01.06.2015 | Original Research | Ausgabe 1-2/2015
The existence of solutions for boundary value problems of two types fractional perturbation differential equations
- Zeitschrift:
- Journal of Applied Mathematics and Computing > Ausgabe 1-2/2015
Wichtige Hinweise
This research is supported by the Natural Science Foundation of China (61374074), Natural Science Outstanding Youth Foundation of Shandong Province (JQ201119) and supported by Shandong Provincial Natural Science Foundation (ZR2012AM009, ZR2013AL003).
Abstract
In this paper, we study the existence of solutions for the boundary value problems of fractional perturbation differential equations or subject to where \(1<\alpha <2,\,D^{\alpha }\) is the standard Caputo fractional derivatives. Using some fixed point theorems, we prove the existence of solutions to the two types. For each type we give an example to illustrate our results.
$$\begin{aligned} D^{\alpha }\left( \frac{x(t)}{f(t,x(t))}\right) =g(t,x(t)),\;\;a.e.\;t\in J=[0,1], \end{aligned}$$
$$\begin{aligned} D^{\alpha }\left( x(t)-f(t,x(t))\right) =g(t,x(t)),\;\;a.e.\;t\in J, \end{aligned}$$
$$\begin{aligned} x(0)=y(x),\;\;x(1)=m, \end{aligned}$$
