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Successive iteration and positive solutions of a fractional boundary value problem on the half-line
© Xie et al.; licensee Springer 2013
Received: 17 April 2013
Accepted: 28 June 2013
Published: 12 July 2013
In this paper, we study the existence of positive solutions for a nonlinear fractional boundary value problem on the half-line. Based on the monotone iterative technique, we obtain the existence of positive solutions of a fractional boundary value problem and establish iterative schemes for approximating the solutions. As application, an example is presented to illustrate the main results.
MSC:34A08, 34A12, 34B40.
The initial and boundary value problems for nonlinear fractional differential equations arise from the study of models of viscoelasticity, control, porous media, etc. [1, 2]. In the past few years, the existence and multiplicity of positive solutions for nonlinear fractional boundary value problems have been widely studied by many authors (see [3–11] and the references therein).
where , is the standard Riemann-Liouville fractional derivative. By applying monotone iterative techniques, we construct some successive iterative schemes to approximate the solutions in this paper. They start off with a simple function and the zero function respectively, which is convenient for application.
Throughout this paper, we assume that the following conditions hold:
(H1) , on any subinterval of and when u is bounded, is bounded on ;
(H2) is not identical zero on any closed subinterval of , and .
We need the following definitions and lemmas that will be used to prove our main results.
Definition 2.1 
where , is called the Riemann-Liouville fractional integral of order α and is the Euler gamma function defined by , .
Definition 2.2 
where , denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order α.
Lemma 2.1 
as a unique solution.
Lemma 2.2 
for some , , .
From , we know that .
The proof is complete. □
is a continuous function and for ;
, for .
The proof is easy, so we omit it here.
where is defined by (2.2).
Lemma 2.5 
Let (), . If is equicontinuous on any compact intervals of and equiconvergent at infinity, then V is relatively compact on E.
Lemma 2.6 Let (H1) and (H2) hold, then is completely continuous.
Then T is continuous.
Hence, T Ω is locally equicontinuous on .
Hence, T Ω is equiconvergent at infinity. By using Lemma 2.5, we obtain that is completely continuous. The proof is complete. □
3 Main results
We will prove the following existence results.
Theorem 3.1 Assume that (H1) and (H2) hold, and there exists such that:
(H3) for any , ;
(H4) , .
Then, in what follows, we first prove that .
Hence, we have proved that .
Let , , then . Let , ; then by Lemma 2.6, we have that and . We denote , . Since , we have that , . It follows from the complete continuity of T that is a sequentially compact set.
Thus, there exists such that as . Applying the continuity of T and , we get that .
Let , ; then . Let , ; then by Lemma 2.6, we know that and . We denote , . Since , we have , . It follows from the complete continuity of T that is a sequentially compact set.
Thus, there exists such that as . Applying the continuity of T and , we get .
If , , then the zero function is not the solution of fractional boundary value problem (1.1). Thus, is a positive solution of fractional boundary value problem (1.1).
It is well known that each fixed point of T in P is a solution of fractional boundary value problem (1.1). Hence, and are two positive solutions on of fractional boundary value problem (1.1), satisfying . □
4 An example
The conditions in Theorem 3.1 are all satisfied. Therefore, fractional boundary value problem (4.1) has two positive solutions.
This work is supported by the National Nature Science Foundation of P.R. China (10871063), partially supported by the National Nature Science Foundation of P.R. China (61170320), the Foundation of Guangdong Natural Science (S2011040002981), and the Nature Science Foundation of Guangdong Medical College (B2012053).
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