On Massera and Bohr-Neugebauer type theorems for some almost automorphic differential equations
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)]).
scientific article; zbMATH DE number 7610091
scientific article