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Shift operators and stability in delayed dynamic equations. (English) Zbl 1227.34096
The authors introduce generalized shift operators, delay functions generated by them, and their properties. Then, for a time scale $\mathbb{T}$ having a delay function $\delta_-(h,t)$, where $h\ge t_0$ and $t_0\in\mathbb{T}$ is nonnegative and fixed, they investigate the general delay dynamic equation $$x^{\Delta}(t)=a(t)x(t)+b(t)x(\delta_-(h,t))\delta_-^{\Delta}(h,t),\quad t\in [t_0,\infty)_{\mathbb{T}}.\tag1$$ By using Lyapunov’s direct method, the authors obtain some inequalities which lead to exponential stability or instability of the zero solution of $(1)$. In this way, they extend and unify the stability analysis of delay differential, delay difference, delay $h$-difference and delay $q$-difference equations, which are the most important particular cases of equation $(1)$. Some applications are also presented.

34N05Dynamic equations on time scales or measure chains
34K20Stability theory of functional-differential equations
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