Abstract
In this paper we establish the existence and uniqueness of solutions for impulsive fractional boundary-value problems with fractional integral jump conditions. By using a variety of fixed-point theorems, some new existence and uniqueness results are obtained. Illustrative examples of our results are also presented.
MSC:34A08, 34B37, 34B15, 34B10.
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1 Introduction
In this paper, we investigate the following boundary-value problem for impulsive fractional differential equations with fractional integral jump conditions:
where is the Caputo fractional derivative of order α, , is a continuous function, , with , , are constants, is the Riemann-Liouville fractional integral of order for and , , a, b, c are given constants such that .
The integral jump conditions are very general and include many conditions as special cases. In particular, if and , then the impulsive fractional integral of equation (1.1) reduces to
Recently, much attention has been paid to the existence of solutions for fractional differential equations due to its wide application in engineering, economics and other fields. A variety of results on initial- and boundary-value problems of fractional differential equations and inclusions can easily be found in the literature on the topic. For some recent results, we can refer to [1–13] and references cited therein.
On the other hand, integer order impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. There has a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments, see for instance [14–24].
In this paper we prove some new existence and uniqueness results by using a variety of fixed-point theorems. In Theorem 3.1 we prove an existence and uniqueness result by using Banach’s contraction principle, in Theorem 3.2 we prove an existence and uniqueness result by using Banach’s contraction principle and Hölder’s inequality, in Theorem 3.3 we prove the existence of a solution by using Krasnoselskii’s fixed-point theorem, while in Theorem 3.4 we prove the existence of a solution via Leray-Schauder’s nonlinear alternative. Leray-Schauder’s degree theory is used in proving the existence result in Theorem 3.5.
The rest of the paper is organized as follows: In Section 2 we recall some preliminaries and present a basic lemma which is used to convert the impulsive fractional boundary-value problem (1.1) into an equivalent integral equation. The main results are presented in Section 3, while illustrative examples are contained in Section 4.
2 Preliminaries
Let = { is continuous everywhere except for some at which and exist and , }. is a Banach space endowed with the norm defined by . Next, we introduce some notations, definitions of fractional calculus [25–27], and we present a preliminary result needed in our proofs later.
Definition 2.1 The Riemann-Liouville fractional integral of order of a function is defined by
where Γ is the Gamma function.
Definition 2.2 The Riemann-Liouville fractional derivative of order of a continuous function is defined by
where , denotes the integral part of real number α, provided the right-hand side is point-wise defined on .
Definition 2.3 For a continuous function , the Caputo derivative of fractional order α is defined as
where , denotes the integral part of real number α, provided exists.
Lemma 2.1 ([28])
Let and be continuous. A function is a solution of the fractional Cauchy problem
if and only if x is a solution of the following integral equation:
Lemma 2.2 Let and . The unique solution of the impulsive fractional boundary-value problem (1.1) is given by
Proof For , Riemann-Liouville fractional integrating of order α, from 0 to t, for the first equation of (1.1), we have
Substituting into (2.2), we get
For , by Lemma 2.1 with the second equation of (1.1), we obtain
If then again from Lemma 2.1, we have
If then again from Lemma 2.1, we get
In particular, for , we have
From the third equation of (1.1) and (2.3), we get
Therefore, we have
This completes the proof. □
As in Lemma 2.2, we define an operator by
with . It should be noticed that problem (1.1) has solutions if and only if the operator A has fixed points.
3 Main results
We are in a position to establish our main results. In the following subsections we prove existence as well as existence and uniqueness results for the impulsive fractional BVP (1.1) by using a variety of fixed-point theorems.
3.1 Existence and uniqueness results via Banach’s fixed-point theorem
In this subsection we give first an existence and uniqueness result for the impulsive fractional BVP (1.1) by using Banach’s fixed-point theorem.
For convenience, we set
Theorem 3.1 Assume the following.
(H1) There exists a constant such that , for each and .
(H2) There exists a constant such that for each , .
If
then impulsive fractional boundary-value problem (1.1) has a unique solution in .
Proof We transform the problem (1.1) into a fixed-point problem, , where the operator A is defined by equation (2.4). Using Banach’s contraction principle, we shall show that A has a fixed point.
Setting , and choosing , we show that , where . For , we have
which proves that .
Now let . Then, for , we have
Therefore,
As follows from equation (3.4), A is a contraction. As a consequence of Banach’s fixed-point theorem, we have A has a fixed point which is a unique solution of the impulsive fractional boundary-value problem (1.1). This completes the proof. □
Now we give another existence and uniqueness result for impulsive fractional BVP (1.1) by using Banach’s fixed-point theorem and Hölder’s inequality. In addition, for , we set
Theorem 3.2 Assume that the following conditions hold:
(H3) , for each , , where , .
(H4) , for each , with constants , .
Denote .
If
then the impulsive fractional boundary-value problem (1.1) has a unique solution.
Proof For and for each , by Hölder’s inequality, we get
Therefore,
It follows that A is a contraction mapping. Hence Banach’s fixed-point theorem implies that A has a unique fixed point, which is the unique solution of the impulsive fractional boundary-value problem (1.1). This completes the proof. □
3.2 Existence result via Krasnoselskii’s fixed-point theorem
Lemma 3.1 (Krasnoselskii’s fixed point theorem) [29]
Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be the operators such that (a) whenever ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists such that .
Theorem 3.3 Let be a continuous function and let (H2) holds. In addition, we assume that:
(H5) , , and .
(H6) There exists a constant such that , , for .
Then the impulsive fractional boundary-value problem (1.1) has at least one solution on if
where Φ is defined by equation (3.2).
Proof We define and choose a suitable constant as
where Ω and Ψ are defined by equations (3.1) and (3.3), respectively. We define the operators and on as
For , we find that
Thus, . It follows from the assumption (H2) together with (3.8) that is a contraction mapping. Continuity of f implies that the operator is continuous. Also, is uniformly bounded on as
Now we prove the compactness of the operator .
We define , with and consequently we have
which is independent of x and tends to zero as . Thus, is equicontinuous. So is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus all the assumptions of Lemma 3.1 are satisfied. So the conclusion of Lemma 3.1 implies that the impulsive fractional boundary-value problem (1.1) has at least one solution on . The proof is completed. □
3.3 Existence result via Leray-Schauder’s Nonlinear Alternative
Lemma 3.2 (Nonlinear alternative for single valued maps) [30]
Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either
-
(i)
F has a fixed point in , or
-
(ii)
there is a (the boundary of U in C) and with .
Theorem 3.4 Assume the following.
(H7) There exist a continuous nondecreasing function and a function such that
(H8) There exists a continuous nondecreasing function such that
(H9) There exists a constant such that
Then the impulsive fractional boundary-value problem (1.1) has at least one solution on .
Proof We show that A maps bounded sets (balls) into bounded sets in . For a positive number r, let be a bounded ball in . Then for we have
Consequently
Next we show that A maps bounded sets into equicontinuous sets of . Let , with , , , and . Then we have
Obviously the right-hand side of the above inequality tends to zero independently of as . As A satisfies the above assumptions, therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
Let x be a solution. Then, for , and following the similar computations as in the first step, we have
Consequently, we have
In view of (H9), there exists such that . Let us set
Note that the operator is continuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.2), we deduce that A has a fixed point which is a solution of the problem (1.1). This completes the proof. □
3.4 Existence result via Leray-Schauder degree
Theorem 3.5 Assume the following.
(H10) There exist constants and such that
(H11) There exist constants and such that
where Ω and Φ are given by equations (3.1) and (3.2), respectively.
Then the impulsive fractional boundary-value problem (1.1) has at least one solution on .
Proof We define an operator as in equation (2.4) and consider the fixed-point problem
We are going to prove that there exists a fixed point satisfying equation (3.9). It is sufficient to show that satisfies
where . We define
As shown in Theorem 3.4, we find that the operator A is continuous, uniformly bounded, and equicontinuous. Then, by the Arzelá-Ascoli theorem, a continuous map defined by is completely continuous. If equation (3.10) is true, then the following Leray-Schauder degrees are well defined and by the homotopy invariance of topological degree, it follows that
where I denotes the identity operator. By the nonzero property of the Leray-Schauder degree, for at least one . In order to prove equation (3.10), we assume that for some . Then
Computing directly for , we have
If , inequality (3.10) holds. This completes the proof. □
4 Examples
In this section we give examples to illustrate our results.
Example 4.1 Consider the following impulsive fractional boundary-value problem:
where , .
Set , , , , , , , , , and .
Since and for , then (H1) and (H2) are satisfied with and . We can show that
Hence, by Theorem 3.1, the boundary-value problem (4.1)-(4.3) has a unique solution on .
Example 4.2 Consider the following impulsive fractional boundary-value problem:
where , .
Set , , , , , , , , , and .
Since , and , then (H3) and (H4) are satisfied with , , , and . We can show that
Hence, by Theorem 3.2, the boundary-value problem (4.4)-(4.6) has a unique solution on .
Example 4.3 Consider the following impulsive fractional boundary-value problem:
where , .
Set , , , , , , , , , , and .
It is easy to see that . Clearly,
and
Choosing , and , we obtain
which implies that . Hence, by Theorem 3.4, the boundary-value problem (4.7)-(4.9) has at least one solution on .
Example 4.4 Consider the following impulsive fractional boundary-value problem:
where , , .
Set , , , , , , , , , , , , , , , and .
Since , for , then (H10) and (H11) are satisfied with , , and . We have
and
Hence, by Theorem 3.5, the boundary value-problem (4.10)-(4.12) has at least one solution on .
Authors’ information
The third author is a Member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
References
Guezane-Lakoud A, Khaldi R: Positive solution to a higher order fractional boundary value problem with fractional integral condition. Rom. J. Math. Comput. Sci. 2012, 2: 41-54.
Kaufmann E: Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Discrete Contin. Dyn. Syst. 2009, 2009: 416-423. suppl.
Wang J, Xiang H, Liu Z: Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Int. J. Differ. Equ. 2010., 2010: Article ID 186928
Benchohra M, Henderson J, Ntouyas SK, Ouahab A: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 2008, 338: 1340-1350. 10.1016/j.jmaa.2007.06.021
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
Sudsutad W, Tariboon J: Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions. Adv. Differ. Equ. 2012., 2012: Article ID 93
Ntouyas SK: Existence results for nonlocal boundary value problems for fractional differential equations and inclusions with fractional integral boundary conditions. Discuss. Math., Differ. Incl. Control Optim. 2013, 33: 17-39. 10.7151/dmdico.1146
Ntouyas SK: Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions. Opusc. Math. 2013, 33: 117-138. 10.7494/OpMath.2013.33.1.117
Guezane-Lakoud A, Khaldi R: Solvability of a fractional boundary value problem with fractional integral condition. Nonlinear Anal. 2012, 75: 2692-2700. 10.1016/j.na.2011.11.014
Ahmad B, Ntouyas SK, Assolani A: Caputo type fractional differential equations with nonlocal Riemann-Liouville integral boundary conditions. J. Appl. Math. Comput. 2013, 41: 339-350. 10.1007/s12190-012-0610-8
Baleanu D, Mustafa OG, Agarwal RP: An existence result for a superlinear fractional differential equation. Appl. Math. Lett. 2010, 23: 1129-1132. 10.1016/j.aml.2010.04.049
Debbouche A, Baleanu D, Agarwal RP: Nonlocal nonlinear integrodifferential equations of fractional orders. Bound. Value Probl. 2012., 2012: Article ID 78
Nyamoradi N, Baleanu D, Agarwal RP: On a multipoint boundary value problem for a fractional order differential inclusion on an infinite interval. Adv. Math. Phys. 2013., 2013: Article ID 823961
Agarwal RP, Ahmad B: Existence of solutions for impulsive anti-periodic boundary value problems of fractional semilinear evolution equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 2011, 18: 535-544.
Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.
Benchohra M, Henderson J, Ntouyas SK: Impulsive Differential Equations and Inclusions. Hindawi Publishing, New York; 2006.
Ahmad B, Nieto JJ: Existence of solutions for impulsive anti-periodic boundary value problems of fractional order. Taiwan. J. Math. 2011, 15: 981-993.
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
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
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
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
Zhang X, Huang X, Liu Z: The existence and uniqueness of mild solutions for impulsive fractional equations with nonlocal conditions and infinite delay. Nonlinear Anal. Hybrid Syst. 2010, 4: 775-781. 10.1016/j.nahs.2010.05.007
Miller KS, Ross B: An Introduction to the Fractional Calculus and Differential Equations. Wiley, New York; 1993.
Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon; 1993.
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Fec̆kan M, Zhou Y, Wang J: On the concept and existence of solution for impulsive fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 3050-3060. 10.1016/j.cnsns.2011.11.017
Krasnoselskii MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 1955, 10: 123-127.
Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.
Acknowledgements
We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research of CT and JT is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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Thaiprayoon, C., Tariboon, J. & Ntouyas, S.K. Impulsive fractional boundary-value problems with fractional integral jump conditions. Bound Value Probl 2014, 17 (2014). https://doi.org/10.1186/1687-2770-2014-17
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DOI: https://doi.org/10.1186/1687-2770-2014-17