Levitan/Bohr almost periodic and almost automorphic solutions of second order monotone differential equations

The aim of this paper is to prove the existence of Levitan/Bohr almost periodic, almost automorphic, recurrent and Poisson stable solutions of the second order differential equationequation(1)x″=f(σ(t,y),x,x′)(y∈Y) where Y   is a complete metric space and (Y,R,σ)(Y,R,σ) is a dynamical system (also called a driving system). When the function f in (1) is increasing with respect to its second variable, the existence of 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 (1) admits at least one solution φ such that φ   and φ′φ′ are bounded on the real axis.