  We study two damped and forced discrete nonlinear Schr\"odinger equations on
the one-dimensional infinite lattice. Without damping and forcing they are
represented by the integrable Ablowitz-Ladik equation (AL) featuring non-local
cubic nonlinear terms, and its standard (nonintegrable) counterpart with local
cubic nonlinear terms (DNLS). The global existence of a unique solution to the
initial value problem for both, the damped and forced AL and DNLS, is proven.
It is further shown that for sufficiently close initial data, their
corresponding solutions stay close for all times. Concerning the asymptotic
behaviour of the solutions to the damped and forced AL and DNLS, for the former
a sufficient condition for the existence of a restricted global attractor is
established while it is shown that the latter possesses a global attractor.
Finally, we prove the congruence of the restricted global AL attractor and the
DNLS attractor for dynamics ensuing from initial data contained in an
appropriate bounded subset in a Banach space.
