- Research
- Open Access
Positive solutions for integral boundary value problem of two-term fractional differential equations
- Mengrui Xu^{1} and
- Zhenlai Han^{1}Email author
- Received: 18 March 2018
- Accepted: 21 June 2018
- Published: 28 June 2018
Abstract
In this paper, we investigate a class of nonlinear two-term fractional differential equations involving two fractional orders \(\delta\in(1,2]\) and \(\tau\in (0,\delta)\) with integral boundary value conditions. By the Schauder fixed point theorem we obtain the existence of positive solutions based on the method of upper and lower solutions. Then we obtain the uniqueness result by the Banach contraction mapping principle. Examples are given to illustrate our main results.
Keywords
- Fractional differential equations
- Boundary value problem
- Positive solutions
MSC
- 34A08
- 34B18
- 34A12
1 Introduction
Fractional differential equations are an important tool to describe many processes and phenomena of science and engineering [1–3]. The boundary value problems of fractional differential equations have received widespread attention in recent years, and there are some attractive results obtained: see [4–9]. The theory of lower and upper solutions is known to be an effective method for proving the existence of solutions to fractional differential equations (see, e.g., [10–14]).
To the best of our knowledge, no paper has considered the existence of positive solutions for nonlinear fractional differential equations with integral boundary conditions (1.1)–(1.2). This equation has two nonlinear terms, one containing the fractional derivative. Compared to many two-term fractional differential equations, the form of the equation we considered is more general in a way. In addition, the other nonlinear term requires no any monotonicity, which can respond better to objective laws.
In this paper, we prove the existence of positive solutions to the boundary value problem (1.1)–(1.2) by the Schauder fixed point theorem and the method of upper and lower solutions. Then, we give a uniqueness result by the Banach contraction mapping principle.
The paper is organized as follows. Section 2 contains some necessary concepts and results. In Sect. 3, our main results on the existence of positive solutions are proved. Section 4 is devoted to study the uniqueness of positive solutions.
2 Preliminaries
In this section, we present some definitions and lemmas, which are required for our theorems.
Definition 2.1
([1])
Definition 2.2
([1])
Lemma 2.1
([2])
Lemma 2.2
([3])
Lemma 2.3
([3])
Lemma 2.4
([3])
Let X be a Banach space, and let \(\Omega\subset X\) be a convex closed bounded set. If \(T: \Omega\rightarrow\Omega\) is a continuous operator such that \(T\Omega\subset X\) and TΩ is relatively compact, then T has at least one fixed point in Ω.
Lemma 2.5
([3])
Lemma 2.6
Proof
The proof is divided into two cases.
Lemma 2.7
Lemma 2.8
The operator \(T: X\rightarrow X\) is completely continuous.
Proof
Definition 2.3
3 Existence of positive solutions
In this section, we establish the existence of positive solutions for the boundary value problem (1.1)–(1.2) by the Schauder fixed point theorem based on the method of upper and lower solutions.
Theorem 3.1
Assume that x̅ and \(\underline{x}\) are a pair of upper and lower solutions of the boundary value problem (1.1)–(1.2). Then the boundary value problem (1.1)–(1.2) has at least one positive solution \(x\in X\), and \(\underline{x}(t)\leq x(t)\leq\overline {x}(t),t\in[0,1]\).
Proof
Corollary 3.1
Proof
Corollary 3.2
Let (3.4) and \(f(t,u)\geq k_{1}(t)\geq0\), \(t\in[0,1]\), hold with at least one of \(k_{1}(t)\) and \(k_{3}(t)\) not identically zero. Assume that \(f(t,u)\) converges uniformly on \([0,1]\) to \(k(t)\) as \(u\rightarrow \infty\). Then the boundary value problem (1.1)–(1.2) has at least one positive solution on P.
Proof
Corollary 3.3
Proof
Example 3.1
4 Uniqueness of positive solution
In this section, we prove the uniqueness of a positive solution for the boundary value problem (1.1)–(1.2) by the Banach contraction mapping principle. In particular, the monotonicity of \(g(t,x)\) is dispensable.
Theorem 4.1
Proof
Remark 4.1
The conditions of Corollary 3.3 imply the conditions of Theorem 4.1, and thus Corollary 3.3 also concludes the uniqueness of a positive solution.
Example 4.1
Declarations
Acknowledgements
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.
Availability of data and materials
The authors declare that all data and material in the paper are available and veritable.
Funding
This research is supported by the Natural Science Foundation of China (61703180) and by Shandong Provincial Natural Science Foundation (ZR2017MA043).
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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