In this paper, we study the (new) concept of S-asymptotically \((N,\lambda )\) -periodic sequences as a generalization of \((N,\lambda )\) -periodic sequences. We present their basic properties and give simple examples that show how they differ from the classical concept of periodicity. We also apply these results to linear and semilinear difference equations, proving the existence and uniqueness of solutions under appropriate sufficient assumptions. Finally, we illustrate the results with an example from population dynamics.
