 Research
 Open Access
Positive solutions of fractional differential equation nonlocal boundary value problems
 Jingjing Tan^{1}Email author,
 Caozong Cheng^{1} and
 Xinguang Zhang^{2, 3}
https://doi.org/10.1186/s1366201505828
© Tan et al. 2015
 Received: 1 June 2015
 Accepted: 23 July 2015
 Published: 19 August 2015
Abstract
In this paper, we study the existence and uniqueness of positive solutions for a class of higherorder nonlocal fractional differential equations with RiemannStieltjes integral boundary conditions. We firstly convert the problem to an equivalent integral equation, and then by applying a fixed point theorem of a sum operator, the existence and uniqueness of positive solutions is established. Furthermore, an iterative scheme to approximate the solution is constructed and an example is given to illuminate the application of the main results.
Keywords
 fractional differential equation
 boundary value problem
 fixed point theorem
 mixed monotone operators
 RiemannStieltjes integral
1 Introduction
Motivated by the work mentioned above, we focus on the existence and uniqueness of positive solutions for the nonlocal BVP (1) based on a fixed point theorem of a sum operator. Our work presented in this work has the following new features. Firstly, the existence and uniqueness of positive solutions are obtained, which possess a nice estimate, i.e., there exist \(\lambda>\mu >0\) such that \(\mu t^{\alpha1} \leq x^{*}(t)\leq\lambda t^{\alpha1}\); secondly, the boundary conditions are nonlocal which involve the RiemannStieltjes integral of x with respect to A, moreover, dA can be a signed measure, this implies that it can cover the multipoint and integral boundary value problems as special cases; thirdly, we also construct an iterative sequence to approximate the positive solution.
The rest of this paper is organized as follows. In Section 2, we recall some definitions and facts. In Section 3, the main results are discussed by using the properties of the Green function and a fixed point theorem of a sum operator. Finally, in Section 4, an illustrative example is also presented.
2 Preliminaries
 (i)
\(x\in P\), \(\lambda\geq0\Rightarrow\lambda x\in P \), and
 (ii)
\(x\in P\), \(x\in P\Rightarrow x=\theta\),
We say that an operator \(A: E\rightarrow E\) is increasing (decreasing) if \(x\leq y\) implies \(Ax\leq Ay\) (\(Ax\geq Ay\)).
Definition 2.1
([28])
Proposition 2.1
([28])
In [29], the authors obtained the following results.
Lemma 2.1
([29])
Lemma 2.2
([29])
 (1)
\(G(t, s)>0\), for all \(t, s\in(0, 1)\);
 (2)$$ (1t)t^{\alpha1}s(1s)^{\alpha1} \leq\Gamma(\alpha)G(t, s) \leq (\alpha1) (1t)t^{\alpha1}, \quad \textit{for } t, s\in[0, 1]. $$
The following lemmas are obtained by Zhang and Han [27].
Lemma 2.3
([27])
Lemma 2.4
 (1)
\(H(t, s)>0\), for all \(t, s\in(0, 1)\);
 (2)$$ \frac{1}{1\Lambda}t^{\alpha1}\mathscr{G}_{A}(s) \leq H(t, s)\leq \biggl(\frac{\Vert \mathscr{G}_{A}(s)\Vert }{1\Lambda} +\frac{1}{\Gamma(\alpha1)} \biggr)t^{\alpha1}, \quad \textit{for } t, s\in[0, 1]. $$
We recall the following lemmas and definitions which are important to prove our main results.
Definition 2.2
([30])
Definition 2.3
([30])
Lemma 2.5
([31])
 (i)
there is a \(h_{0}\in P_{h}\) such that \(A(h_{0}, h_{0})\in P_{h}\) and \(Bh_{0}\in P_{h}\);
 (ii)
there exists a constant \(\delta_{0}>0\) such that \(A(x, y)\geq\delta_{0}Bx\), \(\forall x, y\in P\).
 (1)
\(A: P_{h}\times P_{h}\rightarrow P_{h}\), \(B: P_{h}\rightarrow P_{h}\);
 (2)there exist \(u_{0}, v_{0}\in P_{h}\) and \(\gamma\in(0, 1)\) such that$$\gamma v_{0}\leq u_{0}< v_{0},\qquad u_{0} \leq A(u_{0}, v_{0})+ Bu_{0}\leq A(v_{0}, u_{0})+Bv_{0}\leq v_{0}; $$
 (3)
the operator equation \(A(x, x)+Bx=x\) has a unique solution \(x^{*}\) in \(P_{h}\);
 (4)for any initial values \(x_{0}, y_{0} \in P_{h}\), constructing successively sequencesthen \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*}\) as \(n\rightarrow \infty\).$$x_{n}= A(x_{n1}, y_{n1})+Bx_{n1}, y_{n}=A(y_{n1}, x_{n1})+By_{n1},\quad n=1, 2, \ldots, $$
Lemma 2.6
([32])
 (i)
there exists a \(h>\theta\) such that \(Ah\in P_{h}\) and \(Bh\in P_{h}\);
 (ii)
there exists a constant \(\delta_{0}>0\) such that \(Ax\geq\delta_{0}Bx\), \(\forall x\in P\).
Lemma 2.7
([31])
 (i)
there is a \(h_{0}\in P_{h}\) such that \(A(h_{0}, h_{0})\in P_{h}\) and \(Bh_{0}\in P_{h}\);
 (ii)
there exists a constant \(\delta_{0}>0\) such that \(A(x, y)\leq\delta_{0}Bx\), \(\forall x, y\in P\).
 (1)
\(A: P_{h}\times P_{h}\rightarrow P_{h}\) and \(B: P_{h}\rightarrow P_{h}\);
 (2)there exist \(u_{0}, v_{0}\in P_{h}\) and \(\gamma\in(0, 1)\) such that$$\gamma v_{0}\leq u_{0}< v_{0}, \qquad u_{0}\leq A(u_{0}, v_{0})+ Bu_{0}\leq A(v_{0}, u_{0})+Bv_{0}\leq v_{0}; $$
 (3)
the operator equation \(A(x, x)+Bx=x\) has a unique solution \(x^{*}\) in \(P_{h}\);
 (4)for any initial values \(x_{0}, y_{0} \in P_{h}\), constructing successively the sequenceswe have \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*}\) as \(n\rightarrow\infty\).$$x_{n}= A(x_{n1}, y_{n1})+Bx_{n1},\qquad y_{n}=A(y_{n1}, x_{n1})+By_{n1}, \quad n=1, 2, \ldots, $$
Lemma 2.8
Proof
By using similar method to Lemma 2.3 and standard arguments, we can show the conclusion. □
3 Main results
The basic space used in this paper is the space \(C[0, 1]\), it is a Banach space if it is endowed with the norm \(\Vert x\Vert ={\sup}\{\vert x(t)\vert : t\in[0, 1]\}\) for any \(x\in C[0, 1]\), and E can equip with a partial order \(x, y\in C[0, 1]\), \(x\leq y\Longleftrightarrow x(t)\leq y(t)\) for \(t\in[0, 1]\). Let \(P=\{x\in C[0, 1]\mid x(t)\geq0, t\in [0, 1]\}\). Clear P is a normal cone in \(C[0, 1]\) and the normality constant is 1.
First, we give the existence and uniqueness of positive solutions to the BVP (1).
Theorem 3.1
 (H_{1}):

A is a function of bounded variation such that \(\mathscr{G}_{A}(s)\geq0\) for \(s\in[0, 1]\) and \(\Lambda\in[0, 1)\);
 (H_{2}):

\(f(t, x, y): [0, 1]\times[0, +\infty)\times[0, +\infty )\rightarrow[0, +\infty)\) is continuous and increasing in x and y decreasing, and there exists a constant \(\gamma\in(0, 1) \) such that$$f\bigl(t, \lambda x, \lambda^{1}y\bigr)\geq\lambda^{\gamma}f(t, x, y),\quad \forall t\in[0, 1], x, y\in[0, +\infty); $$
 (H_{3}):

\(g(t, x): [0, 1]\times[0, +\infty)\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\), \(g(t, \lambda x)\geq\lambda g(t, x)\) for \(\lambda\in(0, 1)\), \((t,x)\in [0, 1]\times[0, +\infty)\), and \(g(t, 0) \not\equiv0\);
 (H_{4}):

there exists a constant \(\delta_{0}>0\) such that \(f(t, x, y)\geq\delta_{0}g(t, x)\), \(t\in[0, 1]\), \(x, y\geq0\).
Proof
 (a)
there exist \(a_{1}, a_{2}>0\) such that \(a_{2}h(t)\leq A_{1}(h, h)(t)\leq a_{1}h(t)\), \(t\in[0, 1]\);
 (b)
there exist \(b_{1}, b_{2}>0\) such that \(b_{2}h(t)\leq B_{1}h(t)\leq b_{1}h(t)\), \(t\in[0, 1]\).
Remark 3.1
In Theorem 3.1, we cannot only obtain the existence of unique positive solution, but also construct an iterative sequence for approximate the unique positive solution for any initial value in \(P_{h}\). Moreover, the estimate of unique positive solution is derived with \(\mu t^{\alpha1}\leq x^{*}(t)\leq\lambda t^{\alpha1}\) for some \(\lambda> \mu>0\). Thus the property of the unique positive solution is more clear.
If \(g(t, x(t))\equiv0\), from Remark 2.1, we have the following corollary.
Corollary 3.1
Assume that (H_{1}) and (H_{2}) hold. If \(f(t, 0) \not\equiv0\).
If the nonlinear term \(f(t, x, x)\) is replaced by \(f(t, x)\), we can get the following results.
Theorem 3.2
 (H_{5}):

\(f(t, x): [0, 1]\times[0, +\infty)\rightarrow[0, +\infty)\) is continuous and increasing with respect to the second argument, and there exists a constant \(\gamma\in(0, 1)\) such that \(f(t, \lambda x)\geq\lambda^{\gamma} f(t, x)\), \(\forall t\in[0, 1]\), \(\lambda\in(0, 1)\), \(x\in[0, \infty)\);
 (H_{6}):

there exists a constant \(\delta_{0}>0\) such that \(f(t, x)\geq\delta_{0}g(t, x)\) for \(t\in[0, 1]\), \(x\geq0\).
Proof
Corollary 3.2
Assume that (H_{1}) and (H_{5}) hold.
From the proof of Theorem 3.1 and using Lemma 2.7, we can prove the following conclusion.
Theorem 3.3
 (H_{7}):

\(f(t, x, y): [0, 1]\times[0, +\infty)\times[0, +\infty )\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\) for fixed \(t\in[0, 1]\), \(y\in[0, +\infty)\) decreasing in \(y\in[0, +\infty)\) for fixed \(t\in[0, 1]\), \(x\in[0, +\infty)\), and \(f(t, \lambda x, \lambda^{1}y)\geq\lambda f(t, x, y)\), \(\forall t\in [0, 1]\), \(x, y\in[0, +\infty)\);
 (H_{8}):

\(g(t, x): [0, 1]\times[0, +\infty)\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\) for fixed \(t\in[0, 1]\), and there exists a constant \(\gamma\in(0, 1) \) such that \(g(t, \lambda x)\geq\lambda^{\gamma} g(t, x)\) for \(\lambda\in(0, 1)\), \(t\in[0, 1]\), \(u\in[0, +\infty)\) and \(g(t, 0) \not\equiv0\);
 (H_{9}):

there exists a constant \(\delta_{0}>0\) such that \(f(t, x, y)\leq\delta_{0}g(t, x)\), \(t\in[0, 1]\), \(x, y\geq0\).
If the nonlocal boundary condition \(x(1)=\int_{0}^{1}x(s)\,dA(s)\) replace by local boundary condition \(u(1)=0\), we can obtain the following results.
Corollary 3.3
Corollary 3.4
Corollary 3.5
Corollary 3.6
4 Example
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable suggestions and comments.
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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