- Research Article
- Open Access
Existence of Solutions to Anti-Periodic Boundary Value Problem for Nonlinear Fractional Differential Equations with Impulses
© A. Chen and Y. Chen 2011
- Received: 20 October 2010
- Accepted: 20 January 2011
- Published: 8 February 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.
- Fractional Order
- Fixed Point Theorem
- Mild Solution
- Fractional Differential Equation
- Impulsive Differential Equation
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 ).
where denotes the integer part of the real number .
Definition 2.2 (see ).
provided that the right side is pointwise defined on .
Definition 2.3 (see ).
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 ).
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 ).
Then either is unbounded or has a fixed point.
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.
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 .
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.
maps bounded sets into bounded sets in .
maps bounded sets into equicontinuous sets in .
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.
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 .
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.
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.
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