On the existence and uniqueness of solutions for non-autonomous semi-linear systems with non-instantaneous impulses, delay, and non-local conditions

TL;DR

This paper investigates the existence and uniqueness of solutions for complex non-autonomous semi-linear differential systems with non-instantaneous impulses, delays, and non-local conditions, providing theoretical results and an illustrative example.

Contribution

It introduces new conditions ensuring solution existence and uniqueness for such systems, extending previous work to include non-instantaneous impulses and non-local effects.

Findings

Proved existence and uniqueness of solutions using Karakostas fixed-point theorem.

Developed results on prolongation of solutions.

Provided an example and remarks on infinite-dimensional cases.

Abstract

A non-autonomous evolution semi-linear differential system under non-instantaneous impulses, delays, and perturbed by non-local conditions is studied. Its piece-wise continuous solutions belong to a finite-dimensional Banach space. The existence and uniqueness of solutions on the interval $[-r,\tau]$ are obtained by applying Karakostas fixed-point theorem. Further results concerning solution prolongation are developed. An example is presented, and several remarks on the infinite-dimensional case are included.