On the nonlinear $$\Psi $$-Hilfer hybrid fractional differential equations

Abstract

In this paper, we initially derive the equivalent fractional integral equation to $$\Psi $$ -Hilfer hybrid fractional differential equations and through it, we prove the existence of a solution in the weighted space. The paper’s primary objective is to obtain estimates on $$\Psi $$ -Hilfer fractional derivative and utilize it to derive the hybrid fractional differential inequalities involving $$\Psi $$ -Hilfer fractional derivative. With the assistance of these fractional differential inequalities, we determine the existence of extremal solutions and comparison theorems.