Discrete Melnikov functions

We consider non-autonomous N-periodic discrete dynamical systems of the form (Formula presented.) having when (Formula presented.) an open continuum of initial conditions such that the corresponding sequences are N-periodic. From the study of some variational equations of low order, we obtain successive maps, that we call discrete Melnikov functions, such that the simple zeroes of the first one that is not identically zero control the initial conditions that persist as N-periodic sequences of the perturbed discrete dynamical system. We apply these results to several examples, including some Abel-type discrete dynamical systems and some non-autonomous perturbed globally periodic difference equations.