A coupled fixed point theorem and application to fractional hybrid differential problems
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- 作者：Tahereh Bashiri ; Seyed Mansour Vaezpour…
- 关键词：hybrid initial value problem ; Banach space ; coupled fixed point theorem ; Riemann ; Liouville fractional derivative
- 刊名：Fixed Point Theory and Applications
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：2016
- 期：1
- 全文大小：1,602 KB
- 刊物主题：Analysis; Mathematics, general; Applications of Mathematics; Differential Geometry; Topology; Mathematical and Computational Biology;
- 出版者：Springer International Publishing
- ISSN：1687-1812
文摘
This paper is devoted to the study of the existence of solution to the following system of fractional hybrid differential equations: $$ \textstyle\begin{cases} D^{p} [x(t)- f(t,x(t))] = g (t,y(t),I^{\alpha}(y(t))) ,\quad \mbox{a.e. }t \in J, \\ D^{p} [y(t)- f(t,y(t))] = g (t,x(t),I^{\alpha}(x(t))) ,\quad \mbox{a.e. }t \in J, 0 < p < 1, \alpha>0, \\ x (0) = 0,\qquad y (0) = 0, \end{cases} $$ where \(D^{\alpha}\) is the R-L fractional derivative of order α, \(J=[0,T]\), \(T>0\), and the functions \(f :J\times\mathbb{R} \times\mathbb{R} \rightarrow\mathbb{R}\), \(f(0,0)=0\) and \(g:J \times\mathbb{R} \times\mathbb{R}\rightarrow\mathbb{R}\) satisfy certain conditions. The proof of the existence theorem is based on a coupled fixed point theorem of Krasnoselskii type, which extends a fixed point theorem of Burton (Appl. Math. Lett. 11:85-88, 1998). Finally, our results are illustrated by a concrete example.