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
The integral jump conditions are very general and include many conditions as special
cases. In particular, if
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
Definition 2.1 The RiemannLiouville fractional integral of order
where Γ is the Gamma function.
Definition 2.2 The RiemannLiouville fractional derivative of order
where
Definition 2.3 For a continuous function
where
Lemma 2.1 ([28])
Let
if and only ifxis a solution of the following integral equation:
Lemma 2.2Let
Proof For
Substituting
For
If
If
In particular, for
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
with
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 constant
(H_{2}) There exists a constant
If
then impulsive fractional boundaryvalue problem (1.1) has a unique solution in
Proof We transform the problem (1.1) into a fixedpoint problem,
Setting
which proves that
Now let
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
Theorem 3.2Assume that the following conditions hold:
(H_{3})
(H_{4})
Denote
If
then the impulsive fractional boundaryvalue problem (1.1) has a unique solution.
Proof For
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)
Theorem 3.3Let
(H_{5})
(H_{6}) There exists a constant
Then the impulsive fractional boundaryvalue problem (1.1) has at least one solution on
where Φ is defined by equation (3.2).
Proof We define
where Ω and Ψ are defined by equations (3.1) and (3.3), respectively. We define the
operators and on
For
Thus,
Now we prove the compactness of the operator .
We define
which is independent of x and tends to zero as
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
(i) Fhas a fixed point in
(ii) there is a
Theorem 3.4Assume the following.
(H_{7}) There exist a continuous nondecreasing function
(H_{8}) There exists a continuous nondecreasing function
(H_{9}) There exists a constant
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
Consequently
Next we show that Amaps bounded sets into equicontinuous sets of
Obviously the righthand side of the above inequality tends to zero independently
of
Let x be a solution. Then, for
Consequently, we have
In view of (H_{9}), there exists
Note that the operator
3.4 Existence result via LeraySchauder degree
Theorem 3.5Assume the following.
(H_{10}) There exist constants
(H_{11}) There exist constants
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
We are going to prove that there exists a fixed point
where
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
where I denotes the identity operator. By the nonzero property of the LeraySchauder degree,
Computing directly for
If
4 Examples
In this section we give examples to illustrate our results.
Example 4.1 Consider the following impulsive fractional boundaryvalue problem:
where
Set
Since
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:
where
Set
Since
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:
where
Set
It is easy to see that
and
Choosing
which implies that
Example 4.4 Consider the following impulsive fractional boundaryvalue problem:
where
Set
Since
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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