Existence and uniqueness of positive solutions to fractional boundary value problems with nonlinear boundary conditions
© Nyamoradi et al.; licensee Springer 2013
Received: 17 May 2013
Accepted: 16 August 2013
Published: 8 September 2013
In this manuscript, we consider two problems of boundary value problems for a fractional differential equation. A fixed point theorem in partially ordered sets and a contraction mapping principle are applied to prove the existence of at least one positive solution for both fractional boundary value problems.
MSC:47H10, 26A33, 34A08.
by using the properties of Green’s function of the corresponding problem and the Schauder fixed point theorem, where , , , , are continuous functions, and f, g may be singular at .
where , , , , , are continuous functions, and f, g may be singular at . By using the properties of Green’s function together with the Schauder fixed point theorem, it has been proved that this problem has at least one positive solution.
where , , is the Riemann-Liouville fractional derivative of order α, is a continuous function, and f may be singular at , i.e., .
The rest of the article is organized as follows: in Section 2, we shall recall certain results from the theory of the continuous fractional calculus. In Section 3, we shall provide some conditions, under which problem (1) has at least one positive solution. In Section 4, by suitable conditions, we will prove that problem (2) has at least one positive solution. Finally, in Section 5, we shall provide two numerical examples, which shall explicate the applicability of our results.
In this section, we present some notations and preliminary lemmas that will be used in the proofs of the main results.
, , implies , and
, , implies .
where is the Euler gamma function.
Lemma 1 
The equality , holds for .
Lemma 2 
has a unique solution , , , where .
Lemma 3 
, , where .
3 Existence solution of problem (1)
In this section, we study the existence and uniqueness of solutions of (1). To prove the main result, we need the following definitions and a preliminary lemma.
is measurable on for every ,
is continuous on for all ,
- (iii)for each , there exist and such that
The following two lemmas are fundamental in the proofs of our main results.
Lemma 4 
where is a continuous and nondecreasing function such that is positive in and . If there exists with , then T has a fixed point.
then we have the following lemma in .
Lemma 5 
Adding condition (6) to the hypotheses of Lemma 4, one obtains uniqueness of the fixed point of T.
Lemma 6 (See Lemma 2.1 in )
Proof The proof is straightforward, so we omit it here. □
We state our main result as follows.
and is nondecreasing,
φ is positive in .
Then the boundary value problem (1) has a unique positive solution.
Suppose that and is continuous, nondecreasing, positive in and . Thus, for , . Finally, take into account that for the zero function, , by Lemma 4, the boundary value problem (1) has at least one positive solution. Moreover, this solution is unique since satisfies condition (6) and Lemma 5. This completes the proof. □
Here the authors worked in the space .
4 Existence solution of problem (2)
In this section, we study the existence and uniqueness of solutions of (2). To prove the main result, we need the following assumptions:
(H1) and ;
Theorem 2 Assume that (H1) holds, and is a jointly continuous function, which satisfies (H2) that there exist numbers and such that . Then problem (2) has a unique solution, provided , where is given in (H3).
where is given in (H3). As , therefore, L is a contraction. By the contraction mapping principle, we conclude that L has a unique fixed point, which is a unique solution of problem (2). □
Then by using Theorem 2, problem (13) has a unique solution on .
Therefore, by using Theorem 1, the boundary value problem (14) has one positive solution.
The existence of the positive solutions to fractional boundary value problems involving the nonlinear boundary conditions is an important issue in the area of fractional calculus, and it is a crucial step for finding the correct numerical solutions of these types of equations.
In this paper, by using a fixed point theorem in partially ordered sets and the contraction mapping principle, we have proved the existence of at least one positive solution for two problems of boundary value problems for the fractional differential equation, and we provided two illustrative examples in order to justify our approach.
The authors would like to thank the anonymous referee of this paper for very helpful comments and suggestions.
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