Positive solutions of fractional differential equations involving the Riemann–Stieltjes integral boundary condition
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Abstract
Keywords
Riemann–Stieltjes integral Mixed monotone operator Fixed point theorem Existence and uniquenessMSC
34A08 34B18 35J051 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
 \((H1)\)
 Ais 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 inxfor each\(t\in [0,1]\), \(y\in[0,+\infty)\)and nonincreasing inyfor 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.
Notes
Acknowledgements
Not applicable.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Funding
This work is supported by NSFC (11571207), the Taishan Scholar project and SDUST graduate innovation project SDKDYC170343.
Competing interests
The authors declare that they have no competing interests.
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