About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 785078, 9 pages
http://dx.doi.org/10.1155/2013/785078
Research Article

Existence Results for Fractional Differential Equations with Separated Boundary Conditions and Fractional Impulsive Conditions

1Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, China
2School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China

Received 6 March 2013; Accepted 14 July 2013

Academic Editor: Stanislaw Migorski

Copyright © 2013 Xi Fu and Xiaoyou Liu. 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

This paper is concerned with the fractional separated boundary value problem of fractional differential equations with fractional impulsive conditions. By means of the Schaefer fixed point theorem, Banach fixed point theorem, and nonlinear alternative of Leray-Schauder type, some existence results are obtained. Examples are given to illustrate the results.

1. Introduction

Recently, much attention has been paid to study fractional differential equations due to the fact that they have been proven to be valuable tools in the mathematical modeling of many phenomena in physics, biology, mechanics, and so forth, (see [13]).

The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modeling of a wide variety of practical situations and has emerged as an important area of investigation in recent years. For the general theory and applications of impulsive differential equations, see [410] and so forth. However, impulsive fractional differential equations have not been much studied, and many aspects of these equations are yet to be explored. For some recent work on impulsive fractional differential equations, we can refer to [1126] and the references therein.

In this paper, we consider the existence and uniqueness of solutions for the following fractional separated boundary value problem with fractional impulsive conditions: where is the Caputo fractional derivative of order with the lower limit zero, , , , , with , representing the right and left limits of at , has a similar meaning for , and , , are real constants with and .

We note that the papers on this topic cited above except [24] all deal with the Caputo derivative and the impulsive conditions only involve integer order derivatives. Here we study the fractional differential equations with fractional impulsive conditions and fractional separated boundary conditions.

In [24], the author considered the following two impulsive problems: where is the Caputo fractional derivative of order with the lower limit zero, , and where is the Riemann-Liouville fractional derivative of order with the lower limit zero and .

In [25], Fečkan et al. studied the impulsive problem of the following form: where is jointly continuous, and satisfy , , and .

Furthermore, Wang et al. [26] considered the impulsive fractional differential equations with boundary conditions as follows: where .

To the best of our knowledge, there are few papers concerning fractional differential equations with separated boundary conditions [27, 28].

The rest of the paper is organized as follows. In Section 2 we introduce some preliminary results needed in the sequel. In Section 3 we present the existence results for the problem (1). Two examples are given in Section 4 to illustrate the results.

2. Preliminaries

Let us set , , , and , and introduce the space , , and there exist and , , with . It is clear that is a Banach space with the norm .

Definition 1 (see [3]). The Riemann-Liouville fractional integral of order for a continuous function is defined as which provided that the integral exists.

Definition 2 (see [3]). For times an absolutely continuous function , the Caputo derivative of order is defined as where denotes the integer part of the real number .

Lemma 3 (see [3]). Let . Then the differential equation has solutions and which hold for almost all points on the interval , here , , .

Definition 4. A function with its -derivative existing on is said to be a solution of the problem (1) if satisfies the equation on and the conditions
are satisfied.

By using a similar discussion of [25], we have the following lemma.

Lemma 5. Let . A function is a solution of the fractional integral equation: where if and only if is a solution of the impulsive fractional BVP:

Proof. For , by Lemma 3, we know that a general solution of the equation on each interval () is given by where are arbitrary constants. Since ( is a constant), , and (see [3]), then from (15), we have for . Applying the boundary conditions of (14), we get Next, using the impulsive conditions in (14), we obtain that for Now we can derive the values of , from formulae (17)-(18). That is, for and Hence for , we have Now it is clear that a solution of the problem (14) has the form of (11).
Conversely, assume that satisfies the fractional integral equation (11). That is, for , , we have Since , we have ( is a constant) and . Using the fact that is the left inverse of , we get which means that satisfies the first equation of the impulsive fractional BVP (14). Next we will verify that satisfies the impulsive conditions. Taking fractional derivative of (22), we have, for , From (22), we obtain Hence we have, for , Similarly, from (24), we can obtain that, for , Finally, it follows from (22) and (24) that (since , ) , , and Now we get Therefore given by (11) satisfies the impulsive fractional boundary value problem (14). The proof is complete.

Remark 6. We notice that the expression of (11) does not depend on the parameter appearing in the boundary conditions of the problem (14). Thus by Lemma 5, we conclude that the parameter is of arbitrary nature of the problem (14).

Let be Banach spaces and , and we say that is a compact if the image of each bounded set in under is relatively compact. The following are two fixed point theorems which will be used in the sequel.

Theorem 7 (nonlinear alternative of Leray-Schauder type [29]). Let be a Banach space, a nonempty convex subset of , and a nonempty open subset of with . Suppose that is a continuous and compact map. Then either (a) has a fixed point in or (b) there exist a (the boundary of ) and with .

Theorem 8 (Schaefer fixed point theorem [30]). Let be a normed space and a continuous mapping of into which is compact on each bounded subset of . Then either (I) the equation has a solution for or (II) the set of all such solutions , for , is unbounded.

3. Main Results

This section deals with the existence and uniqueness of solutions for the problem (1).

In view of Lemma 5, we define an operator by with Here , mean that , defined in Lemma 5 are related to . It is obvious that is well defined because of the continuity of , , and . Observe that the problem (1) has solutions if and only if the operator equation has fixed points.

Lemma 9. The operator defined by (31) is completely continuous.

Proof. Since , , and are continuous, it is easy to show that is continuous on .
Let be bounded. Then there exist positive constants , , such that , , and for all , , . Thus, for and , we have Now we can obtain that, for all , and , which implies that the operator is uniformly bounded on .
On the other hand, let and for any , , with , we have By (34) and the above inequality, we deduce that This implies that is equicontinuous on the interval . Hence by PC-type Arzela-Ascoli theorem (see Theorem 2.1 [10]), the operator is completely continuous.

Theorem 10. Assume that (1) there exist and continuous, nondecreasing such that for ; (2) there exist continuous, nondecreasing such that , for all and ; (3) there exists a constant such that where Then, BVP (1) has at least one solution.

Proof. We will show that the operator defined by (31) satisfies the assumptions of the nonlinear alternative of Leray-Schauder type.
From Lemma 9, the operator is continuous and completely continuous.
Let such that for some . Then using the computations in proving that maps bounded sets into bounded sets in Lemma 9, we have Consequently, we have Then by condition (38), . Let us set The operator is continuous and compact. From the choice of the set , there is no such that for some . Therefore by the nonlinear alternative of Leray-Schauder type (see Theorem 7), we deduce that has a fixed point in which is a solution of the problem (1). The proof is complete.

Theorem 11. Assume that there exist and positive constants such that, for , , , Then, BVP (1) has at least one solution on .

Proof. Lemma 9 tells us that the operator defined by (31) is continuous and compact on each bounded subset of .
Let . Since, for each , we know that is bounded. Thus, by Theorem 8, the operator has at least one fixed point. Hence the problem (1) has at least one solution. The proof is completed.

Theorem 12. Assume that there exist and positive constants such that, for , , , Moreover Then, BVP (1) has a unique solution on .

Proof. Let . Then for each , we have Since then combining these two estimations with (47), we obtain Therefore, by (46), the operator is a contraction mapping on . Then it follows Banach's fixed point theorem that the problem (1) has a unique solution on . This completes the proof.

4. Examples

Finally we give two simple examples to show the applicability of our results.

Example 1. Consider the following impulsive fractional separated BVP:
Here , , , and . Clearly, we can take , and such that the relations (45) hold. Moreover Thus, all the assumptions of Theorem 12 are satisfied. Hence, by the conclusion of Theorem 12, the impulsive fractional BVP (50) has a unique solution on .

Example 2. Consider the following impulsive fractional separated BVP:
In the context of this problem, we have Put , , and . Then from Theorem 11, the impulsive fractional BVP (52) has at least one solution on .

References

  1. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989. View at Zentralblatt MATH · View at MathSciNet
  5. Z. Luo and J. J. Nieto, “New results for the periodic boundary value problem for impulsive integro-differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2248–2260, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Z. Luo and J. Shen, “Stability of impulsive functional differential equations via the Liapunov functional,” Applied Mathematics Letters, vol. 22, no. 2, pp. 163–169, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  7. Z. Liu and J. Liang, “A class of boundary value problems for first-order impulsive integro-differential equations with deviating arguments,” Journal of Computational and Applied Mathematics, vol. 237, no. 1, pp. 477–486, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Z. He and J. Yu, “Periodic boundary value problem for first-order impulsive ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 67–78, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. L. Zhang, “Boundary value problem for first order impulsive functional integro-differential equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2442–2450, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. W. Wei, X. Xiang, and Y. Peng, “Nonlinear impulsive integro-differential equations of mixed type and optimal controls,” Optimization, vol. 55, no. 1-2, pp. 141–156, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. Wang, “Existence results for fractional functional differential equations with impulses,” Journal of Applied Mathematics and Computing, vol. 38, no. 1-2, pp. 85–101, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. M. Benchohra, S. Hamani, J. J. Nieto, and B. A. Slimani, “Existence of solutions to differential inclusions with fractional order and impulses,” Electronic Journal of Differential Equations, vol. 2010, no. 80, pp. 1–18, 2010. View at Zentralblatt MATH · View at MathSciNet
  13. B. Ahmad and S. Sivasundaram, “Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 251–258, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. B. Ahmad and S. Sivasundaram, “Existence of solutions for impulsive integral boundary value problems of fractional order,” Nonlinear Analysis: Hybrid Systems, vol. 4, no. 1, pp. 134–141, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Z. Liu and X. Li, “On the controllability of impulsive fractional evolution inclusions in Banach spaces,” Journal of Optimization Theory and Applications, vol. 156, no. 1, pp. 167–182, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. Y. Tian and Z. Bai, “Existence results for the three-point impulsive boundary value problem involving fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2601–2609, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Cao and H. Chen, “Some results on impulsive boundary value problem for fractional differential inclusions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2011, no. 11, pp. 1–24, 2011. View at Zentralblatt MATH · View at MathSciNet
  18. G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 792–804, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. T. L. Guo and J. Wei, “Impulsive problems for fractional differential equations with boundary value conditions,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3281–3291, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. G. Wang, B. Ahmad, and L. Zhang, “Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1389–1397, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Z. Tai and S. Lun, “On controllability of fractional impulsive neutral infinite delay evolution integrodifferential systems in Banach spaces,” Applied Mathematics Letters, vol. 25, no. 2, pp. 104–110, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. X. Wang, “Existence of solutions for nonlinear impulsive higher order fractional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2011, no. 80, pp. 1–12, 2011. View at MathSciNet
  23. C. Bai, “Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance,” Journal of Applied Mathematics and Computing, vol. 39, no. 1-2, pp. 421–443, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. N. Kosmatov, “Initial value problems of fractional order with fractional impulsive conditions,” Results in Mathematics, vol. 63, no. 3-4, pp. 1289–1310, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  25. M. Fečkan, Y. Zhou, and J. R. Wang, “On the concept and existence of solution for impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 7, pp. 3050–3060, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. R. Wang, Y. Zhou, and M. Fečkan, “On recent developments in the theory of boundary value problems for impulsive fractional differential equations,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3008–3020, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. B. Ahmad and S. K. Ntouyas, “A note on fractional differential equations with fractional separated boundary conditions,” Abstract and Applied Analysis, vol. 2012, Article ID 818703, 11 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. B. Ahmad and S. K. Ntouyas, “Fractional differential inclusions with fractional separated boundary conditions,” Fractional Calculus and Applied Analysis, vol. 15, no. 3, pp. 362–382, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  29. A. Granas and J. Dugundji, Fixed Point Theory, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2003. View at MathSciNet
  30. D. R. Smart, Fixed Point Theorems, Cambridge University Press, London, UK, 1974. View at MathSciNet