Almost-periodic solutions of discrete equations

We consider non-linear almost-periodic discrete equations in a metric space that have solutions with precompact sets of values. By using an auxiliary functional $\delta$ defined on the set of such solutions, we obtain conditions for the almost periodicity of these solutions, which, in contrast to the classical papers of Favard and Amerio, do not use the ${\mathscr H}$-classes of the equations under consideration. We conduct similar studies for linear almost-periodic and non-linear Poisson-stable discrete equations.