Summary: The object of this paper is to give sufficient conditions for the existence of bounded solutions that are globally exponentially stable, periodic and almost periodic, for a certain third-order non-linear differential equation. A matrix inequality is obtained and proved to satisfy a generalized frequency domain inequality of

*V. A. Yakubovich* [Sov. Math., Dokl. 6, 98–101 (1965;

Zbl 0142.36301)] through the frequency domain technique.