Center Manifold Theorem and Stability for Integral Equations with Infinite Delay

TL;DR

This paper investigates the center manifold and stability properties of autonomous integral equations with infinite delay using a dynamical systems approach, establishing existence, attractivity, and a stability reduction principle.

Contribution

It introduces a method to analyze stability of such integral equations by reducing it to an ordinary differential equation called the 'central equation.'

Findings

Existence of center manifold for integral equations with infinite delay.

Local exponential attractivity of the center manifold.

Stability of the integral equation is implied by the stability of the central ODE.

Abstract

The present paper deals with autonomous integral equations with infinite delay via dynamical system approach. Existence, local exponential attractivity, and other properties of center manifold are established by means of the variation-of-constants formula in the phase space that is obtained in a previous paper \cite{mur}. Furthermore, we prove a stability reduction principle by which the stability of an autonomous integral equation is implied by that of an ordinary differential equation which we call the "central equation".