Levitan Almost Periodic and Almost Automorphic Solutions of Second-order Monotone Differential Equations

The aim of this paper is the study of problem of existence of Levitan almost periodic, almost automorphic, recurrent and Poisson stable solutions of seconde order differential equation (1) x′′ = f(σ(t, y), x, x′), (y ∈ Y ) where Y is a complete metric space and (Y,R, σ) is a dynamical system (driving system). For equation (1) with increasing (with respect to second variable) function f the existence at least one quasi periodic (respectively, Bohr almost periodic, almost automorphic, recurrent, pseudo recurrent, Levitan almost periodic, almost recurrent, Poisson stable) solution of (1) is proved under the condition that (23) admits at least one bounded on the real axis solution together with its first derivative.