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Monotone iterative technique for nonlinear boundary value problems of fractional order \(p\in(2,3]\)
 Yujun Cui^{1, 2}Email author,
 Qiao Sun^{2} and
 Xinwei Su^{3}
https://doi.org/10.1186/s136620171314z
© The Author(s) 2017
Received: 2 March 2017
Accepted: 8 August 2017
Published: 22 August 2017
Abstract
After establishing a comparison result of the nonlinear RiemannLiouville fractional differential equation of order \({p\in(2, 3]}\), we obtain the existence of maximal and minimal solutions, and the uniqueness result for fractional differential equations. As an application, an example is presented to illustrate the main results.
Keywords
 fractional differential equation
 monotone iterative technique
 comparison result
1 Introduction
In the past years, much attention has been devoted to the study of fractional differential equations due to the fact that they have many applications in a broad range of areas such as physics, chemistry, aerodynamics, electrodynamics of complex medium and polymer rheology. Many existence results of solutions to initial value problems and boundary value problems for fractional differential equations have been established in terms of all sorts of methods; see, e.g., [1–17] and the references therein. Generally speaking, it is difficult to get the exact solution for fractional differential equations. To obtain approximate solutions of nonlinear fractional differential problems, we can use the monotone iterative technique and the lower and upper solutions. This technique is well known and can be used for both initial value problems and boundary value problems for differential equations [18–20]. Recently, this method has also been applied to initial value problems and boundary value problems for fractional differential equations; see [21–33]. To the best of our knowledge, there is still little utilization of the monotone iterative method to a fractional differential equation of order \(p\in(2,3]\).
2 Preliminaries
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the recent literature; see [1–4].
Definition 2.1
[1]
Definition 2.2
[1]
Lemma 2.1
[1]
For brevity, let us take \(E=\{u: D^{p}u(t)\in C(0,1)\cap L(0,1)\}\). In the Banach space \(C[0,1]\), in which the norm is defined by \(\x\=\max_{t\in[0,1]}x(t)\), we set \(P=\{ x\in C[0,1] \mid x(t)\geq0, \forall t\in[0,1]\} \). P is a positive cone in \(C[0,1]\). Throughout this paper, the partial ordering is always given by P.
The following are the existence and uniqueness results of a solution for a linear boundary value problem, which is important for us in the following analysis.
Lemma 2.2
[6]
The following properties of Green’s function play an important part in this paper.
Lemma 2.3
[10]
 (1)
\(t^{p1}(1t)s(1s)^{p1}\leq\Gamma(p)G(t,s)\leq(p1)s(1s)^{p1}\), \(t,s\in(0,1) \),
 (2)
\(t^{p1}(1t)s(1s)^{p1}\leq\Gamma(p)G(t,s)\leq(p1)t^{p1}(1t)\), \(t,s\in(0,1) \).
Lemma 2.4
Proof
Lemma 2.5
Proof
Lemma 2.6
S is a completely continuous operator and \(S(P)\subset P_{1}\).
Proof
Lemma 2.7
Proof
Lemma 2.8
[34]
Suppose that \(S:C[0,1]\rightarrow C[0,1]\) is a completely continuous linear operator and \(S(P)\subset P\). If there exist \(\psi\in C[0,1]\setminus(P)\) and a constant \(c>0\) such that \(cS\psi\geq\psi\), then the spectral radius \(r(S)\neq0\) and S has a positive eigenfunction corresponding to its first eigenvalue \(\lambda_{1}=(r(S))^{1}\), i.e., \(\varphi=\lambda_{1}S\varphi\).
Lemma 2.9
Suppose that S is defined by (2.10), then the spectral radius \(r(S)\neq0\) and S has a positive eigenfunction \(\varphi^{*}(t)\) corresponding to its first eigenvalue \(\lambda_{1}=(r(S))^{1}\).
Proof
3 Main results
 \((H_{1})\) :

There exist \(\alpha_{0}, \beta_{0}\in E\) with \(\alpha_{0}(t)\leq\beta _{0}(t)\) such that$$\begin{gathered} D^{p} \alpha_{0}(t)+f \bigl(t, \alpha_{0}(t) \bigr) \geq0,\quad t\in(0,1), \alpha_{0}(0)= \alpha_{0}'(0)=0, \alpha_{0}(1)\leq0, \\ D^{p} \beta_{0}(t)+f \bigl(t, \beta_{0}(t) \bigr) \leq0, \quad t\in(0,1), \beta_{0}(0)= \beta_{0}'(0)=0, \beta_{0}(1)\geq0. \end{gathered} $$
 \((H_{2})\) :
Theorem 3.1
Suppose that \((H_{1})\) and \((H_{2})\) hold. Then there exist monotone iterative sequences \(\{\alpha_{n}(t)\}, \{\beta_{n}(t)\}\) which converge uniformly on \([0,1]\) to the extremal solutions of problem (1.1) in the sector \(\Omega=\{v\in C[0,1]: \alpha_{0}(t)\leq v(t) \leq\beta_{0}(t), t\in[0,1]\}\).
Proof
The uniqueness results of a solution to problem (1.1) are established in the following theorem.
Theorem 3.2
Proof
Remark 3.1
4 Example
Therefore, by Theorem 3.2, there exist iterative sequences \(\{\alpha_{n}\} \), \(\{\beta_{n}\}\) which converge uniformly to the unique solution in \([\alpha_{0},\beta _{0}]\), respectively.
Declarations
Acknowledgements
The Project supported by the National Natural Science Foundation of China (11371221, 11371364, 11571207), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.
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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