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

The main purpose of this paper is to study the existence of solutions for the following hybrid nonlinear fractional pantograph equation $$\left\{\begin{aligned} &D_{0+}^\alpha \left[\frac{x(t)}{f(t,x(t),x(\varphi(t)))}\right]=g(t,x(t),x(\rho(t))),\,\,0<t<1\\ &x(0)=0, \end{aligned} \right.$$ where $\alpha\in (0,1)$, $\varphi$ 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.