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Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions
Advances in Difference Equations volume 2012, Article number: 93 (2012)
This article studies a boundary value problem of nonlinear fractional differential equations with three-point 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.
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 [1–6]. There has been a significant development in the theory of initial and boundary value problems for nonlinear fractional differential equations; see, for example, [7–15].
Ahmad and co-authors have studied the existence and uniqueness of solutions of nonlinear fractional differential and integro-differential equations for a variety of boundary conditions using standard fixed-point theorems and Leray-Schauder degree theory. Ahmad et al.  discusses the existence and uniqueness of solutions of fractional integro-differential equations for fractional nonlocal integral boundary conditions. Ahmad et al.  and references therein give details of recent work on the properties of solutions of sequential fractional differential equations. Ahmad et al.  considers solutions of fractional differential equations with non-separated type integral boundary conditions. In Ahmad et al. , 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.  discusses the existence and uniqueness of solutions of nonlinear fractional differential equations with three-point integral boundary conditions.
Cabada et al.  have also studied properties of solutions of nonlinear fractional differential equations. They used the properties of the associated Green’s function and the Guo-Krasnosellskii fixed-point theorem to investigate the existence of positive solutions of nonlinear fractional differential equations with integral boundary-value conditions.
Motivated by the papers  and , this article is concerned with the existence and uniqueness of solutions for a boundary value problem of nonlinear fractional differential equations with three-point fractional integral boundary conditions given by
where denotes the Caputo fractional derivative of order q, is the Riemann-Liouville 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 three-point integral condition. In such a case, the boundary condition corresponds to the area under the curve of solutions from to .
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 Riemann-Liouville fractional integral of order q for a continuous function is defined as
provided that such integral exists.
Definition 2.3 The Riemann-Liouville fractional derivative of order q for a continuous function is defined by
provided that the right-hand side is pointwise defined on .
Furthermore, we note that the Riemann-Liouville fractional derivative of a constant is usually nonzero which can cause serious problems in real would applications. Actually, the relationship between the two-types of fractional derivative is as follows
So, we prefer to use Caputo’s definition which gives better results than those of Riemann-Liouville.
Lemma 2.1 
Let , then the fractional differential equation
where n is the smallest integer greater than or equal to q.
Lemma 2.2 
Let , then
for some , where n is the smallest integer greater than or equal to q.
Lemma 2.3 Let , . Then for , the problem
has a unique solution
Proof By applying Lemma 2.2, we may reduce (2.1) to an equivalent integral equation
for some .
From , it follows . Using the Riemann-Liouville integral of order p for (2.4), we have
The second condition of (2.2) implies that
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 3.1 Assume that there exists a constant such that
() , for each , and all .
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.
Let . Then, for each we have
By using the property of beta function, , we have
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 3.2 Assume that:
() The function is continuous.
() There exists a constant such that for each and 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
which in view of () gives
Hence, we deduce that
Step III. We prove that is equicontinuous with defined as in Step II.
Let and , then
Actually, as , the right-hand side of the above inequality tends to zero. As a consequence of Steps I to III together with the Arzela-Ascoli theorem, we get that is completely continuous.
Step IV. A priori bounds.
We show that the set
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). □
In this section, in order to illustrate our results, we consider two examples.
Example 4.1 Consider the following three-point 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 three-point fractional integral boundary value problem
Set , , , and . Clearly .
Hence, all the conditions of Theorem 3.2 are satisfied and consequently the problem (4.1)-(4.2) has at least one solution.
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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.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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Sudsutad, W., Tariboon, J. Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions. Adv Differ Equ 2012, 93 (2012). https://doi.org/10.1186/1687-1847-2012-93
- Caputo fractional derivative
- Riemann-Liouville fractional integral
- boundary value problem