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Ergodic solutions via ergodic sequences. (English) Zbl 0957.34054
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., A. I. Alonso, J. Hong and 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.
34F05ODE with randomness