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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 717235, 26 pages
Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations of Order with Nonlocal Boundary Conditions
1School of Mathematics and Computer Science, Shanxi Normal University, Shanxi, Linfen 041004, China
2Department of Mathematics, China University of Petroleum, Shandong, Qingdao 266555, China
Received 2 February 2012; Revised 13 April 2012; Accepted 28 April 2012
Academic Editor: Lishan Liu
Copyright © 2012 Lihong Zhang et al. 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.
We investigate the existence and uniqueness of solutions to the nonlocal boundary value problem for nonlinear impulsive fractional differential equations of order . By using some well-known fixed point theorems, sufficient conditions for the existence of solutions are established. Some examples are presented to illustrate the main results.
Fractional differential equations arise in many engineering and scientific disciplines such as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, control theory, signal and image processing, biophysics, electrodynamics of complex medium, polymer rheology, and fitting of experimental data [1–6]. For example, one could mention the problem of anomalous diffusion [7–9], the nonlinear oscillation of earthquake can be modeled with fractional derivative , and fluid-dynamic traffic model with fractional derivatives  can eliminate the deficiency arising from the assumption to continuum traffic flow and many other [12, 13] recent developments in the description of anomalous transport by fractional dynamics. For some recent development on nonlinear fractional differential equations, see [14–29] and the references therein.
In this paper, we investigate a three-point boundary value problems for nonlinear impulsive fractional differential equations of order : where is the Caputo fractional derivative, , , , , , , , , where and denote the right and the left limits of at , respectively. has a similar meaning for .
Impulsive differential equations arise in many engineering and scientific disciplines as the important mathematical modeling of systems and processes in the fields of biology, physics, engineering, and so forth. Due to their significance, it is important to study the solvability of impulsive differential equations. The theory of impulsive differential equations of integer order has emerged as an important area of investigation. Recently, the impulsive differential equations of fractional order have also attracted a considerable attention, and a variety of results can be found in the papers [30–42].
The study of multipoint boundary-value problems was initiated by Bicadze and Samarskiĭ in . Many authors since then considered nonlinear multipoint boundary-value problems, see [44–51] and the references therein. The multipoint boundary conditions are important in various physical problems of applied science when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the sensors located at intermediate points. For example, the vibrations of a guy wire of uniform cross-section and composed of parts of different densities can be set up as a multipoint boundary-value problem.
To our knowledge, no paper has considered nonlinear impulsive fractional differential equations of order with nonlocal boundary conditions, that is, problem (1.1). This paper fills this gap in the literature. Our purpose here is to give the existence and uniqueness of solutions for nonlinear impulsive fractional differential equations (1.1). Our results are based on some well-known fixed point theorems.
We introduce the space: with the norm: Obviously, is a Banach space.
Theorem 2.2 (see ). Let be a Banach space. Assume that is a completely continuous operator, and the set is bounded. Then has a fixed point in .
Theorem 2.3 (see ). Let be a Banach space. Assume that is an open bounded subset of with , and let be a completely continuous operator such that Then, has a fixed point in .
Lemma 2.4. Let , is a nonnegative integer, . For a given , a function is a solution of the following impulsive boundary value problem: if and only if is a solution of the impulsive fractional integral equation: where
Proof. Let is a solution of (2.4), it holds
for some . Then, we have
In view of , it follows .
If , then for some .
Thus, we have
In view of and , we have
Similarly, we can get
By (2.13), it follows
In view of the condition , we have
Substituting the value of in (2.7) and (2.13) and letting , we can get (2.5). Conversely, assume that is a solution of the impulsive fractional integral equation (2.5), then by a direct computation, it follows that the solution given by (2.5) satisfies (2.4).
This completes the proof.
3. Main Results
Let , is a nonnegative integer, and . Define the operator as follows:
then (1.1) has a solution if and only if the operator has a fixed point.
Lemma 3.1. The operator is completely continuous.
Proof. Obviously, is continuous in view of continuity of , , and .
Let be bounded. Then, there exist positive constants such that and , for all . Thus, for all , we have which implies
On the other hand, for any , we have