Solutions to partial functional differential equations with infinite delays: periodicity and admissibility
type: Item
instance of: scholarly article
title: Solutions to partial functional differential equations with infinite delays: periodicity and admissibility
zbMATH Open document ID: 1398.34112
author: Thieu Huy Nguyen
author: Quy Dang Ngo
published in: Acta Mathematica Vietnamica
publication date: 2018-07-24T00:00:00Z
review text: The authors consider partial functional differential equations with infinite delay of the form \[ \dot{u}=A(t)u+g(t,u_{t}),\quad t\in\mathbb{R}^{+} \] with \(u_{t}(s):=u(t+s)\) for \(s\in\left(-\infty,0\right]\) on a Banach space \(X\) having a separable predual \(Y\). First they study the existence and uniqueness of periodic solutions when \(A(t)\) is \(1\)-periodic and the nonlinear term \(g(t,\phi)\) is \(1\)-periodic with respect to \(t\) for every fixed \(\phi\) in memory phase spaces, and is \(\phi(t)\)-Lipschitz. Also, they apply their results to the case when the family \(\left(A(t)\right)_{t\geq0}\) generates an evolution family having an exponential dichotomy, to study the existence, uniqueness and conditional stability of periodic solutions. They conclude with the existence of a local stable manifold near the periodic solution in that case.
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K13
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 34K19
zbMATH DE Number: 6909837
zbMATH Keywords: partial functional differential equations
zbMATH Keywords: periodic solutions
zbMATH Keywords: admissibility of function spaces
zbMATH Keywords: conditional stability
zbMATH Keywords: local stable manifolds
reviewed by: Panagiotis N. Koumantos
MaRDI profile type: Publication
full work available at URL: s40306-018-0265-0
OpenAlex ID: W2803622370
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Wikidata QID: Q129813661
DOI: 10.1007/S40306-018-0265-0
description: scientific article; zbMATH DE number 6909837