 Research
 Open Access
Existence and uniqueness of solutions for systems of fractional differential equations with Riemann–Stieltjes integral boundary condition
 Xinqiu Zhang^{1},
 Lishan Liu^{1, 2}Email author,
 Yonghong Wu^{2} and
 Yumei Zou^{3}
https://doi.org/10.1186/s1366201816507
© The Author(s) 2018
 Received: 29 April 2018
 Accepted: 30 May 2018
 Published: 8 June 2018
Abstract
In this article, we first establish an existence and uniqueness result for a class of systems of nonlinear operator equations under more general conditions by means of the cone theory and monotone iterative technique. Furthermore, the iterative sequence of the solution and the error estimation of the system are given. Then we use this new result to study the existence and uniqueness of the solution for boundary value problems of systems of fractional differential equations with a Riemann–Stieltjes integral boundary condition in real Banach spaces. The results obtained in this paper are more general than many previous results and complement them.
Keywords
 Systems of nonlinear binary operator equations
 Real Banach spaces
 Monotone iterative technique
 Existence and uniqueness of positive solutions
 Systems of fractional differential equations
MSC
 47H07
 47H10
 47J25
 34A08
 34B18
 34G20
1 Introduction
However, the upperlower solutions conditions play an important role in [37]. Instead of supposing the upperlower solutions conditions, they used a generating normal cone, which strengthened the conditions. Thus, how to remove these conditions is an interesting and important question. In this paper, compared with [37], we get Lemma 3.2 for which we shall not suppose the upperlower solutions conditions, the generating of the cone and compactness of the operators. Here, by means of the cone theory and the Banach contraction mapping principle, the existence of a unique solution of the system of nonlinear operator binary equations (1.1) is investigated, furthermore, the iterative sequence of the solution and the error estimation of the system are given. The theorems obtained in this paper are more general than many previous results and complement them.
Fractional differential equations, arising in the mathematical modeling of systems and processes, have drawn more and more attention of the research community due to their numerous applications in various fields of science such as engineering, chemistry, physics, mechanics, etc. Boundary value problems of fractional differential equations have been investigated for many years (see [9, 17, 20, 27, 29, 36]). Now, there are many papers dealing with the problem for different kinds of boundary value conditions such as multipoint boundary condition (see [7, 8]), integral boundary condition (see [14–16, 19, 22, 25, 26, 30]), and many other boundary conditions (see [12, 31, 32]). In recent years, the existence and uniqueness theorems of solutions for boundary value problems of nonlinear fractional differential equations have been studied extensively in the literature, mainly by using the fixed point theorem of the mixedmonotone operator (see, for instance, [7, 18, 33, 34] and their references), a priori estimate method and a maximal principle (see, for instance, [2]), the Banach contraction mapping principle and the Krasnose’skii fixed point theorem (see, for instance, [1, 10, 11, 17]).
By means of monotone iterative technique and cone theory, we obtain some new existence theorems of the solutions and iterative approximation of the unique solution for the system of fractional differential equations with a Riemann–Stieltjes integral boundary condition, which does not possess any upper and lower solutions conditions in ordered Banach spaces. From this paper, we can see that the fixed point theorems in this paper have extensive applied background.
2 Preliminaries
Now we present briefly some definitions and basic results that are to be used in the article for convenience of the reader. We refer the reader to [3–6] for more details.
Suppose that \((E,\\cdot\)\) is a real Banach space, θ is the zero element of E. Recall that a nonempty closed convex set \(P\subset E\) is a cone if it satisfies (1) \(x\in P\), \(\lambda\geq0 \Rightarrow\lambda x \in P\); (2) \(x\in P\), \(x\in P \Rightarrow x=\theta\). The real Banach space E can be partially ordered by a cone \(P\subset E\), i.e., \(x\leq y \) if and only if \(yx\in P\). If \(x\leq y\) and \(x\neq y\), then we denote \(x< y\) or \(y>x\). Let \(C[I,E]=\{x(t):I\rightarrow E \mid x(t) \text{ is continuous} \}\). Then \(C[I,E]\) is a Banach space with the norm \(\x\_{c}=\max_{t\in I}\ x(t)\\), for \(x \in C[I,E]\).
Moreover, P is called normal if there exists a constant \(N>0\) such that, for all \(x,y\in E\), \(\theta\leq x \leq y\) implies \(\x \\leq N \y \\), the smallest N is called the normality constant of P. If \(x_{1},x_{2} \in E\) with \(x_{1} \leq x_{2}\) the set \([x_{1},x_{2}]=\{x\in E \mid x_{1} \leq x \leq x_{2}\}\) is called the order interval between \(x_{1}\) and \(x_{2}\).
Definition 2.1
Let D be a subset of a real Banach space E. \(A:D\times D\rightarrow E\) is said to be a mixedmonotone operator if \(A(x,y)\) is increasing in x, and decreasing in y, i.e., for all \(x_{i},y_{i}\in P\) (\(i=1,2\)) with \(x_{1}\leq x_{2}\), \(y_{1}\geq y_{2}\) imply \(A(x_{1},y_{1})\leq A(x_{2},y_{2})\). The element \(x\in D\) is called a fixed point of A if \(A(x,x)=x\).
3 Lemmas
Lemma 3.1
 (H_{0}):

\(u_{0}\leq A (u_{0},v_{0} )\), \(B (v_{0},u_{0} )\leq v_{0} \).
 (H_{1}):

For fixed \(x\in D\), \(A(x,y)\) and \(B(x,y)\) are decreasing in y, and there exist two positive numbers \(M_{i}>0\) (\(i=1,2\)) such that, for fixed \(y\in D\), and for any \(x_{1}, x_{2}\in D\) with \(x_{1}\leq x_{2}\),$$\begin{aligned}& A (x_{2},y )A (x_{1},y )\geqM_{1}(x_{2}x_{1}), \\& B (x_{2},y )B (x_{1},y )\geqM_{2}(x_{2}x_{1}). \end{aligned}$$
 (H_{2}):

There exist a positive number \(M_{3}>0\) and a positive integer \(n_{0}\) such that, for all \(x,y\in D\) with \(x\leq y\),where \(L:E\rightarrow E\) is a positive bounded linear operator with \(r (L )<1\).$$ M_{3}(yx)\leq B (y,x )A (x,y )\leq L^{n_{0}}(yx), $$(3.1)
Proof
(A_{1}) \(F, G: D\times D\rightarrow E\) are two mixedmonotone operators.
The proof will be divided into four steps.
Step 3: We prove that \((x^{\ast},x^{\ast} )\) is the unique solution of operator equations (1.1) in \(D\times D\). In fact, suppose \((\overline{x},\overline{x} )\) is another solution of Eqs. (1.1) in \(D\times D\), then, by (A_{1}) and the method of the introduction, we easily see that \(u_{n}\leq\overline{x}\leq v_{n}\) (\(n=1,2,\ldots\)). Thus, by (3.21) and the normality of P, we have \(\overline{x}=x^{\ast}\). Therefore, the operator equations (1.1) have a unique solution \((x^{\ast},x^{\ast} )\) in \(D\times D\).
Lemma 3.2
 (J_{0}):

For fixed \(x\in P\), \(A(x,y)\) and \(B(x,y)\) are decreasing in y, and there exist two positive numbers \(M_{i}>0\) (\(i=1,2\)) such that, for fixed \(y\in P\), and for any \(x_{1}, x_{2}\in P\) with \(x_{1}\leq x_{2}\),$$\begin{aligned}& A (x_{2},y )A (x_{1},y )\geqM_{1}(x_{2}x_{1}), \\& B (x_{2},y )B (x_{1},y )\geqM_{2}(x_{2}x_{1}). \end{aligned}$$
 (J_{1}):

There exist a positive bounded linear operator \(L_{1}:E\rightarrow E\) with \(r (L_{1} )<1\), a positive integer \(m_{0}\) and \(h\in P\) such that, for all \(x\in P\),$$ B (x,\theta )\leq L^{m_{0}}_{1}x+h. $$(3.30)
 (J_{2}):

There exist a positive bounded linear operator \(L_{2}:E\rightarrow E\) with \(r (L_{2} )<1\), a positive integer \(n_{0}\) and a positive number \(M_{3}>0\) such that, for all \(x,y\in P\), \(x\leq y\),$$ M_{3}(yx)\leq B (y,x )A (x,y )\leq L^{n_{0}}_{2}(yx). $$(3.31)
Proof
Taking \(u_{0}=\theta\), \(v_{0}=w_{0}\), then \(A (\theta,v_{0} )\geq\theta\) and \(B (v_{0},\theta )\leq v_{0}\). Consequently, by Lemma 3.1, the conclusions hold. Therefore, the proof of Lemma 3.2 is completed. □
4 Main result
Let E be a real Banach space, P be a normal cone in E. In this section, we consider the existence and uniqueness of the solution as well as iterative approximation of the system of fractional differential equations (1.3) with a Riemann–Stieltjes integral boundary condition in ordered Banach spaces E:
Now we present briefly some definitions, lemmas, and basic results that are to be used in the article for convenience of the reader. We refer the reader to [13, 14, 21, 23–25, 28, 35] for more details.
Definition 4.1
Definition 4.2
Lemma 4.3
 (1)If \(u\in L^{1}(0,1)\) and \(\alpha>\beta>0\), then$$ I_{0^{+}}^{\alpha}I_{0^{+}}^{\beta}u(t)=I_{0^{+}}^{\alpha+\beta}u(t), \qquad D_{0^{+}}^{\beta}I_{0^{+}}^{\alpha}u(t)=I_{0^{+}}^{\alpha\beta}u(t), \qquad D_{0^{+}}^{\beta}I_{0^{+}}^{\beta}u(t)=u(t). $$(4.1)
 (2)If \(u\in L^{1}(0,1)\) and \(\alpha>\beta>0\), then \(D_{0^{+}}^{\alpha }u(t)=0\) has the unique solutionwhere \(c_{i}\in\mathbb{R}\) (\(i=0,1,2,\ldots,n\)), \(n=[\alpha]+1\).$$ f(t)=c_{1}t^{\alpha1}+c_{2}t^{\alpha2}+ \cdots+c_{n}t^{\alphan}, $$(4.2)
Lemma 4.4
Lemma 4.5
([19])
Lemma 4.6
([19])
 (1)
\(G:[0,1]\times[0,1]\rightarrow\mathbb{R}_{+}\) is continuous and \(G(t,s)>0\) for all \(t,s\in(0,1)\);
 (2)For any \(t,s\in[0,1]\), we have \(t^{\alpha\beta_{n1}1}\phi (s)\leq G(t,s)\leq\phi(s)\), where$$\phi(s)=K(1,s)+\frac{g_{a}(s)}{\frac{\Gamma(\alpha\beta_{n1})}{\Gamma (\alpha\beta)}\delta},\quad s\in[0,1]. $$
 (H_{1}):

\(f_{i}:I\times E^{4}\rightarrow E\) is continuous and satisfies, for all \(x_{i}, y_{i}\in E\) (\(i=1,2,3,4\)), with \(y_{1}\geq x_{1}\), \(y_{2}\leq x_{2}\), \(y_{3}\geq x_{3}\), \(y_{4}\leq x_{4}\),$$ f_{i} (t,y_{1},y_{2},y_{3},y_{4} ) f_{i} (t,x_{1},x_{2},x_{3},x_{4} )\geq0,\quad \forall t\in I, i=1,2; $$(4.7)
 (H_{2}):

There exist three positive Lebesgue integrable functions \(a,b,c \in L^{1}(I,\mathbb{R}_{+})\) such that for all \(x,y\in E\), \(t\in I\),where e is a unit element in E;$$ f_{2} (t,x,\theta,y,\theta )\leq a(t)x+b(t)y+c(t)e, $$(4.8)
 (H_{3}):

There exist four constants \(r_{i}>0\) (\(i=1,2,3,4\)) such that, for any \(t\in I\), \(x_{i},y_{i}\in E\) (\(i=1,2,3,4\)) with \(x_{1}\leq y_{1}\), \(x_{2}\geq y_{2}\), \(x_{3}\leq y_{3}\), \(x_{4}\geq y_{4}\),$$\begin{aligned} 0 \leq& f_{2} (t,y_{1},y_{2},y_{3},y_{4} ) f_{1} (t,x_{1},x_{2},x_{3},x_{4} ) \\ \leq& r_{1}(y_{1}x_{1})+r_{2}(x_{2}y_{2})+r_{3}(y_{3}x_{3})+r_{4}(x_{4}y_{4}); \end{aligned}$$(4.9)
 (H_{4}):

\(\max_{t\in I}\int_{0}^{1}\widetilde {G}(t,s)\,ds<1\), \(\max_{t\in I}\int_{0}^{1} \vert \overline{G}(t,\tau ) \vert \,d\tau<1\), where$$\begin{aligned}& \begin{aligned} \widetilde{G}(t,s)&=\frac{1}{\Gamma(\beta_{n1}n+2)} \biggl( \int _{1}^{\tau}G(t,s)a(s) (s\tau)^{\beta_{n1}n+1}\,ds \biggr) \\ &\quad {}+\frac{1}{\Gamma(\beta_{n1}\beta_{i})} \biggl( \int_{1}^{\tau }G(t,s)b(s) (s\tau)^{\beta_{n1}\beta_{i}1}\,ds \biggr), \end{aligned} \\& \begin{aligned} \overline{G}(t,s)&=\frac{r_{1}+r_{2}}{\Gamma(\beta _{n1}n+2)} \biggl( \int_{1}^{\tau}G(t,s) (s\tau)^{\beta _{n1}n+1}\,ds \biggr) \\ &\quad {}+\frac{r_{3}+r_{4}}{\Gamma(\beta_{n1}\beta_{i})} \biggl( \int_{1}^{\tau }G(t,s) (s\tau)^{\beta_{n1}\beta_{i}1}\,ds \biggr). \end{aligned} \end{aligned}$$
Theorem 4.7
Proof
Thus all conditions of Lemma 3.2 are satisfied, therefore the conclusions of Theorem 4.7 hold. Consequently, the proof of Theorem 4.7 is completed. □
Declarations
Acknowledgements
The authors would like to thank the referees for their useful suggestions, which have significantly improved the paper.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Funding
The work are supported financially by the National Natural Science Foundation of China (11371221, 11571296).
Authors’ contributions
All authors contributed equally to each part of this work. 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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