Unified Massera type theorems for dynamic equations on time scales

Halis Can Koyuncuoğlu


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 $\mathbb{T}=\mathbb{Z}$.


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