Stability and periodicity in dynamic delay equations

Let TT 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 xΔ(t)=−a(t)x(δ(t))δΔ(t).xΔ(t)=−a(t)x(δ(t))δΔ(t). Moreover, if TT is a periodic time scale, then necessary conditions are given for the existence of a unique periodic solution of the above mentioned equation.