On the (k,Ψ)‐Hilfer nonlinear impulsive fractional differential equations

Abstract

In this paper, the ‐Hilfer fractional derivative, the most generalized fractional derivative operator, is used to analyze nonlinear impulsive fractional differential equations. The initial goal of this work is to provide a suitable representation formula for the solutions of the ‐Hilfer nonlinear impulsive fractional differential equations. The second goal is to use the representation formula we came up with to show that there is a solution to the ‐Hilfer nonlinear impulsive fractional differential equations. Analysis of the work done is based on the calculus of the ‐Hilfer fractional derivative, the fixed point theorems, and the measure of noncompactness. We illustrate the outcomes of our work through examples.

Study on a Nonlocal Fractional Coupled System Involving (k,ψ)-Hilfer Derivatives and (k,ψ)-Riemann–Liouville Integral Operators

“…By using Banach's fixed-point theorem, the existence of a unique solution was proven. For some recent results on (k, ψ)-Hilfer fractional derivative operators of orders in (0, 1], see [29,30] and references cited therein. Boundary value problems of the (k, ψ)-Hilfer fractional derivative operator of orders in (1,2] were initiated in [31] by studying the problem…”