Unified Massera type theorems for dynamic equations on time scales

Abstract

In this paper, we aim to obtain Massera type theorems for both linear and nonlinear dynamic equations by using a generalized periodicity notion, namely (T, ?)-periodicity, on time scales. To achieve this task, first we define a new boundedness concept so-called ?-boundedness, and then we establish a linkage between the existence of ?-bounded solutions and (T, ?)-periodic solutions of dynamic equations in both linear and nonlinear cases. In our analysis, we assume that the time scale T is periodic in shifts ?? which does not need to be translation invariant. Thus, outcomes of this work are valid for a large class of time-domains not restricted to T = R or T = Z.