Abstract
This article studies a boundary value problem of nonlinear fractional differential equations with threepoint fractional integral boundary conditions. Some new existence results are obtained by applying standard fixed point theorems. As an application, we give two examples that illustrate our results.
MSC: 26A33, 34B15.
Keywords:
existence; Caputo fractional derivative; RiemannLiouville fractional integral; boundary value problem1 Introduction
Fractional differential equations have recently proved to be valuable tools in the modelling of many phenomena in various field of science and applications, such as physics, mechanics, chemistry, biology, economics, control theory, aerodynamics, engineering, etc. See [16]. There has been a significant development in the theory of initial and boundary value problems for nonlinear fractional differential equations; see, for example, [715].
Ahmad and coauthors have studied the existence and uniqueness of solutions of nonlinear fractional differential and integrodifferential equations for a variety of boundary conditions using standard fixedpoint theorems and LeraySchauder degree theory. Ahmad et al.[16] discusses the existence and uniqueness of solutions of fractional integrodifferential equations for fractional nonlocal integral boundary conditions. Ahmad et al.[17] and references therein give details of recent work on the properties of solutions of sequential fractional differential equations. Ahmad et al.[18] considers solutions of fractional differential equations with nonseparated type integral boundary conditions. In Ahmad et al.[19], the Krasnoselskii fixed point theorem and the contraction mapping principle are used to prove the existence of solutions of the nonlinear Langevin equation with two fractional orders for a number of different intervals. Ahmad et al.[20] discusses the existence and uniqueness of solutions of nonlinear fractional differential equations with threepoint integral boundary conditions.
Cabada et al.[21] have also studied properties of solutions of nonlinear fractional differential equations. They used the properties of the associated Green’s function and the GuoKrasnosellskii fixedpoint theorem to investigate the existence of positive solutions of nonlinear fractional differential equations with integral boundaryvalue conditions.
Motivated by the papers [16] and [20], this article is concerned with the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations with threepoint fractional integral boundary conditions given by
where denotes the Caputo fractional derivative of order q, is the RiemannLiouville fractional integral of order is a continuous function and is such that . By we denote the Banach space of all continuous functions from into ℝ with the norm
We note that if , then condition (1.2) reduces to the usual threepoint integral condition. In such a case, the boundary condition corresponds to the area under the curve of solutions from to .
2 Preliminaries
In this section, we introduce notations, definitions of fractional calculus and prove a lemma before stating our main results.
Definition 2.1 For a continuous function , the Caputo derivative of fractional order q is defined as
provided that exists, where denotes the integer part of the real number q.
Definition 2.2 The RiemannLiouville fractional integral of order q for a continuous function is defined as
provided that such integral exists.
Definition 2.3 The RiemannLiouville fractional derivative of order q for a continuous function is defined by
provided that the righthand side is pointwise defined on .
Furthermore, we note that the RiemannLiouville fractional derivative of a constant is usually nonzero which can cause serious problems in real would applications. Actually, the relationship between the twotypes of fractional derivative is as follows
So, we prefer to use Caputo’s definition which gives better results than those of RiemannLiouville.
Lemma 2.1[3]
Let, then the fractional differential equation
has solution
wherenis the smallest integer greater than or equal toq.
Lemma 2.2[3]
for some, wherenis the smallest integer greater than or equal to q.
Lemma 2.3Let, . Then for, the problem
has a unique solution
Proof By applying Lemma 2.2, we may reduce (2.1) to an equivalent integral equation
From , it follows . Using the RiemannLiouville integral of order p for (2.4), we have
The second condition of (2.2) implies that
Thus,
Substituting the values of and in (2.4), we obtain the solution (2.3). □
In the following, for the sake of convenience, set
3 Main results
Now we are in the position to establish the main results.
Theorem 3.1Assume that there exists a constantsuch that
If, where Λ is defined by (2.5), then the BVP (1.1)(1.2) has a unique solution on.
Proof Transform the BVP (1.1)(1.2) into a fixed point problem. In view of Lemma 2.3, we consider the operator defined by
Obviously, the fixed points of the operator F are solution of the problem (1.1)(1.2). We shall use the Banach fixed point theorem to prove that F has a fixed point. We will show that F is a contraction.
By using the property of beta function, , we have
Thus
Therefore, F is a contraction. Hence, by Banach fixed point theorem, we get that F has a fixed point which is a solution of the problem (1.1)(1.2). □
The following result is based on Schaefer’s fixed point theorem.
Theorem 3.2Assume that:
() There exists a constantsuch thatfor eachand all.
Then the BVP (1.1)(1.2) has at least one solution on.
Proof We shall use Schaefer’s fixed point theorem to prove that F has a fixed point. We divide the proof into four steps.
Step I. Continuity of F.
Let be a sequence such that in . Then for each
Since f is continuous function, then as . This means that F is continuous.
Step II. F maps bounded sets into bounded sets in .
So, let us prove that for any , there exists a positive constant l such that for each , we have . Indeed, we have for any
Hence, we deduce that
Thus,
Step III. We prove that is equicontinuous with defined as in Step II.
Actually, as , the righthand side of the above inequality tends to zero. As a consequence of Steps I to III together with the ArzelaAscoli theorem, we get that is completely continuous.
Step IV. A priori bounds.
We show that the set
is bounded.
Let . Then for some . Thus, for each we have
This implies by () that for all , we get
Hence, we deduce that
This shows that the set E is bounded. As a consequence of Schaefer’s fixed point theorem, we conclude that F has a fixed point which is a solution of the problem (1.1)(1.2). □
4 Examples
In this section, in order to illustrate our results, we consider two examples.
Example 4.1 Consider the following threepoint fractional integral boundary value problem
Set , , , and . Since , then, () is satisfied with . We can show that
Hence, by Theorem 3.1, the boundary value problem (4.1)(4.2) has a unique solution on .
Example 4.2 Consider the following threepoint fractional integral boundary value problem
Hence, all the conditions of Theorem 3.2 are satisfied and consequently the problem (4.1)(4.2) has at least one solution.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
The authors thank the referees for several useful remarks and interesting comments. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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