Abstract
This work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem:
where
MSC: 26A33, 34B18, 34B27.
Keywords:
RiemannLiouville fractional derivative; fractional differential equation; positive solution; existence and uniqueness; fixed point theorem for mixed monotone operator1 Introduction
In this paper, we investigate the existence and uniqueness of positive solutions for the fractional boundary value problem (FBVP for short) of the form:
where
Fractional differential equations arise in many fields such as physics, mechanics, chemistry, economics, engineering and biological sciences, etc.; see [16] 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 [3], Podlubny [5], Kilbas et al.[6], and the papers [716] 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 [18], 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 [19] considered the existence of a positive solution to the following systems of differential equations of fractional order:
where
under the assumptions that
Different from the works mentioned above, motivated by the work [20], 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 [20]. 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.
2 Preliminaries
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 [18])
Let
whenever the righthand side is defined. Similarly, with
where
Lemma 2.2 (See [19])
Let
where
is the Green function for this problem.
Lemma 2.3 (See [19])
Let
(i)
(ii)
Lemma 2.4The function
Proof Evidently, the right inequality holds. So, we only need to prove the left inequality.
If
Hence,
When
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 [2022] for details.
Suppose that
P is called normal if there exists a constant
For all
Definition 2.6 An operator
Definition 2.7 Let
Lemma 2.8 (See Theorem 2.1 in [20])
Let
(i) there is
(ii) there exists a constant
Then:
(1)
(2) there exist
(3) the operator equation
(4) for any initial values
we have
Lemma 2.9 (See Theorem 2.4 in [20])
Let
(i) there is
(ii) there exists a constant
Then:
(1)
(2) there exist
(3) the operator equation
(4) for any initial values
we have
Remark 2.10 (i) If we take
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.
In our considerations, we work in the Banach space
Set
Theorem 3.1Assume that
(H_{1})
(H_{2})
(H_{3})
(H_{4}) there exists a constant
Then:
(1) there exist
where
(2) FBVP (1.1) has a unique positive solution
(3) for any
we have
Proof To begin with, from Lemma 2.2, FBVP (1.1) has an integral formulation given by
where
Define two operators
It is easy to prove that u is the solution of FBVP (1.1) if and only if
Firstly, we prove that A is a mixed monotone operator. In fact, for
That is,
Further, it follows from (H_{2}) and Lemma 2.3 that B is increasing. Next we show that A satisfies the condition (2.5). For any
That is,
that is,
On the other hand, also from (H_{1}), (H_{2}) and Lemma 2.4, for any
From (H_{2}), (H_{4}), we have
Since
and in consequence,
So,
from
In the following, we show the condition (ii) of Lemma 2.8 is satisfied. For
Then we get
we have
FBVP (1.1) has a unique positive solution
satisfy
Theorem 3.2Assume (H_{1}), (H_{2}) and
(H_{5}) there exists a constant
(H_{6})
Then:
(1) there exist
where
(2) FBVP (1.1) has a unique positive solution
(3) for any
we have
Sketch of the proof Consider two operators A, B defined in the proof of Theorem 3.1. Similarly, from (H_{1}), (H_{2}), we obtain that
From (H_{2}), (H_{6}), we have
Since
and in consequence,
So, we can easily prove that
Then we get
we have
FBVP (1.1) has a unique positive solution
satisfy
From Remark 2.10 and similar to the proofs of Theorems 3.13.2, we can prove the following conclusions.
Corollary 3.3Let
where
has a unique positive solution
we have
Corollary 3.4Let
where
has a unique positive solution
we have
4 An example
We now present one example to illustrate Theorem 3.1.
Example 4.1
Consider the following FBVP:
where
Obviously, problem (4.1) fits the framework of FBVP (1.1) with
Obviously,
Moreover, if we take
Hence all the conditions of Theorem 3.1 are satisfied. An application of Theorem 3.1
implies that problem (4.1) has a unique positive solution in
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
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 (20100210021).
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