Roughness of exponential dichotomy under unbounded perturbation in linear partial functional differential equations
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917455053455360 |
|---|---|
| author | Bui, Xuan-Quang Van Minh, Nguyen |
| author_facet | Bui, Xuan-Quang Van Minh, Nguyen |
| contents | 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 $d_Y(U,V):=\limsup_{μ\to +\infty} \| U_μ-V_μ\|$, where $U_μ$ and $V_μ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04873 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Roughness of exponential dichotomy under unbounded perturbation in linear partial functional differential equations Bui, Xuan-Quang Van Minh, Nguyen Dynamical Systems 34G10, 37D10, 34D20, 34C45, 34D09 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 $d_Y(U,V):=\limsup_{μ\to +\infty} \| U_μ-V_μ\|$, where $U_μ$ and $V_μ$ 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. |
| title | Roughness of exponential dichotomy under unbounded perturbation in linear partial functional differential equations |
| topic | Dynamical Systems 34G10, 37D10, 34D20, 34C45, 34D09 |
| url | https://arxiv.org/abs/2310.04873 |