Sequential dichotomies and uniformities for a neutral equation

TL;DR

This paper investigates the uniformity of sequential dichotomies in neutral delay equations, demonstrating that the series of projections diverges, thus providing insights into their non-uniform behavior.

Contribution

It proves that the series of projections of dichotomies in a scalar neutral equation diverges, answering an open question about their uniformity.

Findings

Sequential dichotomies are not uniform in neutral delay equations.

The series of projections of dichotomies diverges.

Provides a negative answer to the uniformity question for these equations.

Abstract

Sequential dichotomies of general delay equations are not uniform, which was proved two decades ago. This however reminds whether the countably infinite many dichotomies of a neutral equation have the sequential uniformity. In this paper, considering a scalar neutral equation, we give a negative answer and prove that the series of the projections of dichotomies is divergent.