- Open Access
Green’s function for Sturm-Liouville-type boundary value problems of fractional order impulsive differential equations and its application
© Zhou and Feng; licensee Springer. 2014
- Received: 6 January 2014
- Accepted: 5 March 2014
- Published: 25 March 2014
In this paper, we discuss the expression and properties of Green’s function for boundary value problems of nonlinear Sturm-Liouville-type fractional order impulsive differential equations. Its applications are also given. Our results are compared with some recent results by Bai and Lü.
- fractional differential equation
- Green’s function
- fixed point theorem
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, and so forth, and involves derivatives of fractional order. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. An excellent account in the study of fractional differential equations can be found in [1–5]. For the basic theory and recent development of the subject, we refer to a text by Lakshmikantham et al. . For more details and examples, see [7–23] and the references therein.
Integer-order impulsive differential equations have become important in recent years as mathematical models of phenomena in both physical and social sciences. There has been a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see, for instance, [24–26]. Recently, the boundary value problems of impulsive differential equations of integer order have been studied extensively in the literature (see [27–36]).
On the other hand, impulsive differential equations of fractional order play an important role in theory and applications, see [37–46] and the references therein. However, as pointed out in [38, 39], the fractional impulsive differential equations have not been addressed so extensively and many aspects of these problems are yet to be explored. For example, the theory using Green’s function to express the solution of fractional impulsive differential equations has not been investigated till now. Now, in this paper, we shall study the expression of the solution of fractional impulsive differential equations by using Green’s function.
where is the Caputo fractional derivative, , is a continuous function, is a continuous function, are continuous functions, and . with , , , . has a similar meaning for .
Some special cases of (1.1) have been investigated. For example, Bai and Lü  considered problem (1.1) with and . By using the fixed point theorem in cones, they proved some existence and multiplicity results of positive solutions of problem (1.1).
where , .
Thus, it is an interesting problem, and so it is worthwhile to study. We will give the answers in the following sections.
The organization of this paper is as follows. In Section 2, we present the expression and properties of Green’s function associated with problem (1.1). In Section 3, we give some preliminaries about the operator and the fixed point theorem. In Section 4, we get some existence results for problem (1.1) by means of some standard fixed point theorems. The final section of the paper contains two examples to illustrate our main results.
where is the Caputo fractional derivative, , are continuous functions, and . with , , , . has a similar meaning for .
where η is defined in (2.4).
Substituting (2.7), (2.8) into (2.6), we obtain (2.2). This completes the proof. □
From (2.3), we can prove the following results.
From the proof of Theorem 2.1 we have the following results.
Proposition 2.3 The solution of fractional impulsive differential equations can be expressed by Green’s function, and it is not Green’s function of the corresponding fractional differential equations, but Green’s function of the corresponding integer order differential equations.
In this section, we give some preliminaries for discussing the solvability of problem (1.1) as follows.
Definition 3.1 A function with its Caputo derivative of order q existing on J is a solution of problem (1.1) if it satisfies (1.1).
We give the following hypotheses:
(H1) is a continuous function, and there exists such that ;
(H2) is a continuous function;
(H3) are continuous functions.
It follows from Theorem 2.1 that:
Using Lemma 3.1, problem (1.1) reduces to a fixed point problem , where T is given by (3.2). Thus problem (1.1) has a solution if and only if the operator T has a fixed point.
From (3.2) and Lemma 3.1, it is easy to obtain the following result.
Lemma 3.2 Assume that (H1)-(H3) hold. Then is completely continuous.
Proof Note that the continuity of f, ω, and together with and ensures the continuity of T.
It follows from (3.3), (3.5) and (3.6) that T is equicontinuous on all subintervals , . Thus, by the Arzela-Ascoli theorem, the operator is completely continuous. □
To prove our main results, we also need the following two lemmas.
Let D be a nonempty, closed, bounded, convex subset of a B-space X, and suppose that is a completely continuous operator. Then T has a fixed point .
here γ is defined in (3.4). Then problem (1.1) has at least one solution .
Proof We shall use Schauder’s fixed point theorem to prove that T has a fixed point. First, recall that the operator is completely continuous (see the proof of Lemma 3.2).
where , are positive constants.
Consequently, Lemma 3.3 implies that T has a fixed point in , and the proof is complete. □
Remark 4.1 Condition (4.1) is certainly satisfied if uniformly in as , as and as ().
Theorem 4.2 Assume that (H1)-(H3) hold. In addition, let f, and satisfy the following conditions:
for all , .
Then problem (1.1) has at least one solution.
Proof It follows from Lemma 3.2 that the operator is completely continuous.
So it follows from (4.13) that the set V is bounded. Thus, as a consequence of Lemma 3.4, the operator T has at least one fixed point. Consequently, the problem (1.1) has at least one solution. This finishes the proof. □
Finally we consider the existence of a unique solution for problem (1.1) by applying the contraction mapping principle.
Theorem 4.3 Assume that (H1)-(H3) hold. In addition, let f, and satisfy the following conditions:
for each and all .
for all , .
here and are defined in (2.10), then problem (1.1) has a unique solution.
Consequently, we have , where Λ is defined by (4.14). As , therefore T is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle. The proof is complete. □
To illustrate how our main results can be used in practice, we present two examples.
where , , and are positive real numbers.
Conclusion Problem (5.1) has at least one solution in , where .
From the definition of ω, f, and , it is easy to see that (H1)-(H3) hold.
Thus, our conclusion follows from Theorem 4.1. □
where , , , , , and are positive real numbers with .
Conclusion Problem (5.4) has at least one solution in , where .
From the definition of ω, f, and , we can obtain that (H1)-(H3) hold.
Therefore, the conditions (H4) and (H5) of Theorem 4.2 are satisfied. Thus, Theorem 4.2 gives our conclusion. □
This work is sponsored by the project NSFC (11301178, 11171032). The authors are grateful to anonymous referees for their constructive comments and suggestions which have greatly improved this paper.
- Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.MATHGoogle Scholar
- Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.MATHGoogle Scholar
- Podlubny I Mathematics in Sciences and Engineering 198. In Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon; 1993.MATHGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge; 2009.MATHGoogle Scholar
- Graef JR, Kong L, Kong Q, Wang M: Theory of fractional functional differential equations. J. Appl. Anal. Comput. 2013, 3: 21-35.MathSciNetMATHGoogle Scholar
- Javidi M, Nyamoradi N: Dynamic analysis of a fractional order phytoplankton model. J. Appl. Anal. Comput. 2013, 3: 343-355.MathSciNetMATHGoogle Scholar
- Lakshmikantham V, Vatsala AS: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 2008, 21: 828-834. 10.1016/j.aml.2007.09.006MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Nieto JJ: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. 2008, 69: 3291-3298. 10.1016/j.na.2007.09.018MathSciNetView ArticleMATHGoogle Scholar
- Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. 2009., 2009: Article ID 708576 10.1155/2009/708576Google Scholar
- Jiang D, Yuan C: The positive properties of Green’s function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Anal. 2010, 72: 710-719. 10.1016/j.na.2009.07.012MathSciNetView ArticleMATHGoogle Scholar
- Bai Z, Lü H: Positive solutions of boundary value problems of nonlinear fractional differential equation. J. Math. Anal. Appl. 2005, 311: 495-505. 10.1016/j.jmaa.2005.02.052MathSciNetView ArticleMATHGoogle Scholar
- Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033MathSciNetView ArticleMATHGoogle Scholar
- Li C, Luo X, Zhou Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations. Comput. Math. Appl. 2010, 59: 1363-1375. 10.1016/j.camwa.2009.06.029MathSciNetView ArticleMATHGoogle Scholar
- Benchohra M, Hamani S, Ntouyas SK: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 2009, 71: 2391-2396. 10.1016/j.na.2009.01.073MathSciNetView ArticleMATHGoogle Scholar
- Salem HAH: On the nonlinear Hammerstein integral equations in Banach spaces and application to the boundary value problem of fractional order. Math. Comput. Model. 2008, 48: 1178-1190. 10.1016/j.mcm.2007.12.015MathSciNetView ArticleMATHGoogle Scholar
- Salem HAH: On the fractional order m -point boundary value problem in reflexive Banach spaces and weak topologies. J. Comput. Appl. Math. 2009, 224: 565-572. 10.1016/j.cam.2008.05.033MathSciNetView ArticleMATHGoogle Scholar
- Kaufmann ER, Mboumi E: Positive solutions of a boundary value problem for a nonlinear fractional differential equation. Electron. J. Qual. Theory Differ. Equ. 2008, 2008(3):1-11.MathSciNetView ArticleMATHGoogle Scholar
- Liu X, Jia M: Multiple solutions for fractional differential equations with nonlinear boundary conditions. Comput. Math. Appl. 2010, 59: 2880-2886. 10.1016/j.camwa.2010.02.005MathSciNetView ArticleMATHGoogle Scholar
- Jia M, Liu X: Three nonnegative solutions for fractional differential equations with integral boundary conditions. Comput. Math. Appl. 2011, 62: 1405-1412. 10.1016/j.camwa.2011.03.026MathSciNetView ArticleMATHGoogle Scholar
- Zhang S: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Comput. Math. Appl. 2010, 59: 1300-1309. 10.1016/j.camwa.2009.06.034MathSciNetView ArticleMATHGoogle Scholar
- Feng M, Zhang X, Ge W: New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions. Bound. Value Probl. 2011., 2011: Article ID 720702 10.1155/2011/720702Google Scholar
- Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.View ArticleMATHGoogle Scholar
- Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.MATHGoogle Scholar
- Benchohra M, Henderson J, Ntouyas SK: Impulsive Differential Equations and Inclusions. Hindawi, New York; 2006.View ArticleMATHGoogle Scholar
- Lin X, Jiang D: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl. 2006, 321: 501-514. 10.1016/j.jmaa.2005.07.076MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, O’Regan D: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput. 2000, 114: 51-59. 10.1016/S0096-3003(99)00074-0MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ, Lópe RR: Boundary value problems for a class of impulsive functional equations. Comput. Math. Appl. 2008, 55: 2715-2731. 10.1016/j.camwa.2007.10.019MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680-690. 10.1016/j.nonrwa.2007.10.022MathSciNetView ArticleMATHGoogle Scholar
- Li J, Nieto JJ: Existence of positive solutions for multipoint boundary value problem on the half-line with impulses. Bound. Value Probl. 2009., 2009: Article ID 834158 10.1155/2009/834158Google Scholar
- Shen J, Wang W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 2008, 69: 4055-4062. 10.1016/j.na.2007.10.036MathSciNetView ArticleMATHGoogle Scholar
- Liu Y, O’Regan D: Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 1769-1775. 10.1016/j.cnsns.2010.09.001MathSciNetView ArticleMATHGoogle Scholar
- Guo D: Multiple positive solutions of impulsive nonlinear Fredholm integral equations and applications. J. Math. Anal. Appl. 1993, 173: 318-324. 10.1006/jmaa.1993.1069MathSciNetView ArticleMATHGoogle Scholar
- Liu B, Yu J: Existence of solution of m -point boundary value problems of second-order differential systems with impulses. Appl. Math. Comput. 2002, 125: 155-175. 10.1016/S0096-3003(00)00110-7MathSciNetView ArticleMATHGoogle Scholar
- Feng M, Xie D: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Comput. Appl. Math. 2009, 223: 438-448. 10.1016/j.cam.2008.01.024MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, Benchohra M, Slimani BA: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys. 2008, 44: 1-21. 10.1134/S0012266108010011