On the Nonlinear $\Psi$-Hilfer Hybrid Fractional Differential Equations

TL;DR

This paper investigates the existence, uniqueness, and comparison of solutions for nonlinear $ ext{ extPsi}$-Hilfer hybrid fractional differential equations by deriving equivalent integral equations and establishing fractional inequalities.

Contribution

It introduces new methods for analyzing $ ext{ extPsi}$-Hilfer hybrid fractional differential equations, including deriving integral equations and solution estimates.

Findings

Existence of solutions proved in weighted spaces

Derived fractional differential inequalities for $ ext{ extPsi}$-Hilfer derivatives

Established comparison theorems and uniqueness results

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 primary objective of the paper is to obtain estimates on $\Psi$-Hilfer derivative and utilize it to derive the hybrid fractional differential inequalities involving $\Psi$-Hilfer derivative. With the assistance of these fractional differential inequalities, we determine the existence of extremal solutions, comparison theorems and uniqueness of the solution.