Boundary value problems for a new class of three-point nonlocal Riemann-Liouville integral boundary conditions
© Tariboon et al.; licensee Springer 2013
Received: 22 April 2013
Accepted: 28 June 2013
Published: 13 July 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.
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.
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.
was studied. Existence and uniqueness results are proved via Banach’s contraction principle and Leray-Schauder’s nonlinear alternative.
where and .
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.
where Γ is the gamma function.
where , denotes the integral part of real number α, provided the right-hand side is point-wise defined on .
Lemma 2.1 (see )
where , and .
where constant Ω is defined by (2.4). Substituting constants and in (2.5), we obtain (2.3). □
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.
which proves that .
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 , .
then boundary value problem (1.6)-(1.7) has a unique solution.
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 .
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) 
F has a fixed point in , or
there is a (the boundary of U in C) and with .
Theorem 3.5 Assume that
Then boundary value problem (1.6)-(1.7) has at least one solution on .
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.
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. □
Hence, by Theorem 3.1, boundary value problem (4.1)-(4.2) has a unique solution on .
Hence, by Theorem 3.2, boundary value problem (4.3)-(4.4) has a unique solution on .
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.
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.
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