Existence of periodic solutions in neutral nonlinear difference systems with delay

We use Krasnoselskii's fixed point theorem to show that the nonlinear neutral system of difference equation with functional delay Delta x(t) = A(t)x(t) + Delta Q(t,x(t - g(t))) + G(t,x(t),x(t - g(t))) has a periodic solution. Also, inverting the mapping and then using the notion of fundamental matrix solution we are able, using the contraction mapping principle, to show that the periodic solution is unique.