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 $$ \left{\begin{aligned} &amp;D_{0+}^\alpha \left[\frac{x(t)}{f(t,x(t),x(\varphi(t)))}\right]=g(t,x(t),x(\rho(t))),\,\,0&lt;t&lt;1\ &amp;x(0)=0, \end{aligned} \right. $$ where $\alpha\in (0,1)$, $\varphi$ and $\rho$ are functions from $[0,1]$ into itself and $D_{0+}<sup>\alpha$ 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.