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 analog of the well-known Favard Theorem states that the linear difference equation (1) x(t + 1) = A(t)x(t) + f(t) (t ∈ Z) 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 (2) x(t + 1) = B(t)x(t), where B ∈ H(A) := {B | B(t) = lim n→+∞ A(t + tn)}. 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 (3) x(t + 1) = A(t)x(t) are homoclinic to zero in the positive direction (i.e., lim t→+∞ |φ(t)| = 0 for all relatively compact solutions φ 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 t → +∞, 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). Dedicated to the memory of José Real