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Zbl 1227.34096
Shift operators and stability in delayed dynamic equations.
(English)
[J] Rend. Semin. Mat., Univ. Politec. Torino 68, No. 4, 369-396 (2010). ISSN 0373-1243

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.
[Rodica Luca Tudorache (Iaşi)]
MSC 2000:
*34N05
34K20 Stability theory of functional-differential equations

Keywords: delay dynamic equation; shift operators; delay functions; stability; instability

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