Abstract
In this paper we establish the existence and uniqueness of solutions for impulsive fractional boundaryvalue problems with fractional integral jump conditions. By using a variety of fixedpoint theorems, some new existence and uniqueness results are obtained. Illustrative examples of our results are also presented.
MSC: 34A08, 34B37, 34B15, 34B10.
Keywords:
impulsive fractional differential equations; fractional integral jump conditions; fixedpoint theorems1 Introduction
In this paper, we investigate the following boundaryvalue 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 RiemannLiouville 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 boundaryvalue 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 [113] 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 [1424].
In this paper we prove some new existence and uniqueness results by using a variety of fixedpoint 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 fixedpoint theorem, while in Theorem 3.4 we prove the existence of a solution via LeraySchauder’s nonlinear alternative. LeraySchauder’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 boundaryvalue 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 [2527], and we present a preliminary result needed in our proofs later.
Definition 2.1 The RiemannLiouville fractional integral of order of a function is defined by
where Γ is the Gamma function.
Definition 2.2 The RiemannLiouville fractional derivative of order of a continuous function is defined by
where , denotes the integral part of real number α, provided the righthand side is pointwise 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])
Letandbe continuous. A functionis a solution of the fractional Cauchy problem
if and only ifxis a solution of the following integral equation:
Lemma 2.2Letand. The unique solution of the impulsive fractional boundaryvalue problem (1.1) is given by
Proof For , RiemannLiouville 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
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 fixedpoint theorems.
3.1 Existence and uniqueness results via Banach’s fixedpoint theorem
In this subsection we give first an existence and uniqueness result for the impulsive fractional BVP (1.1) by using Banach’s fixedpoint theorem.
For convenience, we set
Theorem 3.1Assume the following.
(H_{1}) There exists a constantsuch that, for eachand.
(H_{2}) There exists a constantsuch thatfor each, .
If
then impulsive fractional boundaryvalue problem (1.1) has a unique solution in.
Proof We transform the problem (1.1) into a fixedpoint 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
Therefore,
As follows from equation (3.4), A is a contraction. As a consequence of Banach’s fixedpoint theorem, we have A has a fixed point which is a unique solution of the impulsive fractional boundaryvalue 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 fixedpoint theorem and Hölder’s inequality. In addition, for , we set
Theorem 3.2Assume that the following conditions hold:
(H_{3}) , for each, , where, .
(H_{4}) , for each, with constants, .
If
then the impulsive fractional boundaryvalue 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 fixedpoint theorem implies that A has a unique fixed point, which is the unique solution of the impulsive fractional boundaryvalue problem (1.1). This completes the proof. □
3.2 Existence result via Krasnoselskii’s fixedpoint theorem
Lemma 3.1 (Krasnoselskii’s fixed point theorem) [29]
LetMbe a closed, bounded, convex and nonempty subset of a Banach spaceX. LetA, Bbe the operators such that (a) whenever; (b) Ais compact and continuous; (c) Bis a contraction mapping. Then there existssuch that.
Theorem 3.3Letbe a continuous function and let (H_{2}) holds. In addition, we assume that:
(H_{6}) There exists a constantsuch that, , for.
Then the impulsive fractional boundaryvalue problem (1.1) has at least one solution onif
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
Thus, . It follows from the assumption (H_{2}) 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 ArzeláAscoli 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 boundaryvalue problem (1.1) has at least one solution on . The proof is completed. □
3.3 Existence result via LeraySchauder’s Nonlinear Alternative
Lemma 3.2 (Nonlinear alternative for single valued maps) [30]
LetEbe a Banach space, Ca closed, convex subset ofE, Uan open subset ofCand. Suppose thatis a continuous, compact (that is, is a relatively compact subset ofC) map. Then either
(ii) there is a (the boundary ofUinC) andwith.
Theorem 3.4Assume the following.
(H_{7}) There exist a continuous nondecreasing functionand a functionsuch that
(H_{8}) There exists a continuous nondecreasing functionsuch that
(H_{9}) There exists a constantsuch that
Then the impulsive fractional boundaryvalue problem (1.1) has at least one solution on.
Proof We show that Amaps 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 Amaps bounded sets into equicontinuous sets of. Let , with , , , and . Then we have
Obviously the righthand side of the above inequality tends to zero independently of as . As A satisfies the above assumptions, therefore it follows by the ArzeláAscoli 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 (H_{9}), 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 LeraySchauder 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 LeraySchauder degree
Theorem 3.5Assume the following.
(H_{10}) There exist constantsandsuch that
(H_{11}) There exist constantsandsuch that
where Ω and Φ are given by equations (3.1) and (3.2), respectively.
Then the impulsive fractional boundaryvalue problem (1.1) has at least one solution on.
Proof We define an operator as in equation (2.4) and consider the fixedpoint problem
We are going to prove that there exists a fixed point satisfying equation (3.9). It is sufficient to show that satisfies
As shown in Theorem 3.4, we find that the operator A is continuous, uniformly bounded, and equicontinuous. Then, by the ArzeláAscoli theorem, a continuous map defined by is completely continuous. If equation (3.10) is true, then the following LeraySchauder 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 LeraySchauder degree, for at least one . In order to prove equation (3.10), we assume that for some . Then
Computing directly for , we have
4 Examples
In this section we give examples to illustrate our results.
Example 4.1 Consider the following impulsive fractional boundaryvalue problem:
Since and for , then (H_{1}) and (H_{2}) are satisfied with and . We can show that
Hence, by Theorem 3.1, the boundaryvalue problem (4.1)(4.3) has a unique solution on .
Example 4.2 Consider the following impulsive fractional boundaryvalue problem:
Since , and , then (H_{3}) and (H_{4}) are satisfied with , , , and . We can show that
Hence, by Theorem 3.2, the boundaryvalue problem (4.4)(4.6) has a unique solution on .
Example 4.3 Consider the following impulsive fractional boundaryvalue problem:
It is easy to see that . Clearly,
and
which implies that . Hence, by Theorem 3.4, the boundaryvalue problem (4.7)(4.9) has at least one solution on .
Example 4.4 Consider the following impulsive fractional boundaryvalue problem:
Set , , , , , , , , , , , , , , , and .
Since , for , then (H_{10}) and (H_{11}) are satisfied with , , and . We have
and
Hence, by Theorem 3.5, the boundary valueproblem (4.10)(4.12) has at least one solution on .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in this article. They read and approved the final manuscript.
Authors’ information
The third author is a Member of Nonlinear Analysis and Applied Mathematics (NAAM)Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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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