obtained the existence results for integral boundary value problem of nonlinear impulsive fractional differential equations (see [15]). Tian et al. in [16] have obtained some existence results for the three-point impulsive boundary value problem involving fractional differential equations by the means of fixed points method. Maraaba et al. in [17, 18] have established the existence and uniqueness theorem for the delay differential equations with Caputo fractional derivatives. Wang et al. in [19] have studied the existence and uniqueness of the mild solution for a class of impulsive fractional differential equations with time-varying generating operators and nonlocal conditions.

To the best of our knowledge, few papers exist in the literature devoted to the antiperiodic boundary value problem for fractional differential equations with impulses. This paper studies the existence of solutions of antiperiodic boundary value problem for fractional differential equations with impulses.

The organization of this paper is as follows. In Section 2, we recall some definitions of fractional integral and derivative and preliminary results which will be used in this paper. In Section 3, we will consider the existence results for problem (1.1). We give three results, the first one is based on Banach fixed theorem, the second one is based on Schaefer fixed point theorem, and the third one is based on the nonlinear alternative of Leray-Schauder type. In Section 4, we will give an example to illustrate the main result.

2. Preliminaries

In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper.

Definition 2.1 (see [4]).

The Caputo fractional derivative of order of a function is defined as
(2.1)

where denotes the integer part of the real number .

Definition 2.2 (see [4]).

The Riemann-Liouville fractional integral of order of a function , , is defined as
(2.2)

provided that the right side is pointwise defined on .

Definition 2.3 (see [4]).

The Riemann-Liouville fractional derivative of order of a continuous function is given by
(2.3)

where and denotes the integer part of real number , provided that the right side is pointwise defined on .

For the sake of convenience, we introduce the following notation.

Let . . We define and exists, and . Obviously, is a Banach space with the norm .

Definition 2.4.

A function is said to be a solution of (1.1) if satisfies the equation for , the equations , , and the condition .

Lemma 2.5 (see [20]).

Let ; then
(2.4)

for some , .

Lemma 2.6 (nonlinear alternative of Leray-Schauder type [21]).

Let be a Banach space with closed and convex. Assume that is a relatively open subset of with and is continuous, compact map. Then either

  1. (1)

    has a fixed point in , or

     
  2. (2)

    there exists and with .

     

Lemma 2.7 (Schaefer fixed point theorem [22]).

Let be a convex subset of a normed linear space and . Let be a completely continuous operator, and let
(2.5)

Then either is unbounded or has a fixed point.

Lemma 2.8.

Assume that . A function is a solution of the antiperiodic boundary value problem
(2.6)
if and only if is a solution of the integral equation
(2.7)

Proof.

Assume that satisfies (2.6). Using Lemma 2.5, for some constants , we have
(2.8)
Then, we obtain
(2.9)
If , then we have
(2.10)
where are arbitrary constants. Thus, we find that
(2.11)
In view of and , we have
(2.12)
Hence, we obtain
(2.13)
Repeating the process in this way, the solution for can be written as
(2.14)
On the other hand, by (2.14), we have
(2.15)
By the boundary conditions , we obtain
(2.16)

Substituting the values of and into (2.8), (2.14), respectively, we obtain (2.7).

Conversely, we assume that is a solution of the integral equation (2.7). By a direct computation, it follows that the solution given by (2.7) satisfies (2.6). The proof is completed.

3. Main Result

In this section, our aim is to discuss the existence and uniqueness of solutions to the problem (1.1).

Theorem 3.1.

Assume that

there exists a constant such that , for each and all ;

there exist constants such that , , for each and all .

If
(3.1)

then problem (1.1) has a unique solution on .

Proof.

We transform the problem (1.1) into a fixed point problem. Define an operator by
(3.2)
where is with the norm . Let ; then for each , we have
(3.3)
Therefore,
(3.4)
Since
(3.5)

consequently is a contraction; as a consequence of Banach fixed point theorem, we deduce that has a fixed point which is a solution of the problem (1.1).

Theorem 3.2.

Assume that

the function is continuous and there exists a constant such that for each and all ;

the functions are continuous and there exist constants such that , , for all , .

Then the problem (1.1) has at least one solution on .

Proof.

We will use Schaefer fixed-point theorem to prove has a fixed point. The proof will be given in several steps.

Step 1.

is continuous.

Let be a sequence such that in ; we have
(3.6)
Since are continuous functions, then we have
(3.7)

Step 2.

maps bounded sets into bounded sets in .

Indeed, it is enough to show that for any , there exists a positive constant such that, for each , we have . By and , for each , we can obtain
(3.8)
Therefore,
(3.9)

Step 3.

maps bounded sets into equicontinuous sets in .

Let be a bounded set of as in Step 2, and let . For each , we can estimate the derivative :
(3.10)
Hence, let ; we have
(3.11)

So is equicontinuous in . As a consequence of Steps 1 to 3 together with the Arzela-Ascoli theorem, we can conclude that is completely continuous.

Step 4.

A priori bounds.

Now it remains to show that the set
(3.12)
is bounded. Let for some . Thus, for each , we have
(3.13)
For each , by and , we have
(3.14)

This shows that the set is bounded. As a consequence of Schaefer fixed-point theorem, we deduce that has a fixed point which is a solution of the problem (1.1).

In the following theorem we give an existence result for the problem (1.1) by applying the nonlinear alternative of Leray-Schauder type and by which the conditions and are weakened.

Theorem 3.3.

Assume that and the following conditions hold.

There exists and continuous and nondecreasing such that

(3.15)

There exist continuous and nondecreasing such that

(3.16)

There exists a number such that

(3.17)

where .

Then (1.1) has at least one solution on .

Proof.

Consider the operator defined in Theorem 3.1. It can be easily shown that is continuous and completely continuous. For and each , let . Then from and , and we have
(3.18)
Thus,
(3.19)
Then by , there exists such that . Let
(3.20)

The operator is a continuous and completely continuous. From the choice of , there is no such that for some . As a consequence of the nonlinear alternative of Leray-Schauder type, we deduce that has a fixed point in which is a solution of the problem (1.1). This completes the proof.

4. Example

Let , , . We consider the following boundary value problem:
(4.1)
where
(4.2)
Obviously . Further,
(4.3)

Thus, all the assumptions of Theorem 3.1 are satisfied. Hence, by the conclusion of Theorem 3.1, the impulsive fractional antiperiodic boundary value problem has a unique solution on .

Declarations

Acknowledgments

This work was supported by the Natural Science Foundation of China (10971173), the Natural Science Foundation of Hunan Province (10JJ3096), the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Construct Program of the Key Discipline in Hunan Province.

Authors’ Affiliations

(1)
Department of Mathematics, Xiangnan University
(2)
School of Mathematics and Computational Science, Xiangtan University

References

  1. Bai Z, Lü H: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005,311(2):495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar
  2. Kosmatov N: A singular boundary value problem for nonlinear differential equations of fractional order. Journal of Applied Mathematics and Computing 2009,29(1-2):125-135. 10.1007/s12190-008-0104-xMathSciNetView ArticleMATHGoogle Scholar
  3. Li CF, Luo XN, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Computers & Mathematics with Applications 2010,59(3):1363-1375.MathSciNetView ArticleMATHGoogle Scholar
  4. Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.MATHGoogle Scholar
  5. Tian Y, Chen A: The existence of positive solution to three-point singular boundary value problem of fractional differential equation. Abstract and Applied Analysis 2009, 2009:-18.Google Scholar
  6. Zhang SQ: Positive solutions for boundary value problems of nonlinear fractional differential equtions. Electronic Journal of Differential Equations 2006, 2006: 1-12.Google Scholar
  7. Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar
  8. Samoĭlenko AM, Perestyuk NA: Impulsive Differential Equations, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises. Volume 14. World Scientific, River Edge, NJ, USA; 1995:x+462.Google Scholar
  9. Lin X, Jiang D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. Journal of Mathematical Analysis and Applications 2006,321(2):501-514. 10.1016/j.jmaa.2005.07.076MathSciNetView ArticleMATHGoogle Scholar
  10. Nieto JJ, Rodríguez-López R: Boundary value problems for a class of impulsive functional equations. Computers & Mathematics with Applications 2008,55(12):2715-2731. 10.1016/j.camwa.2007.10.019MathSciNetView ArticleMATHGoogle Scholar
  11. Shen J, Wang W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2008,69(11):4055-4062. 10.1016/j.na.2007.10.036MathSciNetView ArticleMATHGoogle Scholar
  12. Xiao J, Nieto JJ, Luo Z: Multiple positive solutions of the singular boundary value problem for second-order impulsive differential equations on the half-line. Boundary Value Problems 2010, 2010:-13.Google Scholar
  13. Agarwal RP, Benchohra M, Slimani BA: Existence results for differential equations with fractional order and impulses. Memoirs on Differential Equations and Mathematical Physics 2008, 44: 1-21.MathSciNetView ArticleMATHGoogle Scholar
  14. Ahmad B, Sivasundaram S: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Analysis: Hybrid Systems 2009,3(3):251-258. 10.1016/j.nahs.2009.01.008MathSciNetMATHGoogle Scholar
  15. Ahmad B, Sivasundaram S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis: Hybrid Systems 2010,4(1):134-141. 10.1016/j.nahs.2009.09.002MathSciNetMATHGoogle Scholar
  16. Tian Y, Bai Z: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Computers & Mathematics with Applications 2010,59(8):2601-2609. 10.1016/j.camwa.2010.01.028MathSciNetView ArticleMATHGoogle Scholar
  17. Maraaba T, Baleanu D, Jarad F: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. Journal of Mathematical Physics 2008,49(8):-11.Google Scholar
  18. Maraaba TA, Jarad F, Baleanu D: On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Science in China. Series A 2008,51(10):1775-1786. 10.1007/s11425-008-0068-1MathSciNetView ArticleMATHGoogle Scholar
  19. Wang JR, Yang YL, Wei W: Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces. Opuscula Mathematica 2010,30(3):361-381.MathSciNetView ArticleMATHGoogle Scholar
  20. Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies. Volume 204. Elsevier Science B.V., Amsterdam, The Netherlands; 2006:xvi+523.Google Scholar
  21. Granas A, Guenther RB, Lee JW: Some general existence principles in the Carathéodory theory of nonlinear differential systems. Journal de Mathématiques Pures et Appliquées 1991,70(2):153-196.MathSciNetMATHGoogle Scholar
  22. Schaefer H: Über die Methode der a priori-Schranken. Mathematische Annalen 1955, 129: 415-416. 10.1007/BF01362380MathSciNetView ArticleMATHGoogle Scholar

Copyright

© A. Chen and Y. Chen 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.