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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 𝕋 having a delay function δ - (h,t), where ht 0 and t 0 𝕋 is nonnegative and fixed, they investigate the general delay dynamic equation

x Δ (t)=a(t)x(t)+b(t)x(δ - (h,t))δ - Δ (h,t),t[t 0 ,) 𝕋 ·(1)

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.

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