Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

The well-known Favard's theorem states that the linear differential equationequation(1)x′=A(t)x+f(t)x′=A(t)x+f(t) 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 homogeneous equationsequation(2)x′=B(t)x,x′=B(t)x, where B∈H(A):={B|B(t)=limn→+∞A(t+tn)}B∈H(A):={B|B(t)=limn→+∞A(t+tn)}. If there are bounded solutions which are non-separated, sometimes almost periodic solutions do not exist (R. Johnson, R. Ortega and M. Tarallo, V. Zhikov and B. Levitan).In this paper we prove that linear differential 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 equationequation(3)x′=A(t)xx′=A(t)x are homoclinic to zero (i.e. lim|t|→+∞|φ(t)|=0lim|t|→+∞|φ(t)|=0 for all bounded solutions φ of (3)). If the coefficients of (1) are Bohr almost periodic and all bounded solutions of all limiting equations (2) are homoclinic to zero, then Eq. (1) admits a unique almost automorphic solution.The analogue of this result for difference equations is also given.We study the problem of existence of Bohr/Levitan almost periodic solutions of Eq. (1) in the framework of general non-autonomous dynamical systems (cocycles).