Periodic solutions for a neutral nonlinear dynamical equation on a time scale

Let TT be a periodic time scale. We use a fixed point theorem due to Krasnosel'skiĭ to show that the nonlinear neutral dynamic system with delayxΔ(t)=−a(t)xσ(t)+c(t)xΔ(t−k)+q(t,x(t),x(t−k)),t∈T, has a periodic solution. We assume that k   is a fixed constant if T=RT=R and is a multiple of the period of TT if T≠RT≠R. Under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle.