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The paper is dedicated to the study of problem of Poisson stability (in particular, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, Levitan almost periodicity, pseudo-periodicity, almost recurrence in the sense of Bebutov, recurrence in the sense of Birkhoff, pseudo recurrence and Poisson stability) and asymptotically Poisson stability of motions of monotone non-autonomous difference equations which have a strong monotone first integral. We solve this problem in the framework of general non-autonomous dynamical systems with discrete time.

Bohr/Levitan almost periodic and almost automorphic solutions of monotone difference equations with a strict monotone first integral