Periodic solutions of second order differential equations in Banach spaces
type: Item
instance of: scholarly article
title: Periodic solutions of second order differential equations in Banach spaces
zbMATH Open document ID: 1104.34041
author: Valentin Keyantuo
author: Carlos Lizama
published in: Mathematische Zeitschrift
publication date: 2006-11-09T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/177499
review text: The authors consider the maximal regularity for the second order periodic boundary value problem \[ \begin{gathered} u''(t)-aAu(t)-\alpha Au'(t)=f(t), \;0\leq t\leq 2\pi,\\ u(0)=u(2\pi),\;u'(0)=u'(2\pi),\end{gathered} \] on a Banach space \(X\). Necessary and sufficient conditions for the existence and uniqueness of periodic solutions in the spaces \(L_{2\pi}^{p}({\mathbb R},X)\) (\(1<p<\infty\)) and \(C_{2\pi}^{s}({\mathbb R},X)\) \((0<s<1)\) are given. Moreover, results on mild solutions are also presented. Two types of mild periodic solutions are considered. When the operator \(A\) is the generator of a strongly continuous cosine function, characterizations are given in terms of Fourier multipliers and spectral properties of the cosine function. Throughout the paper, examples involving elliptic partial differential operators with Dirichlet boundary conditions are discussed.
reviewed by: Sotiris K. Ntouyas
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 35L90
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47D09
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 47F05
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35G20
zbMATH DE Number: 5071441
zbMATH Keywords: periodic solutions
zbMATH Keywords: maximal regularity
zbMATH Keywords: Besov spaces
zbMATH Keywords: Hölder spaces
zbMATH Keywords: mild solutions
MaRDI profile type: Publication
OpenAlex ID: W2052790771
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DOI: 10.1007/S00209-005-0919-1
description: scientific article; zbMATH DE number 5071441