Conditionally periodic solutions to differential equations (Q2703930)

A solution \(x(t)\) to a nonautonomous 1-periodic ordinary differential equation is called a conditionally periodic solution if it is defined and bounded for all \(t \geq t_0\) and for each \(\varepsilon >0\) there exists a positive integer \(N_{\varepsilon}\) such that \(\|x(t) - x(t+N_{\varepsilon})\|< \varepsilon\) for all \(t \geq t_0.\) NEWLINENEWLINENEWLINEA theorem is proved that an \(\omega\)-limit set of a solution that is defined and bounded for all \(t \geq t_0\) contains a conditionally periodic solution to the system. A similar result for the autonomous system is received. The author supplements results of \textit{J. L. Massera} [Duke Math. J. 17, 457-475 (1950; Zbl 0038.25002)] and \textit{J. Hale} [Theory of functional differential equations (1977; Zbl 0352.34001)].