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Stability and periodicity in dynamic delay equations. (English) Zbl 1189.34143

Summary: Let \(\mathbb{T}\) be an arbitrary time scale that is unbounded above. By means of a variation of Lyapunov’s method and contraction mapping principle this paper handles asymptotic stability of the zero solution of the completely delayed dynamic equations \[ x^{\Delta} (t)= - a(t)x(\delta (t))\delta^{\Delta} (t). \] Moreover, if \(\mathbb{T}\) is a periodic time scale, then necessary conditions are given for the existence of a unique periodic solution of the above mentioned equation.

MSC:

34K20 Stability theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34N05 Dynamic equations on time scales or measure chains
47N20 Applications of operator theory to differential and integral equations
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