Stability and periodicity in dynamic delay equations

Keywords: Delay dynamic equations Fixed point theory Lyapunov Periodic solutions Stability Time scales a b s t r a c t Let 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 ∆ (t) = −a(t)x(δ(t))δ ∆ (t). Moreover, if T is a periodic time scale, then necessary conditions are given for the existence of a unique periodic solution of the above mentioned equation.