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$ x''+f(x)x'+g(x)=F(t), $

where $F: T\to R$ ($ T= R_+$ or
$R$) is an almost periodic or asymptotically almost
periodic function and $g:(a,b)\to R$ is a strictly
decreasing function. We study also this problem for the vectorial
Liénard equation.

We analyze this problem in the framework of general non-autonomous
dynamical systems (cocycles). We apply the general results
obtained in our early papers [3, 7] to prove the existence of almost periodic
(almost automorphic, recurrent, pseudo recurrent) and
asymptotically almost periodic (asymptotically almost automorphic,
asymptotically recurrent, asymptotically pseudo recurrent)
solutions of Liénard equations (both scalar and vectorial).

(i) There exists a complete trajectory of the corresponding homogeneous equation with constant positive norm;

(ii) The trivial solution of the homogeneous equation is uniformly asymptotically stable.

If the second alternative holds, then the non-homogeneous equation with almost periodic (respectively, almost automorphic, recurrent) coefficients possesses a unique almost periodic (respectively, almost automorphic, recurrent) solution. We investigate this problem within the framework of general linear nonautonomous dynamical systems. We apply our general results also to the cases of functional-differential equations and difference equations.

In this paper we study the problem of Levitan/Bohr almost periodicity of solutions for dissipative differential equations (Bronshtein's conjecture for Bohr almost periodic case). We give a positive answer to this conjecture for monotone Levitan/Bohr almost periodic systems of differential/difference equations.

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