  For a general discrete dynamics on a Banach and Hilbert spaces we give
necessary and sufficient conditions of the existence of bounded solutions under
assumption that the homogeneous difference equation admits an discrete
dichotomy on the semi-axes. We consider the so called resonance (critical) case
when the uniqueness of solution is disturbed. We show that admissibility can be
reformulated in the terms of generalized or pseudo-invertibility. As an
application we consider the case when the corresponding dynamical system is
e-trichotomy.
