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Boundary value problems for a new class of three-point nonlocal Riemann-Liouville integral boundary conditions
Advances in Difference Equations volume 2013, Article number: 213 (2013)
In this paper we investigate the existence and uniqueness of solutions for a new class of fractional boundary value problems involving three-point nonlocal Riemann-Liouville integral boundary conditions. Some new results are obtained by using fixed point theorems. Illustrative examples of our results are also presented.
Fractional order differential equations have been of great interest recently because they play a vital role in describing many phenomena related to physics, chemistry, mechanics, control systems, flow in porous media, electrical networks, mathematical biology and viscoelasticity. For a reader interested in the systematic development of the topic, we refer to the books [1–7]. 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 [8–19] and references cited therein.
Bai  discussed the existence of positive solutions for the following three-point fractional boundary value problem:
where denotes the Riemann-Liouville fractional derivative, and . Some existence results for at least one positive solution are obtained by the use of fixed point index theory.
In paper  the authors studied the following boundary value problem of fractional order differential equations with three-point fractional integral boundary conditions:
where denotes the Caputo fractional derivative of order q, is the Riemann-Liouville fractional integral of order p, and , . Existence and uniqueness results are proved via Banach’s contraction principle and Schaefer’s fixed point theorem. The results of  are completed in , by existence results via Krasnoselskii’s fixed point theorem and Leray-Schauder’s degree theory, and extended to cover the multivalued case.
In  existence and uniqueness results are obtained for a single and multivalued case for the following boundary value problem of fractional order differential equations with nonlocal and fractional integral boundary conditions:
Recently, in  the following boundary value problem of fractional differential equations with a fractional integral condition:
was studied. Existence and uniqueness results are proved via Banach’s contraction principle and Leray-Schauder’s nonlinear alternative.
In  existence and uniqueness results are investigated for the following boundary value problem of fractional order differential equations with four-point nonlocal Riemann-Liouville fractional integral boundary conditions:
where and .
In this paper, we study a new class of three-point boundary value problems of fractional order differential equations with nonlocal Riemann-Liouville fractional integral boundary conditions. More precisely, we consider the nonlinear fractional differential equation
subject to nonlocal fractional integral conditions
where is a given constant. The novelty of this boundary value problem lies in the fact that instead of the value , which appeared in all the above mentioned boundary value problems, we have the value for some .
The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel. In Section 3 we prove our main results. Some examples to illustrate our results are presented in Section 4.
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 .
Lemma 2.1 (see )
Let and . Then the fractional differential equation has a unique solution
where , and .
Lemma 2.2 Suppose that , , and . Then the problem
can be written as an integral equation
Proof From (2.1) and Lemma 2.1, we have
The first condition of (2.2) implies
Using the Riemann-Liouville fractional integral of order to (2.5) and applying Dirichlet’s formula [, p.56], we obtain
The second condition of (2.2) implies
Solving the linear equations (2.6)-(2.7) for unknown constants and , we have
where constant Ω is defined by (2.4). Substituting constants and in (2.5), we obtain (2.3). □
Let denote the Banach space of all continuous functions from to ℝ endowed with the norm defined by . As in Lemma 2.2, we define an operator by
It should be noticed that problem (1.6)-(1.7) 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 BVP (1.6)-(1.7) 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 BVP (1.6)-(1.7) by using Banach’s fixed point theorem.
Theorem 3.1 Assume that
(H1) there exists a constant such that for each and .
then problem (1.6)-(1.7) has a unique solution in .
Proof We transform problem (1.6)-(1.7) into a fixed point problem, , where the operator A is defined by (2.8). Obviously, fixed points of the operator A are solutions of problem (1.6)-(1.7). Using the Banach 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
From (3.1), A is a contraction. As a consequence of Banach’s fixed point theorem, we get that A has a fixed point which is a unique solution of problem (1.6)-(1.7). □
Now we give another existence and uniqueness result for BVP (1.6)-(1.7) by using Banach’s fixed point theorem and Hölder’s inequality.
Theorem 3.2 Suppose that the continuous function f satisfies the following assumption:
(H2) , for , , and , .
Denote . If
then boundary value problem (1.6)-(1.7) has a unique solution.
Proof For and for each , by Hölder’s inequality, we have
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 problem (1.6)-(1.7). This completes the proof. □
3.2 Existence result via Krasnoselskii’s fixed point theorem
Lemma 3.1 (Krasnoselskii’s fixed point theorem) 
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 satisfying (H1). Moreover, we assume that
(H3) , , and .
Then boundary value problem (1.6)-(1.7) has at least one solution on if
Proof Letting , we fix
and consider . We define the operators and on as
For , we find that
Thus, . It follows from assumption (H3) together with (3.2) 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 , and consequently we have
which is independent of u. 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 boundary value problem (1.6)-(1.7) has at least one solution on . □
3.3 Existence result via Leray-Schauder’s nonlinear alternative
Theorem 3.4 (Nonlinear alternative for single-valued maps) 
Let E be a Banach space, C be a closed, convex subset of E, U be an open subset of C and . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either
F has a fixed point in , or
there is a (the boundary of U in C) and with .
Theorem 3.5 Assume that
(H4) there exists a continuous nondecreasing function and a function such that
(H5) there exists a constant such that
Then boundary value problem (1.6)-(1.7) has at least one solution on .
Proof We show that A maps bounded sets (balls) into bounded sets in . For a positive number ρ, let be a bounded ball in . Then for we have
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 u be a solution. Then, for , and following similar computations as in the first step, we have
Consequently, we have
In view of (H5), there exists M 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 (Theorem 3.4), we deduce that A has a fixed point which is a solution of problem (1.6)-(1.7). This completes the proof. □
Example 4.1 Consider the following fractional integral boundary value problem:
Here , , , and and . Since , then (H1) is satisfied with . We can show that
Hence, by Theorem 3.1, boundary value problem (4.1)-(4.2) has a unique solution on .
Example 4.2 Consider the following fractional integral boundary value problem:
Set , , , , and choose . It is easy to see that . Since , then (H2) is satisfied with . We can show that
Hence, by Theorem 3.2, boundary value problem (4.3)-(4.4) has a unique solution on .
Example 4.3 Consider the following fractional integral boundary value problem:
Set , , , , . It is easy to see that . Clearly,
Choosing , , we obtain
which implies that . Hence, by Theorem 3.5, boundary value problem (4.5)-(4.6) has at least one solution on .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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The present paper was done while J Tariboon and T Sitthiwirattham visited the Department of Mathematics of the University of Ioannina, Greece. It is a pleasure for them to thank Professor SK Ntouyas for his warm hospitality. This research of J Tariboon and T Sitthiwirattham is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
The authors declare that they have no competing interests.
All authors contributed equally in this article. They read and approved the final manuscript.