A. Pankov Gap Solitons in Periodic Discrete Nonlinear Schroedinger Equations (39K, LaTeX 2e ) ABSTRACT. It is shown that the periodic DNLS, with cubic nonlinearity, possesses gap solutions, i.~e. standing waves, with the frequency in a spectral gap, that are exponentially localized in spatial variable. The proof is based on the linking theorem in combination with periodic approximations.