Ergodic solutions via ergodic sequences (Q1570931)

It is known (G. H.~Meisters, Z.~Opial, and A. M.~Fink) that the existence of almost-periodic solutions to ordinary differential equations is equivalent to the fact that the restriction of a bounded solution to some discrete subgroup of reals is almost-periodic. There are results of this kind [see, e.g., \textit{A. I.~Alonso, J.~Hong} and \textit{R.~Obaya}, Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences, Appl. Math. Lett. 13, No. 2, 131-137 (2000)]on the existence of almost-periodic, asymptotically almost-periodic, and pseudo almost-periodic solutions to differential equations with piecewise constant argument. In the paper, a similar result on the existence of ergodic solutions is obtained under conditions similar to those of G. H.~Meisters, Z.~Opial, and A. M.~Fink.