Abstract and Applied Analysis

Volume 2012 (2012), Article ID 717235, 26 pages

doi:10.1155/2012/717235

Research Article

Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations of Order with Nonlocal Boundary Conditions

Lihong Zhang,¹ Guotao Wang,¹ and Guangxing Song²

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.

Abstract.

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.

1. Introduction

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 [10], and fluid-dynamic traffic model with fractional derivatives [11] 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 [43]. 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.

2. Preliminaries

Let .

We introduce the space:

with the norm:

Obviously, is a Banach space.

Definition 2.1. A function with its Caputo derivative of order existing on is a solution of (1.1) if it satisfies (1.1).

Theorem 2.2 (see [52]). 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 [52]). 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

Consequently,

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

Hence, let , we have

This implies that is equicontinuous on all . The Arzela-Ascoli Theorem implies that is completely continuous.

Theorem 3.2. Assume that the nonlinearity is bounded and that the impulse functions are bounded. Then, the nonlinear impulsive fractional three-point boundary value problems (1.1) have at least one solution.

Proof. Firstly, by Lemma 3.1, we know that the operator is completely continuous.

Let be nonnegative constants such that

For , consider the equation:

If is a solution of (3.7), then for every we have that

which implies for any , it holds that

This shows that all the solutions of (3.7) are bounded independently of . Using Schaeffer's theorem (see Theorem 2.2), we get that has at least one fixed point, which implies that (1.1) has at least one solution.

Now, we present some existence results when the nonlinearity and the impulse functions have sublinear growth.

Theorem 3.3. Assume that

 (H₁) there exist nonnegative constants and such that

 (H₂) there exist nonnegative constants such that

Then, problem (1.1) has at least one solution.

Proof. If , then we can write that

So, if is a solution of (3.7), then, by the similar process used to obtain (3.2), for every we have that

where

In consequence,

where .

Now, taking into account that ,  , we can conclude that there exists a constant such that for any solution of (3.7), . Therefore, by Schaeffer's theorem (Theorem 2.2), we obtain the existence of a fixed point for , which implies the problem (1.1) has at least one solution.

Theorem 3.4. Assume that

 (H₃) there exist nonnegative constants and such that

 (H₄) there exist nonnegative constants such that

If

where

Then problem (1.1) has at least one solution.

Proof. The proof is similar to that of Theorem 3.3, so we omit it.

Theorem 3.5. Let , and , then problem (1.1) has at least one solution.

Proof. Now, in view of , and , there exists a constant such that , and for , where satisfy

Let . Take such that , that is, . By Lemma 3.1, we know that is completely continuous, and

Thus, in view of (3.21), we obtain , . Therefore, by Theorem 2.3, the operator has at least one fixed point, which implies that (1.1) has at least one solution .

For the forthcoming analysis, we need the following assumption:

 (H₅) there exist positive constants such that

for   and  .

Define the constants:

Theorem 3.6. If condition (H₅) holds. Then problem (1.1) has a unique solution provided , where is given by (3.23).

Proof. Let , by a simple computation, we can get

Thus, .

As , therefore, is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.

4. Examples

Example 4.1. Consider the following nonlinear impulsive fractional differential equations:

where ,  and  .

Obviously, for , and , it is easy to verify condition of Theorem 3.2 holds. Hence, by Theorem 3.2, we can get that the above equation (4.1) has at least one solution.

Example 4.2. Consider the following fractional impulsive the three-point boundary value problem:

where  ,  and  .

Obviously . It is easy to verify that for , the conditions of Theorem 3.3 hold. Therefore, by Theorem 3.3, the impulsive the three-point fractional boundary value problem (4.2) has at least one solution.

Example 4.3. Consider the following nonlinear impulsive fractional differential equations:

where , and .

Clearly, all the assumptions of Theorem 3.5 hold. Thus, by the conclusion of Theorem 3.5 we can get that (4.3) has at least one solution.

5. Conclusion

The existence and uniqueness of solutions to a three-point boundary value problem for fractional nonlinear differential equation with impulses have been discussed. We apply the concepts of fractional calculus together with fixed point theorems to establish the existence results. First of all, we investigate a linear three-point boundary value problem involving fractional derivatives and impulses, which in fact provides the platform to prove the existence of solutions for the associated nonlinear fractional equation with impulses. It is worth mentioning that the conditions of our theorems are easily to verify and that our approach is simple, so they are applicable to a variety of real world problems.

Acknowledgments

The authors would like to express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which improve the quality of present paper. They also would like to thank Professor Bashir Ahmad for his significant suggestions on the improvement of this paper. This paper was supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (Grant no. 2012021002-3) and Science Foundation of Shanxi Normal University, China (Grant no. ZR1113).

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