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Successive iteration and positive solutions of a fractional boundary value problem on the half-line
Advances in Difference Equations volume 2013, Article number: 210 (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).
Most of these papers only considered the existence of positive solutions of various boundary value problems. Yet how can we find the solutions when their existence is known? Sun et al.  proved the existence of positive solutions for second-order p-Laplacian boundary value problems which are defined on finite intervals via the iterative technique. Liang and Zhang  investigated the existence of three positive solutions for the following m-point fractional boundary value problem on an infinite interval by means of the Leggett-Williams fixed point theorems on cones. In [14, 15], by employing the standard fixed point theorems and the monotone iterative technique, the authors deal with the existence of solutions for nonlinear fractional boundary value problems on an unbounded domain. Motivated by all the works above, we investigate the iteration and existence of positive solutions for the following fractional boundary value problems on the half-line:
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 
For a function given in the interval , the expression
where , denotes the integer part of number α, is called the Riemann-Liouville fractional derivative of order α.
Lemma 2.1 
Let and . Then the fractional differential equation
as a unique solution.
Lemma 2.2 
Assume that with a fractional derivative of order that belongs to . Then
for some , , .
Lemma 2.3 Let , then the fractional boundary value problem
has a unique solution
Proof By Lemma 2.2, the solution of (2.1) can be written as
From , we know that .
Together with , we have
Therefore, the unique solution of fractional boundary value problem (2.1) is
The proof is complete. □
Lemma 2.4 The function defined by (2.2) satisfies the following:
is a continuous function and for ;
, for .
The proof is easy, so we omit it here.
In this paper, we use the following space E, which is denoted by
to study fractional boundary value problem (1.1). From , we know that E is a Banach space equipped with the norm . Define the cone by
and an integral operator by
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.
Remark 2.5 is called equiconvergent at infinity if and only if for all , there exists such that for all , , the following holds:
Lemma 2.6 Let (H1) and (H2) hold, then is completely continuous.
Proof First, it is easy to check that is well defined. Now, we prove that T is continuous and compact respectively. Let as in P, then there exists such that . Let . By (H2), we have
Then by the Lebesgue dominated convergence theorem and the continuity of f, we can get
Therefore, we have
Then T is continuous.
Let Ω be any bounded subset of P. Then there exists such that for any . Let , therefore, from Lemma 2.4, we have
So, T Ω is bounded. Moreover, for any and , without loss of generality, we may assume that , we have
On the other hand, we have
Similarly, we have
Hence, T Ω is locally equicontinuous on .
Next, we show that is equiconvergent at ∞. For any , we have
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 fractional boundary value problem (1.1) has two positive solutions and on , satisfying , , , and
Proof By Lemma 2.6, we know that is completely continuous. For any with , from the definition of T and (H3), we can easily get that . We denote
Then, in what follows, we first prove that .
If , then , that is, , . Then assumption (H4) implies for all , together with (2.3), (H3), we get
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.
By (2.3) and (H4), we get
So, by (H3), we have
By induction, we get
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.
Since , we have
So, we have
By induction, we get
Thus, there exists such that as . Applying the continuity of T and , we get .
Since , and the operator T is increasing, then
by induction, we have
Together with (3.1), (3.2), we obtain
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
Example 4.1 Consider the boundary value problem of the fractional differential equation.
In this case, , , , . It is clear that (H1)-(H3) hold. Select , by direct calculation, we can obtain
The conditions in Theorem 3.1 are all satisfied. Therefore, fractional boundary value problem (4.1) has two positive solutions.
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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).
The authors declare that they have no competing interests.
WX and JX conceived of the studies, and drafted the manuscript. ZL participated in the discussion. All authors read and approved the final manuscript.