Exponential decay for nonlinear abstract evolution equations with a countably infinite number of time-dependent time delays

TL;DR

This paper studies a nonlinear evolution equation with infinitely many time-dependent delays, proving existence and exponential decay of solutions under certain conditions, with applications demonstrated through examples.

Contribution

It introduces a novel analysis of equations with infinitely many delays, establishing existence and decay results using fixed point and Gronwall methods.

Findings

Existence of mild solutions under Lipschitz conditions

Exponential decay established with smallness assumptions

Applications demonstrated through illustrative examples

Abstract

In this paper we analyze a nonlinear abstract evolution equation with an infinite number of time-dependent time delays and a Lipschitz continuous nonlinear term. By using a fixed point argument we prove the existence of a mild solution. This allows us to take into account also nonnegative time delays. Furthermore, by using Gronwall estimates, exponential decay of the solution is also proved under some smallness assumptions on the parameters appearing in the system and on the initial data. Finally some examples are illustrated.