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Positive solutions of fractional differential equations involving the Riemann–Stieltjes integral boundary condition
 Qilin Song^{1} and
 Zhanbing Bai^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s1366201816338
© The Author(s) 2018
Received: 18 February 2018
Accepted: 5 May 2018
Published: 16 May 2018
Abstract
Keywords
 Riemann–Stieltjes integral
 Mixed monotone operator
 Fixed point theorem
 Existence and uniqueness
MSC
 34A08
 34B18
 35J05
1 Introduction
The rest of this paper is organized as follows. In Sect. 2, we recall some definitions, theorems, and lemmas. In Sect. 3, we investigate the existence and uniqueness of positive solution for problem (1.1), (1.2). In Sect. 4, we present some examples to illustrate our main results.
2 Preliminaries and lemmas
Suppose that \((E, \\cdot\)\) is a real Banach space, \(P \subset E\) is a normal cone. For all \(x, y\in E\), the notation \(x \sim y\) means that there exist \(\lambda, \mu>0\) such that \(\lambda x\leq y \leq\mu x\). Clearly, ∼ is an equivalence relation. Given \(h>\theta\) (i.e., \(h\geq\theta\), \(h\neq\theta\) ), we denote \(P_{h}=\{x\in Ex\sim h\}\). It is easy to see that \(P_{h}\subset P\) is convex and \(\lambda P_{h}=P_{h}\) for all \(\lambda>0\). We refer the readers to the references [9] and [19] for details.
Definition 2.1
([19])
\(T: P\times P\rightarrow P\) is said to be a mixed monotone operator if \(T(x, y)\) is increasing in x and decreasing in y, i.e., \(u_{i}, v_{i}\ (i=1, 2)\in P\), \(u_{1}\leq u_{2}\), \(v_{1}\geq v_{2}\) imply \(T(u_{1}, v_{1})\leq T(u_{2}, v_{2})\). The element \(x\in P\) is called a fixed point of T if \(T(x, x)=x\).
Theorem 2.1
([9])
 \((A1)\) :

There exists \(h\in P\) with \(h\neq\theta\) such that \(T(h, h)\in P_{h}\).
 \((A2)\) :

For any \(u, v\in P\) and \(t\in(0, 1)\), there exists \(\varphi(t)\in(t, 1]\) such that \(T(tu, t^{1}v) \geq\varphi(t)T(u,v)\).
Lemma 2.1
([22])
Lemma 2.2
([22])
 (1)
\(H(t,s)>0\) for all \(t,s\in(0,1)\);
 (2)The following relation holds:where the constants \(c=\frac{1}{1\delta}\), \(d=\frac{\G_{A}(s)\}{1\delta }+\frac{1}{\Gamma(\alpha1)}\).$$ ct^{\alpha1}G_{A}(s)\leq H(t,s) \leq\,dt^{\alpha1}\leq d,\quad t, s\in[0,1], $$(2.2)
3 Main results
Theorem 3.1
 \((H1)\) :

A is a function of bounded variation such thatfor \(s \in[0,1]\);$$\int _{0}^{1}G(t,s)\,dA(t) \ge0\quad \textit{and}\quad \int_{0}^{1}t^{\alpha1}\,dA(t) < 1$$
 \((H2)\) :

\(f \in C([0,1]\times[0,+\infty)\times[0,+\infty), [0,+\infty))\), \(f(t, x, y)\) is nondecreasing in x for each \(t\in [0,1]\), \(y\in[0,+\infty)\) and nonincreasing in y for each \(t\in[0,1]\), \(x\in[0, +\infty)\);
 \((H3)\) :

\(f(t,0,1)\neq0\), \(t\in[0,1]\);
 \((H4)\) :

for any \(\gamma\in(0,1)\), there exists a constant \(\varphi(\gamma)\in(\gamma,1]\) such that \(f(t,\gamma x,\gamma ^{1}y)\geq\varphi(\gamma)f(t,x,y)\) for any \(x, y\in[0,+\infty)\).
Proof
4 Examples
Example 4.1
Example 4.2
5 Conclusion
The research of fractional calculus and integral boundary value conditions has become a new area of investigation. By the use of fixed point theorem and the properties of mixed monotone operator theory, the existence and uniqueness of positive solutions for the problem are acquired. Two examples are presented to illustrate the main results. The conclusion obtained in this paper will be very useful in the application point of view. Also, we expect to find some applications in more nonlinear problems.
Declarations
Acknowledgements
Not applicable.
Funding
This work is supported by NSFC (11571207), the Taishan Scholar project and SDUST graduate innovation project SDKDYC170343.
Authors’ contributions
All authors contributed equally to the writing of this paper. All 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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