Abstract
This work is concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problem:
where is the standard RiemannLiouville 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:
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 is the standard RiemannLiouville fractional derivative of order ν, and , .
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 , for and , and , with the following boundary value conditions:
under the assumptions that , , f, g are nonnegative and continuous. But the uniqueness of positive solutions is not treated in these papers.
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 with . Suppose that . Then the νth RiemannLiouville fractional integral is defined to be
whenever the righthand side is defined. Similarly, with and , we define the νth RiemannLiouville fractional derivative to be
where is the unique positive integer satisfying and .
Lemma 2.2 (See [19])
Letbe given. Then the unique solution to problemtogether with the boundary conditions, whereand, is
where
is the Green function for this problem.
Lemma 2.3 (See [19])
Letbe as given in the statement of Lemma 2.2. Then we have
(i) is a continuous function on the unit square;
Lemma 2.4The functiondefined by (2.2) satisfies the following conditions:
Proof Evidently, the right inequality holds. So, we only need to prove the left inequality. If , then we have , and thus
Hence,
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 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 nonempty 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 ().
For all , the notation means that there exist and such that . Clearly, ∼ is an equivalence relation. Given (i.e., and ), we denote by the set . It is easy to see that .
is said to be a mixed monotone operator if is increasing in x and decreasing in y, i.e., , , imply . Element is called a fixed point of A if .
Definition 2.6 An operator is said to be subhomogeneous if it is satisfies
Definition 2.7 Let and β be a real number with . An operator is said to be βconcave if it satisfies
Lemma 2.8 (See Theorem 2.1 in [20])
Letand. is a mixed monotone operator and satisfies
is an increasing subhomogeneous operator. Assume that
(ii) there exists a constantsuch that, .
Then:
(3) the operator equationhas a unique solutionin;
(4) for any initial values, constructing successively the sequences
Lemma 2.9 (See Theorem 2.4 in [20])
Letand. is a mixed monotone operator and satisfies
is an increasingβconcave operator. Assume that
(ii) there exists a constantsuch that, .
Then:
(3) the operator equationhas a unique solutionin;
(4) for any initial values, constructing successively the sequences
Remark 2.10 (i) If we take in Lemma 2.8, then the corresponding conclusion is still true (see Corollary 2.2 in [20]); (ii) if we take in Lemma 2.9, then the conclusion obtained is also true (see Theorem 2.7 in [23]).
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 with the standard norm . Notice that this space can be equipped with a partial order given by
Set , the standard cone. It is clear that P is a normal cone in and the normality constant is 1.
Theorem 3.1Assume that
(H_{1}) is continuous andis continuous;
(H_{2}) is increasing infor fixedand, decreasing infor fixedand, andis increasing infor fixed;
(H_{3}) andfor, , , and there exists a constantsuch that, , , ;
(H_{4}) there exists a constantsuch that, , .
Then:
(1) there existandsuch thatand
where, andis given as in (2.2);
(2) FBVP (1.1) has a unique positive solutionin;
(3) for any, constructing successively the sequences
Proof To begin with, from Lemma 2.2, FBVP (1.1) has an integral formulation given by
It is easy to prove that u is the solution of FBVP (1.1) if and only if . From (H_{1}), we know that and . In the sequel, we check that A, B satisfy all the assumptions of Lemma 2.8.
Firstly, we prove that A is a mixed monotone operator. In fact, for , with , , we know that , , , and by (H_{2}) and Lemma 2.3,
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 and , by (H_{3}) we have
That is, for , . So, the operator A satisfies (2.5). Also, for any , , from (H_{3}) we know that
that is, for , . That is, the operator B is subhomogeneous. Now we show that and . On the one hand, from (H_{1}), (H_{2}) and Lemma 2.4, for any , we have
On the other hand, also from (H_{1}), (H_{2}) and Lemma 2.4, for any , we obtain
From (H_{2}), (H_{4}), we have
and in consequence,
So, , ; and hence we have . Similarly,
from , we easily prove . Hence the condition (i) of Lemma 2.8 is satisfied.
In the following, we show the condition (ii) of Lemma 2.8 is satisfied. For , and any , from (H_{4}),
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
FBVP (1.1) has a unique positive solution in ; for , the sequences
Theorem 3.2Assume (H_{1}), (H_{2}) and
(H_{5}) there exists a constantsuch that, , , , andfor, , ;
(H_{6}) forand there exists a constantsuch that, , .
Then:
(1) there existandsuch thatand
where, andis given as in (2.2);
(2) FBVP (1.1) has a unique positive solutionin;
(3) for any, constructing successively the sequences
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 is a mixed monotone operator and is increasing. From (H_{5}), we have
From (H_{2}), (H_{6}), we have
and in consequence,
So, we can easily prove that , . For , and any , from (H_{6}),
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
FBVP (1.1) has a unique positive solution in ; for , the sequences
From Remark 2.10 and similar to the proofs of Theorems 3.13.2, we can prove the following conclusions.
Corollary 3.3Let. Assume thatfsatisfies the conditions of Theorem 3.1 and. Then: (i) there existandsuch thatand
where, andis given as in (2.2); (ii) the FBVP
has a unique positive solutionin; (iii) for any, constructing successively the sequences
Corollary 3.4Let. Assume thatgsatisfies the conditions of Theorem 3.2 andfor. Then: (i) there existandsuch thatand
where, andis given as in (2.2); (ii) the FBVP
has a unique positive solutionin; (iii) for any, constructing successively the sequences
4 An example
We now present one example to illustrate Theorem 3.1.
Example 4.1
Consider the following FBVP:
where is a constant, are continuous with .
Obviously, problem (4.1) fits the framework of FBVP (1.1) with , . (Note that , therefore, in this case.) In this example, we take and let
Obviously, ; is continuous and is continuous with . And is increasing in for fixed and , decreasing in for fixed and , and is increasing in for fixed . Besides, for , , , we have
Moreover, if we take , then we obtain
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 , where , .
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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