Electron. J. Diff. Equ., Vol. 2016 (2016), No. 06, pp. 1-8.

Monotone iterative method for fractional differential equations

Zhanbing Bai, Shuo Zhang, Sujing Sun, Chun Yin

Abstract:
In this article, by using the lower and upper solution method, we prove the existence of iterative solutions for a class of fractional initial value problem with non-monotone term
$$\displaylines{ D_{0+}^\alpha u(t)=f(t, u(t)), \quad t \in (0, h), \cr t^{1-\alpha}u(t)\big|_{t=0} = u_0 \neq 0, }$$
where $0<h<+\infty$, $f\in C([0, h]\times \mathbb{R}, \mathbb{R})$, $D_{0+}^\alpha u (t) $ is the standard Riemann-Liouville fractional derivative, $0<\alpha< 1$. A new condition on the nonlinear term is given to guarantee the equivalence between the solution of the IVP and the fixed-point of the corresponding operator. Moreover, we show the existence of maximal and minimal solutions.

Submitted October 14, 2015. Published January 6, 2016.
Math Subject Classifications: 34B15, 34A08.
Key Words: Fractional initial value problem; lower and upper solution method; existence of solutions.

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Zhanbing Bai
College of Mathematics and System Science
Shandong University of Science and Technology
Qingdao 266590, China
email: zhanbingbai@163.com
Shuo Zhang
College of Mathematics and System Science
Shandong University of Science and Technology
Qingdao 266590, China
email: 18353250431@163.com
Sujing Sun
College of Information Science and Engineering
Shandong University of Science and Technology
Qingdao 266590, China
email: kdssj@163.com
Chun Yin
School of Automation Engineering
University of Electronic Science and Technology of China
Chengdu 611731, China
email: yinchun.86416@163.com

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