 Research
 Open Access
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
References
 Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) MATHGoogle Scholar
 Miller, KS, Ross, B: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York (1993) MATHGoogle Scholar
 Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999) MATHGoogle Scholar
 Samko, SG, Kilbas, AA, Marichev, OI: Fractional Integrals and Derivatives, Theory and Applications. Gordon & Breach, Yverdon (1993) MATHGoogle Scholar
 Bai, Z, Chen, Y, Lian, H, Sun, S: On the existence of blow up solutions for a class of fractional differential equations. Fract. Calc. Appl. Anal. 17(4), 11751187 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Cui, Y: Uniqueness of solution for boundary value problems for fractional differential equations. Appl. Math. Lett. 51, 4854 (2016) MathSciNetView ArticleMATHGoogle Scholar
 Guo, L, Liu, L, Wu, Y: Existence of positive solutions for singular fractional differential equations with infinitepoint boundary conditions. Nonlinear Anal., Model. Control 21(5), 635650 (2016) MathSciNetView ArticleGoogle Scholar
 Liu, L, Sun, F, Zhang, X, Wu, Y: Bifurcation analysis for a singular differential system with two parameters via to degree theory. Nonlinear Anal., Model. Control 22(1), 3150 (2017) MathSciNetGoogle Scholar
 Jiang, J, Liu, L, Wu, Y: Positive solutions to singular fractional differential system with coupled boundary conditions. Commun. Nonlinear Sci. Numer. Simul. 18(11), 30613074 (2013) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, X, Liu, L, Wu, Y: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 12631274 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, X, Liu, L, Wu, Y: Variational structure and multiple solutions for a fractional advectiondispersion equation. Comput. Math. Appl. 68, 17941805 (2014) MathSciNetView ArticleGoogle Scholar
 Zhang, X, Liu, L, Wu, Y, Wiwatanapataphee, B: The spectral analysis for a singular fractional differential equation with a signed measure. Appl. Math. Comput. 257, 252263 (2015) MathSciNetMATHGoogle Scholar
 Zhang, X: Positive solutions for a class of singular fractional differential equation with infinitepoint boundary value conditions. Appl. Math. Lett. 39, 2227 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, X, Wang, L, Wang, S: Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions. Appl. Math. Comput. 226, 708718 (2014) MathSciNetMATHGoogle Scholar
 Zhao, Y, Sun, S, Han, Z, Li, Q: Positive solutions to boundary value problems of nonlinear fractional differential equations. Abstr. Appl. Anal. 2011, Article ID 390543 (2011) MathSciNetMATHGoogle Scholar
 Yuan, C, Jiang, D, O’Regan, D, Agarwal, RP: Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions. Electron. J. Qual. Theory Differ. Equ. 2012, Article ID 13 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Zou, Y, Liu, L, Cui, Y: The existence of solutions for fourpoint coupled boundary value problems of fractional differential equations at resonance. Abstr. Appl. Anal. 2014, Article ID 314083 (2014) MathSciNetGoogle Scholar
 Coster, CD, Habets, P: TwoPoint Boundary Value Problems: Lower and Upper Solution. Elsevier, Amsterdam (2006) MATHGoogle Scholar
 Guo, D: Extreme solutions of nonlinear second order integrodifferential equations in Banach spaces. J. Appl. Math. Stoch. Anal. 8, 319329 (1995) MathSciNetView ArticleMATHGoogle Scholar
 Ladde, GS, Lakshmikantham, V, Vatsala, AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston (1985) MATHGoogle Scholar
 Bai, Z, Zhang, S, Sun, S, Yin, C: Monotone iterative method for a class of fractional differential equations. Electron. J. Differ. Equ. 2016, Article ID 6 (2016) View ArticleGoogle Scholar
 AlRefai, M, Hajji, MA: Monotone iterative sequences for nonlinear boundary value problems of fractional order. Nonlinear Anal. 74, 35313539 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Cui, Y, Zou, Y: Existence of solutions for secondorder integral boundary value problems. Nonlinear Anal., Model. Control 21(6), 828838 (2016) MathSciNetView ArticleGoogle Scholar
 Cui, Y, Zou, Y: Monotone iterative technique for \((k, nk)\) conjugate boundary value problems. Electron. J. Qual. Theory Differ. Equ. 2015, Article ID 69 (2015) MathSciNetView ArticleMATHGoogle Scholar
 Jankowski, T: Boundary problems for fractional differential equations. Appl. Math. Lett. 28, 1419 (2014) MathSciNetView ArticleMATHGoogle Scholar
 Lin, L, Liu, X, Fang, H: Method of upper and lower solutions for fractional differential equations. Electron. J. Differ. Equ. 2012, Article ID 1 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Liu, X, Jia, M, Ge, W: The method of lower and upper solutions for mixed fractional fourpoint boundary value problem with pLaplacian operator. Appl. Math. Lett. 65, 5662 (2017) MathSciNetView ArticleMATHGoogle Scholar
 Ramirez, JD, Vatsala, AS: Monotone iterative technique for fractional differential equations with periodic boundary conditions. Opusc. Math. 29, 289304 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Sun, Y, Sun, Y: Positive solutions and monotone iterative sequences for a fractional differential equation with integral boundary conditions. Adv. Differ. Equ. 2014, Article ID 29 (2014) MathSciNetView ArticleGoogle Scholar
 Syam, M, AlRefai, M: Positive solutions and monotone iterative sequences for a class of higher order boundary value problems of fractional order. J. Fract. Calc. Appl. 4, Article ID 14 (2013) Google Scholar
 Wang, G: Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments. J. Comput. Appl. Math. 236, 24252430 (2012) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, S, Su, X: The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reversed order. Comput. Math. Appl. 62, 12691274 (2011) MathSciNetView ArticleMATHGoogle Scholar
 Zhang, S: Monotone iterative method for initial value problem involving RiemannLiouville fractional derivatives. Nonlinear Anal. 71, 20872093 (2009) MathSciNetView ArticleMATHGoogle Scholar
 Guo, D, Sun, J: Nonlinear Integral Equations. Shandong Science and Technology Press, Jinan (1987) (in Chinese) Google Scholar