Sequential dichotomies and uniformities for a neutral equation

TL;DR

The paper addresses the existence and uniformity of sequential exponential dichotomies for a scalar neutral delay equation $x'(t)+c x'(t-1)+a x(t)+b x(t-1)=0$ with $c\neq0$. It develops a spectral framework based on the roots of the characteristic equation $h(\lambda)=0$, introducing the dichotomous spectrum $\\Sigma$ and resolvent set $\\Omega$, and shows that the number of dichotomies equals the number of connected components of $\\Omega$, which can be finite or countably infinite depending on $(a,b,c)$. When countably many dichotomies occur (under certain parameter conditions), the paper proves that the sequential uniformity fails: the norms of the dichotomy projections become unbounded and there exist initial data for which the projection series fails to converge. A key part of the analysis uses an auxiliary neutral equation to build an associated orthogonal system, enabling resolvent perturbation arguments that establish the non-uniformity result. These findings extend previous results on delay equations to the neutral setting and highlight fundamental limitations on approximating solutions via finite-dimensional dichotomy projections.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 3.1