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 $(T,\lambda )$-periodicity, on time scales. To achieve this task, first we define a new boundedness concept so-called $\lambda $-boundedness, and then we establish a linkage between the existence of $\lambda $-bounded solutions and $(T,\lambda )$-periodic solutions of dynamic equations in both linear and nonlinear cases. In our
analysis, we assume that the time scale $\mathbb{T}$ is periodic in shifts $\delta _{\pm}$ 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 $\mathbb{T}=\mathbb{R}$ or $\mathbb{T}=\mathbb{Z}$.