Well-posedness and asymptotic behavior of difference equations with a time-dependent delay and applications

TL;DR

This paper studies the well-posedness and long-term behavior of difference equations with time-dependent delays, providing results across different function spaces and applying findings to related delay and transport equations.

Contribution

It offers new insights into the well-posedness and asymptotic analysis of difference equations with variable delays across multiple function spaces.

Findings

Results for continuous, regulated, and L^p spaces are compared.

Applications include difference equations with state-dependent delays.

Extensions to transport equations with time-dependent velocity are provided.

Abstract

In this paper, we investigate the well-posedness and asymptotic behavior of difference equations of the form $x(t) = A x(t - \tau(t))$, $t \geq 0$, where the unknown function $x$ takes values in $\mathbb R^d$ for some positive integer $d$, $A$ is a $d \times d$ matrix with real coefficients, and $\tau\colon [0, +\infty) \to (0, +\infty)$ is a time-dependent delay. We provide our investigations for three spaces of functions: continuous, regulated, and $L^p$. We compare our results for these three cases, showing how the hypotheses change according to the space that we are treating. Finally, we provide applications of our results to difference equations with state-dependent delays for the cases of continuous and regulated function spaces, as well as to transport equations in one space dimension with time-dependent velocity.