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

TL;DR

This paper investigates the existence of solutions for a complex hybrid nonlinear fractional pantograph equation using advanced fixed point theorems and provides an illustrative example.

Contribution

It introduces a generalized fixed point theorem approach to establish solution existence for a new class of hybrid fractional pantograph equations.

Findings

Existence of solutions proved under certain conditions.

Application of a generalized Darbo's fixed point theorem.

An example demonstrating the theoretical results.

Abstract

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