- Research Article
- Open Access
Existence of Solutions to Anti-Periodic Boundary Value Problem for Nonlinear Fractional Differential Equations with Impulses
Advances in Difference Equations volume 2011, Article number: 915689 (2011)
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.
In this paper, we consider an antiperiodic boundary value problem for nonlinear fractional differential equations with impulses
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 [1–6] 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 [9–12].
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  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  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 ). Tian et al. in  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  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.
In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper.
Definition 2.1 (see ).
The Caputo fractional derivative of order of a function is defined as
where denotes the integer part of the real number .
Definition 2.2 (see ).
The Riemann-Liouville fractional integral of order of a function , , is defined as
provided that the right side is pointwise defined on .
Definition 2.3 (see ).
The Riemann-Liouville fractional derivative of order of a continuous function is given by
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 .
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 ).
Let ; then
for some , .
Lemma 2.6 (nonlinear alternative of Leray-Schauder type ).
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
has a fixed point in , or
there exists and with .
Lemma 2.7 (Schaefer fixed point theorem ).
Let be a convex subset of a normed linear space and . Let be a completely continuous operator, and let
Then either is unbounded or has a fixed point.
Assume that . A function is a solution of the antiperiodic boundary value problem
if and only if is a solution of the integral equation
Assume that satisfies (2.6). Using Lemma 2.5, for some constants , we have
Then, we obtain
If , then we have
where are arbitrary constants. Thus, we find that
In view of and , we have
Hence, we obtain
Repeating the process in this way, the solution for can be written as
On the other hand, by (2.14), we have
By the boundary conditions , we obtain
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).
there exists a constant such that , for each and all ;
there exist constants such that , , for each and all .
then problem (1.1) has a unique solution on .
We transform the problem (1.1) into a fixed point problem. Define an operator by
where is with the norm . Let ; then for each , we have
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).
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 .
We will use Schaefer fixed-point theorem to prove has a fixed point. The proof will be given in several steps.
Let be a sequence such that in ; we have
Since are continuous functions, then we have
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
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 :
Hence, let ; we have
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.
A priori bounds.
Now it remains to show that the set
is bounded. Let for some . Thus, for each , we have
For each , by and , we have
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.
Assume that and the following conditions hold.
There exists and continuous and nondecreasing such that
There exist continuous and nondecreasing such that
There exists a number such that
Then (1.1) has at least one solution on .
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
Then by , there exists such that . Let
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.
Let , , . We consider the following boundary value problem:
Obviously . Further,
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 .
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.2005.02.052
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-x
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.
Podlubny I: Fractional Differential Equations, Mathematics in Science and Engineering. Volume 198. Academic Press, San Diego, Calif, USA; 1999:xxiv+340.
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.
Zhang SQ: Positive solutions for boundary value problems of nonlinear fractional differential equtions. Electronic Journal of Differential Equations 2006, 2006: 1-12.
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.
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.
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.2005.07.076
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.2007.10.019
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.2007.10.036
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.
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.
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.2009.01.008
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.2009.09.002
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.2010.01.028
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.
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/s11425-008-0068-1
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.
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.
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.
Schaefer H: Über die Methode der a priori-Schranken. Mathematische Annalen 1955, 129: 415-416. 10.1007/BF01362380
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.