Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations
type: Item
instance of: scholarly article
title: Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations
zbMATH Open document ID: 1194.47047
author: Jing-Huai Liu
author: Xiaoqiu Song
published in: Journal of Functional Analysis
publication date: 2010-01-11T00:00:00Z
review text: The present paper discusses the existence and uniqueness of an almost automorphic (a weighted pseudo almost automorphic) mild solution to a class of semilinear evolution equations \(x'(t)=A(t)x(t)+f(t,x(t))\) in a Banach space. The main results are Theorems 3.2 and 4.2. However, Theorem 3.2 can be seen from [\textit{H.-S.\thinspace Ding, W.\,Long} and \textit{G.\,M.\,N'Guérékata}, Nonlinear Anal., Theory Methods Appl.\ 70, No.\,12 (A), 4158--4164 (2009; Zbl 1161.43301)]. Moreover, the authors use the Banach contraction mapping principle to obtain the conclusion in the proof of Theorem 4.2; thus, the completeness of the space \(WPAA(R,\rho)\) is needed. But from Lemma 2.10 in [\textit{J.\,Blot, G.\,M.\thinspace Mophou, G.\,M.\,N'Guérékata} and \textit{D.\,Pennequin}, Nonlinear Anal., Theory Methods Appl.\ 71, No.\,3--4 (A), 903--909 (2009, Zbl 1177.34077)], one only knows that the space \(WPAA(R,\rho)\) is a Banach space if \(\rho\in U_{b}\). Actually, to the best of the reviewer's knowledge, there is no proof in the literature that says that \(WPAA(R,\rho)\) is complete in the case when \(\rho\) is not necessarily bounded. On the other hand, when \(\rho\) is bounded, Theorem 4.2 is known from [\textit{T.-J.\thinspace Xiao, X.-X.\thinspace Zhu} and \textit{J.\,Liang}, Nonlinear Anal., Theory Methods Appl.\ 70, No.\,11 (A), 4079--4085 (2009, Zbl 1175.34076)] since \(WPAA(R,\rho) = PAA(X)\) in this case.
reviewed by: Jin Liang
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 34G10
zbMATH DE Number: 5655674
zbMATH Keywords: almost automorphic
zbMATH Keywords: weighted pseudo almost automorphic
zbMATH Keywords: semilinear evolution equation
MaRDI profile type: Publication
full work available at URL: j.jfa.2009.06.007
OpenAlex ID: W1966051183
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DOI: 10.1016/J.JFA.2009.06.007
description: scientific article; zbMATH DE number 5655674