On Massera and Bohr-Neugebauer type theorems for some almost automorphic differential equations
type: Item
instance of: scholarly article
title: On Massera and Bohr-Neugebauer type theorems for some almost automorphic differential equations
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-10-31T00:00:00Z
review text: This paper is dedicated to the study the problem of existence of almost automorphic solutions of linear differential equations \[ x'=A(t)x+f(t)\ \ (t\in \mathbb R)\tag{1} \] with almost automorphic coefficients \(A(t)\) and \(f(t)\). A function \(f:\mathbb R\to \mathbb R^{n}\) is said to be almost automorphic if for every sequence \(\{s_n\}\) there exist a subsequence \(\{s_{n_k}\}\) and a function \(g\), such that for each \(t\in \mathbb R\) \(f(t + s_{n_k})\to g(t)\) and \(g(t -s_{n_k})\to f(t)\) as \(k\to \infty\). If these limits hold uniformly in compact subsets of \(\mathbb R\), then \(f\) is said to be compactly almost automorphic. Note that if the function \(f\) is almost automorphic and uniformly continuous on \(\mathbb R\), then it is compactly almost automorphic. The authors prove the following two statements: \textbf{Theorem 1.} Assume that the matrix-function \(A(t)\) is almost automorphic and \(A(t)\ge 0\) for any \(t\in \mathbb R\). Then every bounded on \(\mathbb R\) solution of equation \[ x'=A(t)x \] is constant. \textbf{Theorem 2.} Suppose that the following conditions hold: \begin{itemize} \item[1.] the matrix-function \(A(t)\) and function \(f(t)\) are almost automorphic; \item[2.] equation (1) satisfies Favad's separation condition, i.e., for any bounded on \(\mathbb R\) solution \(x(t)\) we have \[ \inf\limits_{t\in \mathbb R}|x(t)|>0. \] \end{itemize} Then equation (1) has at least a compactly almost automorphic solution. \textbf{Remark.} In the case, when the matrix-function \(A(t)\) and function \(f(t)\) are compactly almost automorphic Theorem 2. was established in the work [\textit{B. A. Shcherbakov}, An. Științ. Univ. Al. I. Cuza Iași, N. Ser., Secț. Ia 21, 57--59 (1975; Zbl 0413.34008)] (see also [\textit{L. Faxing}, Ann. Differ. Equations 3, 329--349 (1987; Zbl 0638.34029)]).
reviewed by: David Cheban
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 43A60
zbMATH DE Number: 7610091
zbMATH Keywords: almost periodic solutions
zbMATH Keywords: positive semi-definite matrix
zbMATH Keywords: almost automorphic solutions
zbMATH Keywords: Massera
zbMATH Keywords: Bohr-Neugebauer
zbMATH Keywords: Favard's condition
author: EL Hadi Ait Dads
author: Brahim Es-Sebbar
author: L. Lhachimi
MaRDI profile type: Publication
full work available at URL: j.jmaa.2022.126761
OpenAlex ID: W4304893411
Wikidata QID: Q115188792
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DOI: 10.1016/J.JMAA.2022.126761
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description: scientific article; zbMATH DE number 7610091