About the existence of solutions for a hybrid nonlinear generalized fractional pantograph equation

Abstract

The main purpose of this paper is to study the existence of solutions for the following hybrid nonlinear fractional pantograph equation $$\{\begin{array}{rl}& D_{0+}^{\alpha }\left[\frac{x(t)}{f(t,x(t),x(\phi (t)))}\right]=g(t,x(t),x(\rho (t))),0<t<1\\ & x(0)=0,\end{array}$$ where $\alpha \in (0,1)$, $\phi$ and $\rho$ are functions from $[0,1]$ into itself and $D_{0+}^{\alpha }$ denotes the Riemann-Liouville fractional derivative. The main tool of our study is a generalization of Darbo's fixed point theorem associated to measures of non-compactness. Also, we present an example illustrating our results.

15 pages