Turkish Journal of Mathematics
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.
DOI
10.3906/mat-2102-106
Keywords
Delay dynamic equation, time scale, equi-bounded, asymptotic stability, shift operator, periodicity
First Page
1985
Last Page
2007
Recommended Citation
KOYUNCUOĞLU, HALİS CAN
(2021)
"Some qualitative results for functional delay dynamic equations on time scales,"
Turkish Journal of Mathematics: Vol. 45:
No.
5, Article 8.
https://doi.org/10.3906/mat-2102-106
Available at:
https://journals.tubitak.gov.tr/math/vol45/iss5/8
