Discussion on iterative process of nonlocal controllability exploration for Hilfer neutral impulsive fractional integro-differential equation

This manuscript primarily focuses on the nonlocal controllability results of Hilfer neutral impulsive fractional integro-differential equations of order $0\le w\le 1$ and $0<g<1$ in a Banach space. The outcomes are derived from the strongly continuous operator, Wright function, linear operator, and bounded operator. First, we explore the existence and uniqueness of the results of the mild solution of Hilfer's neutral impulsive fractional integro-differential equations using Schauder's fixed point theorem and an iterative process. In order to determine nonlocal controllability, the Banach fixed point technique is used. We employed some specific numerical computations and applications to examine the effectiveness of the results.