In this paper, we introduce the class of asymptotically \((N,\lambda )\) -periodic sequences in abstract spaces and establish some important properties related to this new class. We note that the class of asymptotically \((N,\lambda )\) -periodic sequences here is not the same as some known generalized \((N,\lambda )\) -periodic sequences in abstract spaces. We also obtain some existence results for asymptotically \((N,\lambda )\) -periodic mild solutions of semilinear difference equations in Banach spaces. Some examples are given to support the feasibility of our abstract outcomes.
