On the non-uniqueness of continuous solutions to differential equations with a discrete state-dependent delay

TL;DR

This paper investigates the non-uniqueness of continuous solutions in differential equations with discrete state-dependent delays, providing explicit examples and an initial classification approach.

Contribution

It introduces a method to classify non-unique solutions and offers explicit examples, advancing understanding of solution multiplicity in such differential equations.

Findings

Explicit examples demonstrate non-uniqueness of solutions.

A simple classification approach using three colors is proposed.

Further research is needed for a complete classification.

Abstract

We discuss the non-uniqueness of continuous solutions to differential equations with a {\it discrete } state-dependent delay and continuous initial functions. We are interested not only in the fact (conditions) of non-uniqueness, but in additional information on the number of non-unique solutions and discuss an approach to classify them. We provide a few explicit (easy to verify) examples of the non-uniqueness of continuous solutions and propose an approach to their classification. This partial classification may be illustrated by using just three collors and their simple and intuitive combination. We recognize that this initial classification is not exhaustive, and further study is necessary to build a complete picture.