Abstract
This paper is concerned with the existence and uniqueness of solutions for impulsive nonlinear differential equations of fractional order with closed boundary conditions. By applying some standard fixed point theorems, we obtain the sufficient conditions for the existence and uniqueness of solutions of the problem at hand. An illustrative example is presented.
MSC:26A33, 34B15, 34B37.
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1 Introduction
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 [1, 2], sampled data or digital control system [3, 4], and impulsive switched system [5]. 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 [6] and the references therein.
Fractional calculus (differentiation and integration of arbitrary order) has proved to be an important tool in the modeling of dynamical systems associated with phenomena such as fractals and chaos. In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, fitting of experimental data, etc. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. With this advantage, the fractional-order models become more realistic and practical than the classical integer-order models in which such effects are not taken into account. For some recent details and examples, see [7–22] and the references therein.
Impulsive differential equations are found to be important mathematical tools for better understanding of several real world problems in biology, physics, engineering, etc. In fact, 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; for instance, see [23–25] and references therein. The recent surge in developing the theory of differential equations of fractional order has led several researchers to study the fractional differential equations with impulse effects. For some recent work on impulsive differential equations of fractional order, see [26–31] and the references therein.
In this paper, we investigate the existence of solutions for the following impulsive fractional differential equations with closed boundary conditions:
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 the boundary conditions in (1.1) include quasi-periodic boundary conditions () and interpolate between periodic (, ) and antiperiodic (, ) boundary conditions. For more details and applications of closed boundary conditions, see [14].
2 Preliminaries
Let , and we introduce the spaces: with the norm , and with the norm . Obviously, and are Banach spaces.
In passing, we remark that indeed stands for for t in the subinterval .
Definition 2.1 A function with its Caputo derivative of order q existing on J is a solution of (1.1) if it satisfies (1.1).
Define
where .
Lemma 2.1 For a given , a function x is a solution of the impulsive closed boundary value problem
if and only if x is a solution of the impulsive fractional integral equation
where
Proof Let x be a solution of (2.1). Then, for , there exist constants such that
For , there exist constants , such that
Then we have
In view of the impulse conditions and , we have that
Consequently,
By a similar process, we can get
Using the conditions and , we find that
Substituting the value of , in (2.3) and (2.4), we obtain (2.2). Conversely, assume that u is a solution of the impulsive fractional integral equation (2.2), then by a direct computation, it follows that the solution given by (2.2) satisfies (2.1). □
3 Main results
Define an operator as
Observe that the problem (1.1) has a solution if and only if the operator T has a fixed point.
Lemma 3.1 The operator defined by (3.1) is completely continuous.
Proof It is obvious that is continuous in view of continuity of f, and .
Let be bounded. Then, there exist positive constants () such that , and , . Thus, , we have
which implies that .
On the other hand, for any , , we have
Hence, for , , , we have
This implies that is equicontinuous on all , . Thus, by the Arzela-Ascoli theorem, the operator is completely continuous. □
For the sake of convenience, we set the following notations:
Theorem 3.1 Assume that
(H1) there exist nonnegative functions and positive constants () such that
for , and .
Then the problem (1.1) has at least one solution.
Proof Define a ball , we just need to show that the operator , as it has already been proved that the operator is completely continuous in the previous lemma. Let us choose . For any , by the assumption , we have
which implies that
where τ and ν are given by (3.3) and (3.4). So, . Thus is completely continuous. Therefore, by the Schauder fixed point theorem, the operator has at least one fixed point. Consequently, the problem (1.1) has at least one solution in . □
Theorem 3.2 Assume that
() there exist nonnegative functions and positive constants () such that
for , and .
Then the problem (1.1) has at least one solution.
Proof The proof is similar to that of Theorem 3.1, so we omit it. □
Theorem 3.3 ([32])
Let E be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then has a fixed point in E.
Theorem 3.4 If . In addition, assume that
() there exist nonnegative functions and positive constants () such that
for , and .
Then the problem (1.1) has at least one solution.
Proof Let us consider the set
where the operator is defined by (3.1). We just need to show that the set V is bounded as it has already been proved that the operator is completely continuous in the previous lemma. Let , then , . For any , we have
which implies that is bounded for any . So, the set V is bounded. Thus, by the conclusion of Theorem 3.3, the operator has at least one fixed point, which implies that (1.1) has at least one solution. □
Corollary 3.1 Assume that functions f, , () are bounded. Then the nonlinear problem (1.1) has at least one solution.
Theorem 3.5 Assume that
() there exist a nonnegative function and positive constants () such that
for , and .
Then the problem (1.1) has a unique solution if
Proof For , we can get
Consequently, we have , where is given by (3.6). As , the conclusion of the theorem follows by the contraction mapping principle. This completes the proof. □
4 Examples
Example 4.1 Consider the following impulsive fractional boundary value problem with closed boundary conditions
where , and .
In this case, , , , , and the conditions of Theorem 3.1 can readily be verified. Thus, by the conclusion of Theorem 3.1, the problem (4.1) has at least one solution.
References
Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.
Krogh B, Lynch N (Eds): Lecture Notes in Computer Science 1790 In Hybrid Systems: Computation and Control. Springer, New York; 2000.
Vaandrager F, Schuppen J (Eds): Lecture Notes in Computer Science 1569 In Hybrid Systems: Computation and Control. Springer, New York; 1999.
Engell S, Frehse G, Schnieder E Lecture Notes in Control and Information Sciences. In Modelling, Analysis and Design of Hybrid Systems. Springer, Heidelberg; 2002.
Antsaklis PJ, Nerode A: Hybrid control systems: an introductory discussion to the special issue. IEEE Trans. Autom. Control 1998, 43: 457–460. 10.1109/TAC.1998.664148
Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Sabatier J, Agrawal OP, Machado JAT (Eds): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht; 2007.
Lakshmikantham V, Leela S, Devi JV: Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge; 2009.
Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 2009, 58: 1838–1843. 10.1016/j.camwa.2009.07.091
Wei Z, Li Q, Che J: Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 2010, 367: 260–272. 10.1016/j.jmaa.2010.01.023
Zhang SQ: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 2010, 59: 1300–1309. 10.1016/j.camwa.2009.06.034
Zhou Y, Jiao F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl. 2010, 11: 4465–4475. 10.1016/j.nonrwa.2010.05.029
Ahmad B, Nieto JJ, Pimentel J: Some boundary value problems of fractional differential equations and inclusions. Comput. Math. Appl. 2011, 62: 1238–1250. 10.1016/j.camwa.2011.02.035
Wang G, Ntouyas SK, Zhang L: Positive solutions of the three-point boundary value problem for fractional-order differential equations with an advanced argument. Adv. Differ. Equ. 2011., 2011: Article ID 2
Ford NJ, Morgado ML: Fractional boundary value problems: analysis and numerical methods. Fract. Calc. Appl. Anal. 2011, 14: 554–567.
Aghajani A, Jalilian Y, Trujillo JJ: On the existence of solutions of fractional integro-differential equations. Fract. Calc. Appl. Anal. 2012, 15: 44–69.
Ahmad B, Nieto JJ: Anti-periodic fractional boundary value problem with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal. 2012, 15: 451–462.
Baleanu D, Diethelm K, Scalas E, Trujillo JJ Series on Complexity, Nonlinearity and Chaos. In Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.
Ahmad B, Nieto JJ, Alsaedi A, El-Shahed M: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 2012, 13: 599–606. 10.1016/j.nonrwa.2011.07.052
Wang G, Agarwal RP, Cabada A: Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations. Appl. Math. Lett. 2012, 25: 1019–1024. 10.1016/j.aml.2011.09.078
Ahmad B, Ntouyas SK: Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary conditions. Electron. J. Differ. Equ. 2012., 2012: Article ID 98
Chen L, Sun J: Nonlinear boundary value problem of first order impulsive functional differential equations. J. Math. Anal. Appl. 2006, 318(2):726–741. 10.1016/j.jmaa.2005.08.012
Luo Z, Nieto JJ: New results for the periodic boundary value problem for impulsive integro-differential equations. Nonlinear Anal. 2009, 70(6):2248–2260. 10.1016/j.na.2008.03.004
Li C, Sun J, Sun R: Stability analysis of a class of stochastic differential delay equations with nonlinear impulsive effects. J. Franklin Inst. 2010, 347: 1186–1198. 10.1016/j.jfranklin.2010.04.017
Tian Y, Bai Z: Existence results for the three-point impulsive boundary value problem involving fractional differential equations. Comput. Math. Appl. 2010, 59: 2601–2609. 10.1016/j.camwa.2010.01.028
Wang G, Ahmad B, Zhang L: Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order. Nonlinear Anal. 2011, 74: 792–804. 10.1016/j.na.2010.09.030
Ahmad B, Wang G: A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations. Comput. Math. Appl. 2011, 62: 1341–1349. 10.1016/j.camwa.2011.04.033
Bai C: Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative. J. Math. Anal. Appl. 2011, 384: 211–231. 10.1016/j.jmaa.2011.05.082
Ahmad B, Nieto JJ: Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan. J. Math. 2011, 15: 981–993.
Mu J, Li Y: Monotone iterative technique for impulsive fractional evolution equations. J. Inequal. Appl. 2011., 2011: Article ID 125
Sun JX: Nonlinear Functional Analysis and Its Application. Science Press, Beijing; 2008.
Acknowledgements
We would like to express our gratitude to the anonymous reviewers and editors for their valuable comments and suggestions that led to the improvement of the original manuscript. The research of B. Ahmad was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The research of Guotao Wang and Lihong Zhang was supported by the Natural Science Foundation for Young Scientists of Shanxi Province (2012021002-3), China.
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Wang, G., Ahmad, B. & Zhang, L. Existence results for nonlinear fractional differential equations with closed boundary conditions and impulses. Adv Differ Equ 2012, 169 (2012). https://doi.org/10.1186/1687-1847-2012-169
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DOI: https://doi.org/10.1186/1687-1847-2012-169