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 $\left\{\begin{aligned} &D_{0+}^α\left[\frac{x(t)}{f(t,x(t),x(φ(t)))}\right]=g(t,x(t),x(ρ(t))),\,\,0<t<1\\ &x(0)=0, \end{aligned} \right.$ where $α\in (0,1)$, $φ$ and $ρ$ are functions from $[0,1]$ into itself and $D_{0+}^α$ 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.

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