Roughness of exponential dichotomy under unbounded perturbation in linear partial functional differential equations

TL;DR

This paper investigates the stability of exponential dichotomies in linear partial functional differential equations under unbounded perturbations, introducing the Yosida distance to measure perturbation size.

Contribution

It establishes conditions under which exponential dichotomies are preserved despite unbounded perturbations using the Yosida distance, without domain restrictions.

Findings

Small Yosida distance ensures the persistence of exponential dichotomies.

The approach does not require domain relations between operators.

Provides estimates of the Yosida distance between solution semigroup generators.

Abstract

This paper is concerned with the roughness of exponential dichotomies under unbounded perturbations of a class of linear partial functional differential equations \begin{equation}\label{pfde-000-1star} u'(t)=Au(t)+Bu_t, \end{equation} where $A$ is a linear operator on a Banach space $\mathbb{X}$ and $B$ is a linear operator from $C([-r,0],\mathbb{X})$ into $\mathbb{X}$, where $r>0$ is a given constant. To quantify the size of unbounded perturbations, we introduce the \textit{Yosida distance} between linear operators $U$ and $V$, defined by $d_Y(U,V):=\limsup_{\mu\to +\infty} \| U_\mu-V_\mu\|$, where $U_\mu$ and $V_\mu$ are the Yosida approximations of $U$ and $V$, respectively. We show that if $d_Y(A, A_1)$ and $d_Y(B, B_1)$ are sufficiently small, then the perturbed equation \begin{equation}\label{pfde-000-2star} u'(t)=A_1u(t)+B_1u_t \end{equation} also admits an exponential dichotomy whenever \eqref{pfde-000-1star} admits one. The proofs are based on estimates of the Yosida distance between the generators of the solution semigroups associated with \eqref{pfde-000-1star} and \eqref{pfde-000-2star} in the phase space $C([-r,0],\mathbb{X})$, without assuming any relation between their domains.