General Cauchy-Lipschitz theory for shifted and non shifted Delta-Cauchy problems on time scales

TL;DR

This paper extends the classical Cauchy-Lipschitz theory to nonlinear systems on time scales, establishing existence and uniqueness of solutions for shifted and non-shifted Delta-Cauchy problems in a general framework.

Contribution

It introduces the concepts of absolutely continuous solutions and maximal solutions on time scales, and proves a Cauchy-Lipschitz theorem under regressivity and local Lipschitz conditions.

Findings

Existence and uniqueness of maximal solutions established.

Framework applicable to general nonlinear systems on time scales.

Analysis of solution behavior at terminal points.

Abstract

This article is devoted to completing some aspects of the classical Cauchy-Lipschitz (or Picard-Lindel\"of) theory for general nonlinear systems posed on time scales, that are closed subsets of the set of real numbers. Partial results do exist but do not cover the framework of general dynamics on time-scales encountered e.g. in applications to control theory. In the present work, we first introduce the notion of absolutely continuous solution for shifted and non shifted Delta-Cauchy problems, and then the notion of a maximal solution. We state and prove a Cauchy-Lipschitz theorem, providing existence and uniqueness of the maximal solution of a given Delta-Cauchy problem under suitable assumptions like regressivity and local Lipschitz continuity, and discuss some related issues like the behavior of maximal solutions at terminal points.