A fixed-point approach to history-dependent sweeping processes

TL;DR

This paper introduces a fixed-point method for analyzing history-dependent sweeping processes, extending existing results and applying the approach to a viscoelastic model with long memory.

Contribution

It develops a novel fixed-point framework using an enhanced Gronwall's lemma for well-posedness of history-dependent sweeping processes.

Findings

Unified approach for Volterra sweeping processes

New insights into history-dependent processes

Application to viscoelastic models with long memory

Abstract

In this paper, we study the well-posedness of state-dependent and state-independent sweeping processes driven by prox-regular sets and perturbed by a history-dependent operator. Our approach, based on an enhanced version of Gronwall's lemma and fixed-point arguments, provides an efficient framework for analyzing sweeping processes. In particular, our findings recover all existing results for the class of Volterra sweeping processes and provide new insights into history-dependent sweeping processes. Finally, we apply our theoretical results to establish the well-posedness of a viscoelastic model with long memory.