Iterative positive solutions for singular nonlinear fractional differential equation with integral boundary conditions
 Lily Li Liu^{1},
 Xinqiu Zhang^{1},
 Lishan Liu^{1, 2}Email author and
 Yonghong Wu^{2}
https://doi.org/10.1186/s1366201608765
© Liu et al. 2016
Received: 2 March 2016
Accepted: 26 May 2016
Published: 10 June 2016
Abstract
In this article, we study the existence of iterative positive solutions for a class of singular nonlinear fractional differential equations with RiemannStieltjes integral boundary conditions, where the nonlinear term may be singular both for time and space variables. By using the properties of the Green function and the fixed point theorem of mixed monotone operators in cones we obtain some results on the existence and uniqueness of positive solutions. We also construct successively some sequences for approximating the unique solution. Our results include the multipoint boundary problems and integral boundary problems as special cases, and we also extend and improve many known results including singular and nonsingular cases.
Keywords
MSC
1 Introduction
Fractional differential equations have attracted more and more attention in recent decades, which is partly due to their numerous applications in many branches of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability, etc. We refer the reader to [1–9] and the references therein. On the other hand, boundary value problems with integral boundary conditions for ordinary differential equations arise often in many fields of applied mathematics and physics such as heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. The existence and uniqueness of positive solutions for such problems have become an important area of investigation in recent years.
However, up to now, the singular fractional differential equations with RiemannStieltjes integral conditions have seldom been considered by using fixed point theorem. In particular, we consider that \(f(t,u,v)\) has singularity at \(t=0\) or 1 and \(v=0\), \(g(t,v)\) has singularity at \(t=0\) or 1 and \(v=0\). In this article, we apply the fixed point theorem of mixed monotone operators to get the existence and uniqueness of the iterative solutions for singular fractional differential equations (1.1) without using the above condition (b).
Obviously, what we discuss is different from those in [1, 10, 13, 15, 17–20]. Comparing with the results in [10], we are based on a new method dealing with problem (1.1). Moreover, \(f(t,u,v)\) not only has three variables, but also is singular both for time and space variables. Comparing with the results in [13, 15, 17], we do not need \(f(t,u,v)\) to be continuous at \(t=0\) or 1 and at \(u=v=0\). The main new features presented in this article are as follows. Firstly, the boundary value problem has a more general form in which \(p(t)\), \(q(t)\) are allowed to be singular at \(t=0,1\) and f may be singular for time and space variables, that is, \(f(t,u,v)\) and \(g(t,v)\) may be singular at \(t=0\) or 1 and \(v=0\). Secondly, by using the fixed point theorem of mixed monotone operators, we obtain a unique positive solution of the boundary value problem (1.1), and we also construct successively some sequences for approximating the unique positive solution. Thirdly, let \(\int_{0}^{1}x(s)\,dA(s)\) denote the RiemannStieltjes integral, where A is a function of bounded variation, and dA may be a signed measure. As applications, the multipoint problems and integral problems are particular cases. In this paper, we also extend and improve many known results including singular and nonsingular cases.
The rest of the paper is organized as follows. In Section 2, we present some preliminaries and lemmas that are to be used to prove our main results. In Section 3, we discuss the existence and uniqueness positive solution of the BVP (1.1) and also construct successively some sequences for approximating the unique positive solution. In Section 4, we give an example to demonstrate the application of our theoretical results.
2 Preliminaries and lemmas
In this section, we present some definitions, lemmas, and basic results that will be used in the article. For convenience of readers, we refer to [6, 8, 21, 22] for details.
Let \((E,\\cdot\)\) be a Banach space. We denote the zero element of E by θ. 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) \(x\in P\), \(x\in P \Rightarrow x=\theta\). In this paper, suppose that \((E,\\cdot\)\) is a Banach space partially ordered by a cone \(P\subset E\), that is, \(x\leq y \) if and only if \(yx\in P\).
For \(x_{1},x_{2}\in E\), 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 \(x_{2}\). For \(x, y \in E\), the notation \(x\sim y\) means that there exist \(\lambda>0\) and \(\mu>0\) such that \(\lambda x\leq y\leq\mu x\). Clearly, ∼ is an equivalence relation. Given \(h>0\), we denote by \(P_{h}\) the set \(P_{h}= \{x\in P \mid x \sim h\}\). It is easy to see that \(P_{h}\subset P\) is a component of P. A cone 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 such N is called the normality constant of P.
Definition 2.1
([6])
Definition 2.2
([6])
Definition 2.3
([23])
Definition 2.4
([23])
Definition 2.5
([24])
Suppose that \((E,\\cdot\)\) is a Banach space, P is a cone in E, and \(D\subset P\). An operator \(A:D\times D\rightarrow P\) is said to be a mixed monotone operator if \(A(x,y)\) is increasing in x and decreasing in y, that is, for all \(x_{i},y_{i}\in D \) (\(i=1,2\)), \(x_{1}\leq x_{2}\), \(y_{1}\geq y_{2}\) imply \(A(x_{1},y_{1})\leq A(x_{2},y_{2})\).
Lemma 2.1
([6])
From the definition of the RiemannLiouville derivative we obtain the following result.
Lemma 2.2
([6])
In the following, we present the Green function of the fractional differential equation boundary value problem.
Lemma 2.3
([10])
Lemma 2.4
 (1)
\(G:[0,1]\times[0,1]\rightarrow[0,\infty)\) is continuous;
 (2)
For any \(t,s\in[0,1]\), we have \(\frac{t^{\alpha 1}}{1M}g_{A}(s)\leq G(t,s) \leq\frac{t^{\alpha1}}{(1M)\Gamma(\alpha)}(1s)^{\alpha1}\).
Proof
 (1)
By Lemma 2.4 in [10] we have that \(G:[0,1]\times[0,1]\rightarrow[0, \infty)\) is continuous.
 (2)From (2.2) we have$$ 0\leq G_{0}(t,s)\leq\frac{1}{\Gamma(\alpha )}\bigl(t(1s) \bigr)^{\alpha1},\quad \forall t,s \in[0,1]. $$(2.3)
Taking \(A:P_{h}\times P_{h}\rightarrow P_{h}\), \(B:P_{h}\rightarrow P_{h}\) in Theorem 3.1 in [23] and Corollary 3.7 in [25], and also taking \(A:P_{h}\times P_{h}\rightarrow P_{h}\), \(B:P_{h}\rightarrow P_{h}\), \(C=\theta\) \(\varphi(t)=t^{\gamma}\) in Corollary 3.4 in [26], it is easy to get the following lemma.
Lemma 2.5
 (1)
There exists \(\gamma\in(0,1)\) such that \(A(tx,t^{1}y)\geq t^{\gamma}A(x,y)\), \(t\in(0,1)\), \(x,y\in P_{h}\).
 (2)
\(B(t^{1}y)\geq tBy\), \(t\in(0,1)\), \(y\in P_{h}\).
 (3)
There exists a constant \(\delta_{0}>0\) such that \(A(x,y)\geq \delta_{0}By\), \(\forall x,y\in P_{h}\).
3 Main result
Theorem 3.1
 (H_{1}):

\(p,q:(0,1)\rightarrow[0,\infty)\) are continuous, and \(p(t)\), \(q(t)\) are allowed to be singular at \(t=0\) or \(t=1\).
 (H_{2}):

\(f:(0,1)\times(0,\infty)\times(0,\infty) \rightarrow [0,\infty)\), \(g:(0,1)\times(0,\infty)\rightarrow[0,\infty)\) are continuous, and \(f(t,u,v)\), \(g(t,v)\) may be singular at \(t=0\) and \(v=0\).
 (H_{3}):

For fixed \(t\in(0,1)\), and \(v\in(0,\infty)\), \(f(t,u,v)\) is increasing in \(u\in(0,\infty)\); for fixed \(t\in(0,1)\) and \(u\in(0,\infty)\), \(f(t,u,v)\) is decreasing in \(v\in(0,\infty)\); and for fixed \(t\in(0,1)\), \(g(t,v)\) is decreasing in \(v\in(0,\infty)\).
 (H_{4}):

There exists a constant \(\gamma\in(0,1)\) such that for all \(\lambda,t\in(0,1)\) and \(u,v\in(0,\infty)\),$$ f\bigl(t,\lambda u,\lambda^{1}v\bigr)\geq \lambda^{\gamma} f(t,u,v), \qquad g\bigl(t,\lambda^{{1}} v\bigr) \geq\lambda g(t,v). $$(3.1)
 (H_{5}):

\(\int_{0}^{1}(1s)^{\alpha1}p(s)s^{\gamma(1\alpha)}f(s,1,1)\,ds<\infty\) and \(\int_{0}^{1}(1s)^{\alpha1}q(s)s^{1\alpha}g(s,1)\,ds<\infty\).
 (H_{6}):

There exists a constant \(\delta>0\) such that, for all \(t\in(0,1)\) and \(u,v\in(0,\infty)\), \(f(t,u,v)\geq\delta g(t,v)\).
Proof
(3) Next, by (H_{3}) it is easy to prove that A is a mixed monotone operator and B is an decreasing operator.
Remark 3.1
The fractional differential equation with RiemannStieltjes integral conditions considered in Theorem 3.1 is singular, that is, \(f(t,u,v)\) has singularity at \(t=0\) or \(t=1\) and \(v=0\), and \(g(t,v)\) has singularity at \(t=0\) or \(t=1\) and \(v=0\), which generalizes and improves the known results for continuous functions in [16, 27–29].
Remark 3.2
The function \(g(t,v)\) we considered in Theorem 3.1 is decreasing and has singularity at \(t=0\) or \(t=1\) and \(v=0\), which generalizes and improves the results in [30].
Remark 3.3
Comparing with the main results in [31, 32], the nonlinear fractional differential equation we considered is also continuous. However, we get the iterative positive solutions for boundary value problem (1.1) by using the fixed point theorem of the mixed monotone operator, which generalizes and improves the results including singular and nonsingular cases in [17, 31–33].
4 An example
Let \(\alpha=\frac{7}{6}\), \(p(t)=q(t)=t^{\frac{1}{5}}\), \(f(t,u,v)=\sqrt[6]{\frac{u}{tv}}+\frac{1}{\sqrt{tv}}\), \(g(t, v)=\frac {1}{\sqrt{tv(v+1)}}\).
(1) It is obvious that \(p(t)\), \(q(t)\) are singular at \(t=0\). The functions \(f:(0,1)\times(0,\infty)\times(0,\infty)\rightarrow[0,\infty)\) and \(g: (0,1)\times(0,\infty)\rightarrow[0,\infty)\) are continuous. So the conditions (H_{1}) and (H_{2}) hold.
(2) It is obvious that, for fixed \(t\in(0,1)\) and \(v\in(0,\infty)\), \(f(t,u,v)\) is increasing in \(u\in(0,\infty)\), for fixed \(t\in(0,1)\) and \(u\in(0,\infty)\), \(f(t,u,v)\) is decreasing in \(v\in(0,\infty)\), and, for fixed \(t\in(0,1)\), \(g(t,v)\) is decreasing in \(v\in(0,\infty)\).
Therefore, the assumptions of Theorem 3.1 are satisfied. Thus, the conclusions follow from Theorem 3.1.
Declarations
Acknowledgements
The authors would like to thank the referees for their very important comments that improved the results and the quality of the paper. The authors were supported financially by the National Natural Science Foundation of China (11371221, 11571296).
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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