Almost periodic motions in semi-group dynamical systems and Bohr/Levitan almost periodic solutions of linear difference equations without Favard's separation condition

The discrete analogue of the well-known Favard theorem states that the linear difference equation with Bohr almost periodic coefficients admits at least one Bohr almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of the homogeneous equations where In this paper, we prove that the linear difference equation (1) with Levitan almost periodic coefficients has a unique Levitan almost periodic solution, if it has at least one bounded solution, and the bounded solutions of the homogeneous equation are homoclinic to zero in the positive direction (i.e. for all relatively compact solutions of (3)). If the coefficients of (1) are Bohr almost periodic and all relatively compact solutions of all limiting equations (2) tend to zero as , then equation (1) admits a unique almost automorphic solution. We study the problem of existence of Bohr/Levitan almost periodic solutions of equation (1) in the framework of general non-autonomous dynamical systems (cocycles).