- Research
- Open access
- Published:
Green’s function for Sturm-Liouville-type boundary value problems of fractional order impulsive differential equations and its application
Boundary Value Problems volume 2014, Article number: 69 (2014)
Abstract
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.
1 Introduction
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. [6]. 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.
Consider the following nonlinear boundary value problem of fractional impulsive differential equations:
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 [13] 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).
At the end of this section, it is worth mentioning that it is an important method to express the solution of differential equations by Green’s function. According to the previous work, we find that the solution of impulsive differential equations with integer order can be expressed by Green’s function of the case without impulse. For example, Green’s function of the following boundary value problem
can be expressed by
where . The solution of problem (1.2) can be expressed by
If we consider impulsive differential equations
then the solution of problem (1.3) can be expressed by
where , .
Naturally, one wishes to know whether or not the same result holds for the fractional order case. We first study the fractional order differential equations with Caputo derivatives
where is the Caputo fractional derivative. Then
where
Then we study whether the solution of fractional order impulsive differential equations with Caputo derivatives
can be expressed by
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.
2 Expression and properties of Green’s function
Consider the following fractional impulsive boundary value problem:
where is the Caputo fractional derivative, , are continuous functions, and . with , , , . has a similar meaning for .
Theorem 2.1 The solution of problem (2.1) can be expressed by
where
and
Proof Suppose that x is a solution of (2.1). Then, for some constants , we have
It follows from (2.5) that
If , then, for some constants , we can write
Using the impulse conditions and , we find that
Thus
If , repeating the above procedure, we obtain
It follows that , and
By the boundary conditions, we have
where η is defined in (2.4).
Substituting (2.7), (2.8) into (2.6), we obtain (2.2). This completes the proof. □
Remark 2.1 It is clear that is Green’s function of the boundary value problem
Remark 2.2 The expression of the solution of problem (2.1) is simpler than that of [37–46].
From (2.3), we can prove the following results.
Proposition 2.1 For all , we have
Proposition 2.2 For all , we have
and
where
and
For the sake of convenience, let
Then it follows from (2.9) and (2.10) that
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.
3 Preliminaries
In this section, we give some preliminaries for discussing the solvability of problem (1.1) as follows.
Let and
Then is a Banach space with the norm
is a Banach space with the norm
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:
Lemma 3.1 If (H1)-(H3) hold, then a function is a solution of problem (1.1) if and only if is a solution of the impulsive fractional integral equation
Define by
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.
Let be bounded. Then there exist positive constants , and such that , and , . Thus, , we have
where
Furthermore, for any , , we obtain
On the other hand, from (3.5), for with , we have
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.
Lemma 3.3 (See [47, 48]) (Schauder fixed point theorem)
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 .
Lemma 3.4 (See [47, 48]) (Leray-Schauder fixed point theorem)
Let X be a real Banach space and be a completely continuous operator. If
is bounded, then T has a fixed point , where
4 Existence of solutions
In this section, we apply Lemma 3.3, Lemma 3.4 and the contraction mapping principle to establish the existence of solutions of problem (1.1). Let us begin by introducing some notation. Define
Theorem 4.1 Assume that (H1)-(H3) hold. Suppose further that
where
and
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).
On account of (4.1), we can choose , and such that
and
By the definition of ξ, there exists such that
so
where
Similarly, we have
and
where , are positive constants.
It follows from (3.2) and (4.4)-(4.6) that
where is defined by (4.2) and is defined by
Similarly, from (3.2) and (4.4)-(4.6), we get
where is defined by (4.3) and is defined by
It follows from (4.7) and (4.8) that
where
Hence, we can choose a sufficiently large such that , where
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:
(H4) There exists a nonnegative function with on a subinterval of such that
(H5) There exist constants such that
for all , .
Then problem (1.1) has at least one solution.
Proof It follows from Lemma 3.2 that the operator is completely continuous.
Next, we show that the set
is bounded.
Let . Then for . For any , we have
It follows from (H4), (H5), (3.2) and (4.9) that
where
It follows from (4.10) that
Furthermore, for any , , we obtain
which, for any , yields
It follows from (4.11) and (4.12) that
where , here
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:
(H6) There exists a constant such that
for each and all .
(H7) There exist constants such that
for all , .
If
where
here and are defined in (2.10), then problem (1.1) has a unique solution.
Proof Let . Then, for each , it follows from (H6), (H7) and (3.2) that
and
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. □
5 Examples
To illustrate how our main results can be used in practice, we present two examples.
Example 5.1 Let , . We consider the following boundary value problem:
where , , and are positive real numbers.
Conclusion Problem (5.1) has at least one solution in , where .
Proof It follows from (5.1) that
From the definition of ω, f, and , it is easy to see that (H1)-(H3) hold.
On the other hand, it follows from (5.2) and (5.3) that
and
So , , . Noticing that , , we have and
Therefore, , , and therefore (4.1) is satisfied because
Thus, our conclusion follows from Theorem 4.1. □
Example 5.2 Let , . We consider the following boundary value problem:
where , , , , , and are positive real numbers with .
Conclusion Problem (5.4) has at least one solution in , where .
Proof It follows from (5.4) that
From the definition of ω, f, and , we can obtain that (H1)-(H3) hold.
On the other hand, from (5.5) and (5.6) we have
and
Therefore, the conditions (H4) and (H5) of Theorem 4.2 are satisfied. Thus, Theorem 4.2 gives our conclusion. □
References
Miller KS, Ross B: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York; 1993.
Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.
Podlubny I Mathematics in Sciences and Engineering 198. In Fractional Differential Equations. Academic Press, San Diego; 1999.
Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon; 1993.
Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Lakshmikantham V, Leela S, Vasundhara Devi J: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge; 2009.
Graef JR, Kong L, Kong Q, Wang M: Theory of fractional functional differential equations. J. Appl. Anal. Comput. 2013, 3: 21-35.
Javidi M, Nyamoradi N: Dynamic analysis of a fractional order phytoplankton model. J. Appl. Anal. Comput. 2013, 3: 343-355.
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.2007.09.006
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.2007.09.018
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 10.1155/2009/708576
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.2009.07.012
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.2005.02.052
Bai Z: On positive solutions of a nonlocal fractional boundary value problem. Nonlinear Anal. 2010, 72: 916-924. 10.1016/j.na.2009.07.033
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.2009.06.029
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.2009.01.073
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.2007.12.015
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.2008.05.033
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.
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.2010.02.005
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.2011.03.026
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.2009.06.034
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 10.1155/2011/720702
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.
Benchohra M, Henderson J, Ntouyas SK: Impulsive Differential Equations and Inclusions. Hindawi, New York; 2006.
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.2005.07.076
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)00074-0
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.2007.10.019
Nieto JJ, O’Regan D: Variational approach to impulsive differential equations. Nonlinear Anal., Real World Appl. 2009, 10: 680-690. 10.1016/j.nonrwa.2007.10.022
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 10.1155/2009/834158
Shen J, Wang W: Impulsive boundary value problems with nonlinear boundary conditions. Nonlinear Anal. 2008, 69: 4055-4062. 10.1016/j.na.2007.10.036
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.2010.09.001
Guo D: Multiple positive solutions of impulsive nonlinear Fredholm integral equations and applications. J. Math. Anal. Appl. 1993, 173: 318-324. 10.1006/jmaa.1993.1069
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)00110-7
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.2008.01.024
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
Ahmad B, Sivasundaram S: Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlinear Anal. Hybrid Syst. 2009, 3: 251-258. 10.1016/j.nahs.2009.01.008
Ahmad B, Sivasundaram S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Anal. Hybrid Syst. 2010, 4: 134-141. 10.1016/j.nahs.2009.09.002
Mophou GM: Existence and uniqueness of mild solutions to impulsive fractional differential equations. Nonlinear Anal. 2010, 72: 1604-1615. 10.1016/j.na.2009.08.046
Chang YK, Nieto JJ: Existence of solutions for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. Numer. Funct. Anal. Optim. 2009, 30: 227-244. 10.1080/01630560902841146
Abbas S, Benchohra M: Upper and lower solutions method for impulsive partial hyperbolic differential equations with fractional order. Nonlinear Anal. Hybrid Syst. 2010, 4: 406-413. 10.1016/j.nahs.2009.10.004
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
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
Wang G, Ahmad B, Zhang L: Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions. Comput. Math. Appl. 2011, 62: 1389-1397. 10.1016/j.camwa.2011.04.004
Yang L, Chen H: Nonlocal boundary value problem for impulsive differential equations of fractional order. Adv. Differ. Equ. 2011., 2011: Article ID 404917 10.1155/2011/404917
Zeidler E: Nonlinear Functional Analysis and Its Applications - I: Fixed-Point Theorems. Springer, New York; 1986.
Guo D: Nonlinear Functional Analysis. Shandong Science and Technology Press, Jinan; 1985. (in Chinese)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All results belong to JZ and MF. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zhou, J., Feng, M. Green’s function for Sturm-Liouville-type boundary value problems of fractional order impulsive differential equations and its application. Bound Value Probl 2014, 69 (2014). https://doi.org/10.1186/1687-2770-2014-69
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2014-69