Existence and Asymptotic Stability of Periodic Solutions for Impulsive Delay Evolution Equations

TL;DR

This paper investigates the existence and stability of periodic solutions in impulsive delay evolution equations within Banach spaces, providing new theorems and conditions for their existence and asymptotic stability.

Contribution

It introduces novel existence theorems for periodic mild solutions and establishes conditions for their asymptotic stability using operator semigroup theory and integral inequalities.

Findings

Existence of periodic mild solutions under new conditions

Conditions guaranteeing asymptotic stability of solutions

Application of fixed point theorem and integral inequalities

Abstract

In this paper, we are devoted to consider the periodic problem for the impulsive evolution equations with delay in Banach space. By using operator semigroups theory and fixed point theorem, we establish some new existence theorems of periodic mild solutions for the equations. In addition, with the aid of an integral inequality with impulsive and delay, we present essential conditions on the nonlinear and impulse functions to guarantee that the equations have an asymptotically stable $\omega$-periodic mild solution.