This work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem:
where is the standard Riemann-Liouville fractional derivative of order ν, and , . Our analysis relies on two new fixed point theorems for mixed monotone operators with perturbation. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate the main result.
MSC: 26A33, 34B18, 34B27.
Keywords:Riemann-Liouville fractional derivative; fractional differential equation; positive solution; existence and uniqueness; fixed point theorem for mixed monotone operator
In this paper, we investigate the existence and uniqueness of positive solutions for the fractional boundary value problem (FBVP for short) of the form:
Fractional differential equations arise in many fields such as physics, mechanics, chemistry, economics, engineering and biological sciences, etc.; see [1-6] for example. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Miller and Ross , Podlubny , Kilbas et al., and the papers [7-16] and the references therein. In these papers, many authors have investigated the existence of positive solutions for nonlinear fractional differential equation boundary value problems. On the other hand, the uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems has been studied by some authors; see [10,14,17] for example.
In , Goodrich utilized the Krasnoselskii’s fixed point theorem to study a FBVP of the form:
and established the existence of at least one positive solution for FBVP (1.2). By using the same fixed point theorem, Goodrich  considered the existence of a positive solution to the following systems of differential equations of fractional order:
Different from the works mentioned above, motivated by the work , we will use two fixed point theorems for mixed monotone operators with perturbation to show the existence and uniqueness of positive solutions for FBVP (1.1). To our knowledge, there are still very few to utilize the fixed point results on mixed monotone operators with perturbation to study the existence and uniqueness of a positive solution for nonlinear fractional differential equation boundary value problems. So, it is worthwhile to investigate FBVP (1.1) by using our new fixed point theorems in . Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it.
With this context in mind, the outline of this paper is as follows. In Section 2 we recall certain results from the theory of fractional calculus and some definitions, notations and results of mixed monotone operators. In Section 3 we provide some conditions, under which the problem FBVP (1.1) has a unique positive solution. Finally, in Section 4, we provide an example, which explicates the applicability of our result.
For the convenience of the reader, we present here some definitions, lemmas and basic results that will be used in the proofs of our theorems.
Definition 2.1 (See )
Lemma 2.2 (See )
is the Green function for this problem.
Lemma 2.3 (See )
So, the proof is complete. □
In the sequel, we present some basic concepts in ordered Banach spaces for completeness and two fixed point theorems which we will be used later. For convenience of readers, we suggest that one refers to [20-22] for details.
Suppose that is a real Banach space which is partially ordered by a cone , i.e., if and only if . If and , then we denote or . By θ we denote the zero element of E. Recall that a non-empty closed convex set is a cone if it satisfies (i) , ; (ii) , .
P is called normal if there exists a constant such that, for all , implies ; in this case, N is called the normality constant of P. If , the set is called the order interval between and . We say that an operator is increasing (decreasing) if implies ().
Lemma 2.8 (See Theorem 2.1 in )
Lemma 2.9 (See Theorem 2.4 in )
Remark 2.10 (i) If we take in Lemma 2.8, then the corresponding conclusion is still true (see Corollary 2.2 in ); (ii) if we take in Lemma 2.9, then the conclusion obtained is also true (see Theorem 2.7 in ).
3 Main results
In this section, we apply Lemma 2.8 and Lemma 2.9 to study FBVP (1.1), and we obtain some new results on the existence and uniqueness of positive solutions. The method used here is relatively new to the literature and so are the existence and uniqueness results to the fractional differential equations.
Theorem 3.1Assume that
Proof To begin with, from Lemma 2.2, FBVP (1.1) has an integral formulation given by
From (H2), (H4), we have
and in consequence,
Then we get , for . Finally, an application of Lemma 2.8 implies: there exist and such that , ; the operator equation has a unique solution in ; for any initial values , constructing successively the sequences
Theorem 3.2Assume (H1), (H2) and
From (H2), (H6), we have
and in consequence,
Then we get , for . Finally, an application of Lemma 2.9 implies: there exist and such that , ; the operator equation has a unique solution in ; for any initial values , constructing successively the sequences
From Remark 2.10 and similar to the proofs of Theorems 3.1-3.2, we can prove the following conclusions.
4 An example
We now present one example to illustrate Theorem 3.1.
Consider the following FBVP:
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
The authors are grateful to the anonymous referee for his/her valuable suggestions. The first author was supported financially by the Youth Science Foundations of China (11201272) and Shanxi Province (2010021002-1).
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