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Turkish Journal of Mathematics

DOI

10.3906/mat-2102-106

Abstract

Let $\mathbb{T}$ be a nonempty, closed, and arbitrary set of real numbers, namely a time scale, and consider the following delay dynamical equation% \[ x^{\Delta}\left( t\right) =a\left( t\right) x\left( t\right) +f\left( t,x\left( \mathbb{\vartheta}\left( t\right) \right) \right) ,\text{ }t\in\mathbb{T}, \] where $\mathbb{\vartheta}$ stands for the abstract delay function. The main goal of this study is three-fold: obtaining the existence of an equi-bounded solution, proving the asymptotic stability of the zero solution, and showing the existence of a periodic solution based on new periodicity concept on time scales for the given delayed equation under certain conditions. In our analysis, we propose an alternative variation of parameters formulation by using an auxiliary function to invert a mapping for the utilization of fixed point theory.

Keywords

Delay dynamic equation, time scale, equi-bounded, asymptotic stability, shift operator, periodicity

First Page

1985

Last Page

2007

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