- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 356132, 18 pages

http://dx.doi.org/10.1155/2012/356132

## On Impulsive Boundary Value Problems of Fractional Differential Equations with Irregular Boundary Conditions

^{1}School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China^{2}Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 25 February 2012; Accepted 4 September 2012

Academic Editor: Yong H. Wu

Copyright © 2012 Guotao Wang 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 study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.

#### 1. Introduction

Boundary value problems of nonlinear fractional differential equations have recently been studied by several researchers. Fractional differential equations appear naturally in various fields of science and engineering and constitute an important field of research. As a matter of fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes [1–4]. Some recent work on boundary value problems of fractional order can be found in [5–23] and the references therein. In [24], some existence and uniqueness results were obtained for an irregular boundary value problem of fractional differential equations.

Dynamical systems with impulse effect are regarded as a class of general hybrid systems. Impulsive hybrid systems are composed of some continuous variable dynamic systems along with certain reset maps that define impulsive switching among them. It is the switching that resets the modes and changes the continuous state of the system. There are three classes of impulsive hybrid systems, namely, impulsive differential systems [25, 26], sampled data or digital control system [27, 28], and impulsive switched system [29, 30]. Applications of such systems include air traffic management [31], automotive control [32, 33], real-time software verification [34], transportation systems [35, 36], manufacturing [37], mobile robotics [38], and process industry [39]. In fact, hybrid systems have a central role in embedded control systems that interact with the physical world. Using hybrid models, one may represent time and event-based behaviors more accurately so as to meet challenging design requirements in the design of control systems for problems such as cut-off control and idle speed control of the engine. For more details, see [40] and the references therein.

The theory of impulsive differential equations of integer order has found its extensive applications in realistic mathematical modelling of a wide variety of practical situations and has emerged as an important area of investigation. The impulsive differential equations of fractional order have also attracted a considerable attention and a variety of results can be found in the papers [41–50].

In this paper, motivated by [24], we study a nonlinear impulsive hybrid system of fractional differential equations with irregular boundary conditions given by where is the Caputo fractional derivative, , where and denote the right and the left limits of at , respectively. have a similar meaning for .

Here, we remark that irregular boundary value problems for ordinary and partial differential equations occur in scientific and engineering disciplines and have been addressed by many authors, for instance, see [24] and the references.

The paper is organized as follows. Section 2 deals with some definitions and preliminary results, while the main results are presented in Section 3.

#### 2. Preliminaries

Let us fix with and introduce the spaces: with the norm , and with the norm . Obviously, and are Banach spaces.

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

To prove the existence of solutions of problem (1.1), we need the following fixed-point theorems.

Theorem 2.2 (see [51]). *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 2.3 (see [1]). * For , the general solution of fractional differential equation is
**
where ( denotes integer part of ). *

Lemma 2.4 (see [1]). * Let . Then
**
for some .*

Lemma 2.5. *For a given , a function is a solution of the following impulsive irregular boundary value problem
**
if and only if is a solution of the impulsive fractional integral equation
**
where
*

*Proof. *Let be a solution of (2.6). Then, by Lemma 2.4, we have
for some . Differentiating (2.9), we get

If , then
for some . Thus,
Using the impulse conditions
we find that
Consequently, we obtain
By a similar process, we get

Applying the boundary conditions and , we find that
Substituting the value of in (2.9) and (2.16), we obtain (2.7). Conversely, assume that is a solution of the impulsive fractional integral equation (2.7), then by a direct computation, it follows that the solution given by (2.7) satisfies (2.6). This completes the proof.

*Remark 2.6. *With , the first five terms of the solution (2.7) correspond to the solution for the problem without impulses [24].

#### 3. Main Results

Define an operator by Notice that problem (1.1) has a solution if and only if the operator has a fixed point.

For the sake of convenience, we set the following notations:

Theorem 3.1. *Assume that ** there exists a nonnegative function such that
where is a nonnegative constant;** there exist positive constants and such that
**Then problem (1.1) has at least one solution. *

* Proof. *As a first step, we show that the operator is completely continuous. Observe that continuity of follows from the continuity of and .

Let be bounded. Then, there exist positive constants such that , and . Thus, , we have
which implies

On the other hand, for any , we get

Hence, for with , we have
This implies that is equicontinuous on all and hence, by the Arzela-Ascoli theorem, the operator is completely continuous.

Next, we prove that . For that, let us choose and define a ball . For any , by the assumptions and , we have
Thus,
where and are given by (3.2). This implies . Hence, is completely continuous. Therefore, by the Schauder fixed-point theorem, the operator has at least one fixed point. Consequently, problem (1.1) has at least one solution in .

*Remark 3.2. *For in (), if , we can take , then the conclusion of Theorem 3.1 holds.

Theorem 3.3. *Suppose that there exist a nonnegative functions and a nonnegative number such that for . Furthermore, the assumption holds. Then problem (1.1) has at least one solution. *

* Proof. *The proof is similar to that of Theorem 3.1, so we omit it.

Theorem 3.4. *Suppose that
**
Then problem (1.1) has at least one solution. *

*Proof. *By Theorem 3.1, we know that the operator is completely continuous. In view of (3.11), we can find a constant such that and for , where satisfy

Let . Take such that , which means . Then, as in the proof of Theorem 3.1, we have
which, in view of (3.12), implies that , . Therefore, by Theorem 2.2, the operator has at least one fixed point. Thus we conclude that problem (1.1) has at least one solution .

Theorem 3.5. *Assume that** there exist positive constants such that
for .** Then problem (1.1) has a unique solution if
*

*Proof. *For , we have
which, by (3.15), yields . So, is a contraction. Therefore, by the Banach contraction mapping principle, problem (1.1) has a unique solution.

*Example 3.6. *Consider the following fractional impulsive irregular boundary value problem
where and .

Observe that Clearly, , and the conditions of Theorem 3.1 hold for . Thus, by Theorem 3.1, problem (3.17) has at least one solution. In a similar way, for , the impulsive irregular fractional boundary value problem (3.17) has at least one solution by means of Theorem 3.3.

*Example 3.7. *Consider the impulsive fractional irregular boundary value problem given by
where and .

It can easily be verified that all the assumptions of Theorem 3.4 are satisfied. Thus, by the conclusion of Theorem 3.4, we deduce that the problem (3.19) has at least one solution.

*Example 3.8. *Consider

Here , and . With we find that Thus, all the conditions of Theorem 3.5 are satisfied. Consequently, the conclusion of Theorem 3.5 applies and the fractional order impulsive irregular boundary value problem (3.20) has a unique solution on .

#### Acknowledgment

The research of G. Wang and L. Zhang was supported by the Natural Science Foundation for Young Scientists of Shanxi Province (2012021002-3), China.

#### References

- 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. - I. Podlubny,
*Fractional Differential Equations*, Academic Press, San Diego, Calif, USA, 1999. - 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. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives*, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993, Theory and Applications. - R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. P. Agarwal, Y. Zhou, and Y. He, “Existence of fractional neutral functional differential equations,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1095–1100, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Ahmad and S. Sivasundaram, “On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order,”
*Applied Mathematics and Computation*, vol. 217, no. 2, pp. 480–487, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Ahmad, S. K. Ntouyas, and A. Alsaedi, “New existence results for nonlinear fractional differential equations with three-point integral boundary conditions,”
*Advances in Difference Equations*, vol. 2011, Article ID 107384, 11 pages, 2011. View at Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 494720, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions,”
*Boundary Value Problems*, vol. 2009, Article ID 708576, 11 pages, 2009. View at Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,”
*Computers & Mathematics with Applications*, vol. 58, no. 9, pp. 1838–1843, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Zhang, L. Liu, and Y. H. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,”
*Applied Mathematics and Computation*, vol. 218, pp. 8526–8536, 2012. View at Publisher · View at Google Scholar - X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. H. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 512127, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Zhang, L. Liu, and Y. H. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,”
*Mathematical and Computer Modelling*, vol. 55, pp. 1263–1274, 2012. View at Publisher · View at Google Scholar - Y. Wang, L. Liu, and Y. Wu, “Positive solutions of a fractional boundary value problem with changing sign nonlinearity,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 149849, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,”
*Applied Mathematics Letters*, vol. 23, no. 9, pp. 1038–1044, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Jia, X. Zhang, and X. Gu, “Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions,”
*Boundary Value Problems*, vol. 2012, article 70, 2012. View at Publisher · View at Google Scholar - Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 72, no. 2, pp. 916–924, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Băleanu, O. G. Mustafa, and R. P. Agarwal, “An existence result for a superlinear fractional differential equation,”
*Applied Mathematics Letters*, vol. 23, no. 9, pp. 1129–1132, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Băleanu and O. G. Mustafa, “On the global existence of solutions to a class of fractional differential equations,”
*Computers & Mathematics with Applications*, vol. 59, no. 5, pp. 1835–1841, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,”
*Computers and Mathematics with Applications*, vol. 59, no. 3, pp. 1300–1309, 2010. View at Publisher · View at Google Scholar · View at Scopus - J. Wang, Y. Zhou, W. Wei, and H. Xu, “Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls,”
*Computers and Mathematics with Applications*, vol. 62, no. 3, pp. 1427–1441, 2011. View at Publisher · View at Google Scholar · View at Scopus - J. Wang, Y. Zhou, and W. Wei, “Impulsive fractional evolution equations and optimal controls in infinite dimensional spaces,”
*Topological Methods in Nonlinear Analysis*, vol. 38, pp. 17–43, 2011. - B. Ahmad, “Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations,”
*Applied Mathematics Letters*, vol. 23, no. 4, pp. 390–394, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov,
*Theory of Impulsive Differential Equations*, vol. 6, World Scientific, Singapore, 1989. - A. M. Samoĭlenko and N. A. Perestyuk,
*Impulsive Differential Equations*, vol. 14, World Scientific, Singapore, 1995. View at Publisher · View at Google Scholar - B. Krogh and N. Lynch, Eds.,
*Hybrid Systems: Computation and Control*, vol. 1790 of*Lecture Notes in Computer Science*, Springer, New York, NY, USA, 2000. - F. Vaandrager and J. Van Schuppen, Eds.,
*Hybrid Systems: Computation and Control*, vol. 1569 of*Lecture Notes in Computer Science*, Springer, New York, NY, USA, 1999. - P. Egbunonu and M. Guay, “Identification of switched linear systems using subspace and integer programming techniques,”
*Nonlinear Analysis. Hybrid Systems*, vol. 1, no. 4, pp. 577–592, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Engell, G. Frehse, and E. Schnieder,
*Modelling, Analysis and Design of Hybrid Systems*, Lecture Notes in Control and Information Sciences, Springer, Heidelberg, Germany, 2002. - C. Tomlin, G. J. Pappas, and S. Sastry, “Conflict resolution for air traffic management: a study in multiagent hybrid systems,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 4, pp. 509–521, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Altafini, A. Speranzon, and K. H. Johansson, “Hybrid control of a truck and trailer vehicle,” in
*Hybrid Systems: Computation and Control*, C. J. Tomlin and M. R. Greenstreet, Eds., vol. 2289 of*Lecture Notes in Computer Science*, Springer, New York, NY, USA, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Balluchi, L. Benvenuti, M. D. di Benedetto, C. Pinello, and A. L. Sangiovanni-Vincentelli, “Automotive engine control and hybrid systems: challenges and opportunities,”
*Proceedings of the IEEE*, vol. 88, no. 7, pp. 888–911, 2000. View at Publisher · View at Google Scholar · View at Scopus - R. Alur, C. Courcoubetis, and D. Dill, “Model checking for real-time systems,” in
*Proceedings of the 5th Annual IEEE Symposium on Logic in Computer Science*, pp. 414–425, Philadelphia, Pa, USA, 1990. - J. Lygeros, D. N. Godbole, and S. Sastry, “Verified hybrid controllers for automated vehicles,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 4, pp. 522–539, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - P. Varaiya, “Smart cars on smart roads: problems of control,”
*IEEE Transactions on Automatic Control*, vol. 38, no. 2, pp. 195–207, 1993. View at Publisher · View at Google Scholar - D. L. Pepyne and C. G. Cassandras, “Optimal Control of hybrid systems in manufacturing,”
*Proceedings of the IEEE*, vol. 88, no. 7, pp. 1108–1122, 2000. View at Publisher · View at Google Scholar · View at Scopus - A. Balluchi, P. Soueres, and A. Bicchi, “Hybrid feedback control for path tracking by a bounded-curvature vehicle,” in
*Hybrid Systems: Computation and Control*, M. Di Benedetto and A. L. Sangiovanni-Vincentelli, Eds., vol. 2034 of*Lecture Notes in Computer Science*, Springer, New York, NY, USA, 2001. - S. Engell, S. Kowalewski, C. Schulz, and O. Stursberg, “Continuous-discrete interactions in chemical processing plants,”
*Proceedings of the IEEE*, vol. 88, no. 7, pp. 1050–1068, 2000. View at Publisher · View at Google Scholar · View at Scopus - P. J. Antsaklis and A. Nerode, “Hybrid control systems: an introductory discussion to the special issue,”
*IEEE Transactions on Automatic Control*, vol. 43, pp. 457–460, 1998. - R. P. Agarwal and B. Ahmad, “Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations,”
*Dynamics of Continuous, Discrete & Impulsive Systems A*, vol. 18, no. 4, pp. 535–544, 2011. View at Zentralblatt MATH - B. Ahmad and J. J. Nieto, “Existence of solutions for impulsive anti-periodic boundary value problems of fractional order,”
*Taiwanese Journal of Mathematics*, vol. 15, no. 3, pp. 981–993, 2011. - B. Ahmad and G. Wang, “A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1341–1349, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - 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 - G. M. Mophou, “Existence and uniqueness of mild solutions to impulsive fractional differential equations,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 72, no. 3-4, pp. 1604–1615, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - G. Wang, B. Ahmad, and L. Zhang, “Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 74, no. 3, pp. 792–804, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - G. Wang, B. Ahmad, and L. Zhang, “Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions,”
*Computers and Mathematics with Applications*, vol. 62, no. 3, pp. 1389–1397, 2011. View at Publisher · View at Google Scholar · View at Scopus - X. Zhang, X. Huang, and Z. Liu, “The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay,”
*Nonlinear Analysis: Hybrid Systems*, vol. 4, no. 4, pp. 775–781, 2010. View at Publisher · View at Google Scholar · View at Scopus - J. X. Sun,
*Nonlinear Functional Analysis and Its Application*, Science Press, Beijing, China, 2008.