  Local and global existence of localized solutions of a discrete nonlinear
Schrodinger (DNLS) equation, with arbitrary on-site nonlinearity, is proved. In
particular, it is shown that an initially localized excitation persists
localized during infinite time. Moreover, if initial localization is stronger
than |n|^{-d} with any power d, it maintains itself as such during infinite
time. The results are generalized to various types of inter-side and saturable
nonlinearities, to lattices with long range interactions, as well as DNLS with
dissipation.
