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Fractional boundary value problems with RiemannLiouville fractional derivatives
 Jingjing Tan^{1}Email author and
 Caozong Cheng^{1}
https://doi.org/10.1186/s136620150413y
© Tan and Cheng; licensee Springer. 2015
 Received: 3 November 2014
 Accepted: 10 February 2015
 Published: 7 March 2015
Abstract
In this paper, by employing two fixed point theorems of a sum operators, we investigate the existence and uniqueness of positive solutions for the following fractional boundary value problems: \(D_{0+}^{\alpha}x(t)=f(t, x(t), x(t))+g(t, x(t))\), \(0< t <1\), \(1< \alpha<2\), where \(D_{0+}^{\alpha}\) is the standard RiemannLiouville fractional derivative, subject to either the boundary conditions \(x(0)=x(1)=0\) or \(x(0)=0\), \(x(1)=\beta x(\eta)\) with \(\eta, \beta\eta^{\alpha1} \in(0,1)\). We also construct an iterative scheme to approximate the solution. As applications of the main results, two examples are given.
Keywords
 fractional differential equation
 boundary value problem
 fixed point theorem
 mixed monotone operators
1 Introduction
Fractional differential equations are important mathematical models of some practical problems in many fields such as polymer rheology, chemistry physics, heat conduction, fluid flows, electrical networks, and many other branches of science (see [1–7]). Consequently, the fractional calculus and its applications in various fields of science and engineering have received much attention, and many papers and books on fractional calculus, fractional differential equations have appeared (see [8–15]). It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of the existence of positive solutions for nonlinear fractional differential equation boundary value problems (see [16–21]). However, there are few papers to deal with the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.
Motivated by the results mentioned above, in this paper, we study the existence and uniqueness of positive solutions for the BVP (1) and (2). We have found that no result has been established for the existence and uniqueness of positive solutions for the problem (1) and (2) of fractional differential equation. This paper aims to establish the existence and uniqueness of positive solutions for the problem (1) and (2). The technique relies on two fixed point theorems of a sum operator. The method used in this paper is different from the ones in the papers mentioned above. In addition, we also construct an iterative sequence to approximate the 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 the fixed point theorem on mixed monotone operators. Finally, in Section 4, we give two examples to demonstrate our results.
2 Preliminaries
 (1)
\(\lceil\alpha\rceil\): the smallest integer greater than or equal to α;
 (2)
\([\alpha]\): the integer part of the number α;
 (3)
\(\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha1}e^{t}\,dt\).
Definition 2.1
([27])
Definition 2.2
([27])
Lemma 2.1
([23])
Denote by E a real Banach space. Recall that a nonempty closed convex set \(P\subset E\) is a cone if it satisfies (i) \(x\in P\), \(\lambda\geq0\Rightarrow\lambda x\in P \) and (ii) \(x\in P\), \(x\in P\Rightarrow x=\theta\). Suppose that \((E, \\cdot\)\) is a real Banach space which is partially ordered by a cone \(P \subset E\), i.e., \(x\leq y\) if and only if \(yx\in P\). The 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\\), and N is called the normal constant. Putting \(P^{o}=\{x\in P\mid x \mbox{ is an interior point of } P \}\), the cone P is said to be solid if its interior \(P^{o}\) is nonempty. If \(x_{1}, x_{2}\in E\), 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 \(x_{2}\). We say that an operator \(A: E\rightarrow E\) is increasing (decreasing) if \(x\leq y\) implies \(Ax\leq Ay\) (\(Ax\geq Ay\)).
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 \(h> \theta\) (i.e., \(h\geq\theta\) and \(h\neq\theta\)), let \(P_{h}=\{x\in E\mid x\sim h\}\). If \(P_{h}\subset P^{o}\), then \(P_{h}=P^{o}\).
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 \(\x\=\sup\{ x(t): t\in[0, 1]\}\) for any \(x\in C[0, 1]\). Notice that this space can be equipped with a partial order given by \(x, y\in C[0, 1]\), \(x\leq y\Longleftrightarrow x(t)\leq y(t)\) for \(t\in[0, 1]\). Let \(P\subset C[0, 1]\) by \(P=\{x\in C[0, 1]\mid x(t)\geq0, t\in[0, 1]\}\). Clearly P is a normal cone in \(C[0, 1]\) and the normality constant is 1.
Lemma 2.2
([22])
Lemma 2.3
([22])
 (1)
\(G(t, s)\in C([0, 1]\times[0, 1])\);
 (2)
\(G(t, s)>0\) for \(t, s \in(0, 1)\);
 (3)for all \(t, s \in(0, 1)\),$$\frac{\alpha1}{\Gamma(\alpha)}t^{\alpha1}(1t) (1s)^{\alpha1}s\leq G(t, s)\leq\frac{1}{\Gamma(\alpha)}t^{\alpha1}(1t) (1s)^{\alpha2}. $$(5)
Lemma 2.4
Proof
The proof is similar to Lemma 2.2 and omitted. □
Lemma 2.5
([25])
Lemma 2.6
([25])
 (1)
\(H(t, s)\in C([0, 1]\times[0, 1])\);
 (2)
\(H(t, s)>0\) for \(t, s \in(0, 1)\);
 (3)for all \(t, s \in(0, 1)\),where \(0< M_{0}=\min\{1\beta\eta^{\alpha1}, \beta\eta^{\alpha1}\}<1\).$$ \frac{M_{0}t^{\alpha1}s(1s)^{\alpha1}}{\Gamma(\alpha)(1\beta\eta ^{\alpha1})}\leq H(t, s)\leq\frac{t^{\alpha1}(1s)^{\alpha1}}{\Gamma (\alpha)(1\beta\eta^{\alpha1})}, $$(8)
Lemma 2.7
Proof
The proof is similar to Lemma 2.5 and omitted. □
Definition 2.3
([28])
\(A: P\times P\rightarrow P\) is said to be a mixed monotone operator if \(A(u, v)\) is increasing in u and decreasing in v, i.e., \(u_{i}, v_{i}\ (i=1, 2)\in P\), \(u_{1}\leq u_{2}\), \(v_{1}\geq v_{2}\) imply \(A(u_{1}, v_{1})\leq A(u_{2}, v_{2})\). The element \(x\in P\) is called a fixed point of A if \(A(x, x)=x\).
Definition 2.4
([29])
Definition 2.5
([29])
Lemma 2.8
([28])
 (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 \(\gamma v_{0}\leq u_{0}< v_{0}\), \(u_{0}\leq A(u_{0}, v_{0})+ Bu_{0}\leq A(v_{0}, u_{0})+Bv_{0}\leq v_{0}\);
 (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 sequences \(x_{n}= A(x_{n1}, y_{n1})+Bx_{n1}\), \(y_{n}=A(y_{n1}, x_{n1})+By_{n1}\), \(n=1, 2, \ldots\) , we have \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*}\) as \(n\rightarrow \infty\).
Lemma 2.9
([28])
 (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 \(\gamma v_{0}\leq u_{0}< v_{0}\), \(u_{0}\leq A(u_{0}, v_{0})+ Bu_{0}\leq A(v_{0}, u_{0})+Bv_{0}\leq v_{0}\);
 (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 sequences \(x_{n}= A(x_{n1}, y_{n1})+Bx_{n1}\), \(y_{n}=A(y_{n1}, x_{n1})+By_{n1}\), \(n=1, 2, \ldots\) , we have \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*}\) as \(n\rightarrow \infty\).
Lemma 2.10
([28])
Let \(\alpha\in(0, 1)\) and \(A: P\times P\rightarrow P\) be a mixed monotone operator. Assume that (12) holds and there is \(h_{0}> \theta\) such that \(A(h_{0}, h_{0})\in P_{h}\).
 (1)
\(A: P_{h} \times P_{h}\rightarrow P_{h}\);
 (2)
there exist \(u_{0}, v_{0} \in P_{h}\) and \(r\in(0, 1)\) such that \(rv_{0}\leq u_{0}\leq v_{0}\), \(u_{0}\leq A(u_{0}, v_{0})\leq A(v_{0}, u_{0})\leq v_{0}\);
 (3)
the operator equation \(A(x, x)=x\) has a unique solution \(x^{*}\) in \(P_{h}\);
 (4)
for any initial values \(x_{0}, y_{0} \in P_{h}\), constructing successively the sequences \(x_{n}= A(x_{n1}, y_{n1})\), \(y_{n}=A(y_{n1}, x_{n1})\), \(n=1, 2, \ldots\) , we have \(x_{n}\rightarrow x^{*}\) and \(y_{n}\rightarrow x^{*}\) as \(n\rightarrow\infty\).
3 Main results
In this section, we establish the existence and uniqueness of positive solutions results for the problems (1) and (2), respectively.
First, we give the existence and uniqueness of positive solutions to the problem (1).
Theorem 3.1
 (H_{1}):

\(f(t, x, y): [0, 1]\times[0, +\infty)\times[0, +\infty )\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\) for fixed \(t\in[0, 1]\) and \(y\in[0, +\infty)\) decreasing in \(y\in[0, +\infty)\) for fixed \(t\in[0, 1]\) and \(x\in[0, +\infty)\);
 (H_{2}):

\(g(t, x): [0, 1]\times[0, +\infty)\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\) for fixed \(t\in[0, 1]\), \(g(t, 0) \not\equiv0\);
 (H_{3}):

there exists a constant \(\delta_{0}>0\) such that \(f(t, x, y)\geq \delta_{0}g(t, x)\), \(t\in[0, 1]\), \(x, y\geq0\);
 (H_{4}):

\(g(t, \lambda x)\geq\lambda g(t, x)\) for \(\lambda\in(0, 1)\), \(t\in[0, 1]\), \(u\in[0, +\infty)\) and there exists a constant \(\xi\in (0, 1) \) such that \(f(t, \lambda x, \lambda^{1}y)\geq\lambda^{\xi }f(t, x, y)\), \(\forall t\in[0, 1]\), \(x, y\in[0, +\infty)\).
Proof
 (a)
\(\exists a_{1}>0\) and \(a_{2}>0\), such that, for all \(t\in[0, 1]\), \(a_{2}h(t)\leq A_{1}(h, h)(t)\leq a_{1}h(t)\);
 (b)
\(\exists b_{1}>0\) and \(b_{2}>0\), such that, for all \(t\in[0, 1]\), \(b_{2}h(t)\leq B_{1}h(t)\leq b_{1}h(t)\).
Combining the proof of Theorem 3.1 with Lemma 2.10, we can prove the following conclusion.
Corollary 3.1
 \((\mathrm{H}_{1})'\) :

\(f(t, x, y): [0, 1]\times[0, +\infty)\times[0, +\infty )\rightarrow[0, +\infty)\) is continuous and increasing in \(x\in[0, +\infty)\) for fixed \(t\in[0, 1]\) and \(y\in [0, +\infty)\) decreasing in \(y\in[0, +\infty)\) for fixed \(t\in[0, 1]\) and \(x\in[0, +\infty)\), \(f(t, 0, 1) \not\equiv0\);
 \((\mathrm{H}_{2})'\) :

there exists a constant \(\xi\in(0, 1)\) such that \(f(t, \lambda x, \lambda^{1}y)\geq\lambda^{\xi}f(t, x, y)\), \(t\in[0, 1]\), \(\lambda\in(0, 1)\), \(x, y\in[0, +\infty)\).
By using Lemma 2.9 we can also prove the following theorem.
Theorem 3.2
 (H_{5}):

there exists a constant \(\delta_{0}>0\) such that \(f(t, x, y)\leq \delta_{0}g(t, x)\), \(t\in[0, 1]\), \(x, y\geq0\);
 (H_{6}):

there exists a constant \(\xi\in(0, 1)\) such that \(g(t, \lambda x)\geq\lambda^{\xi}g(t, x)\) for \(\lambda\in(0, 1)\), \(t\in[0, 1]\), \(x\in[0, +\infty)\), and \(f(t, \lambda x, \lambda^{1}y)\geq\lambda f(t, x, y)\), \(\forall t\in[0, 1]\), \(\lambda\in(0,1)\), \(x, y\in[0, +\infty)\).
Proof
The proof is similar to that given for Theorem 3.1. We omit it. □
Next, we present the existence and uniqueness of a positive solution to the problem (2).
Theorem 3.3
Assume that (H_{1})(H_{4}) hold. Then
Proof
 (a)
\(\exists a'_{1}>0\) and \(a'_{2}>0\), such that, for all \(t\in[0, 1]\), \(a'_{2}h(t)\leq A_{2}(h, h)(t)\leq a'_{1}h(t)\);
 (b)
\(\exists b'_{1}>0\) and \(b'_{2}>0\), such that, for all \(t\in[0, 1]\), \(b'_{2}h(t)\leq B_{2}h(t)\leq b'_{1}h(t)\).
From Lemma 2.9 we also prove the following theorem.
Theorem 3.4
Assume that (H_{1}), (H_{2}), (H_{5}), and (H_{6}) hold. Then
Proof
The proof is similar to that given for Theorem 3.3. We omit it. □
4 Example
In this section, we give two examples to illustrate our results.
Example 4.1
In this case, \(\alpha=\frac{5}{3}\). Problem (14) can be regarded as a boundary value problem of the form (1) with \(f(t, x, y)=x^{\frac{1}{3}}+y^{\frac{1}{3}}+t^{2}+\frac{\pi}{2}\) and \(g(t, x)=\arctan x+t^{3}\). Now we verify that conditions (H_{1})(H_{4}) are satisfied.
Example 4.2
In this case, \(\alpha=\frac{3}{2}\). Problem (15) can be regard as a boundary value problem of form (2) with \(f(t, x, y)=x^{\frac{1}{2}}+y^{\frac{1}{2}}+t^{2}\) and \(g(t, x)=x^{\frac{1}{2}}+t^{3}\). Now we verify that conditions (H_{1})(H_{4}) are satisfied.
Declarations
Acknowledgements
The authors would like to thank the referees for their valuable suggestions and comments.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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