Massera's theorem on arbitrary discrete time domains

We present a general version of Massera's theorems for arbitrary discrete domains, based on a newly introduced definition for both linear and nonlinear equations. For scalar nonlinear equations, we identify sufficient conditions that ensure each u-bounded solution approaches a periodic solution asymptotically. In the case of linear systems, we prove that the presence of a u-bounded solution necessarily leads to a periodic solution. We also provide some examples to show the practical implications of our findings.