Abstract:  In this paper, we study the existence and uniqueness of periodic solutions of  a totally nonlinear neutral difference system expressed as △x(n)=A(n)h (x(n−τ(n)))+△Q(n,x(n−g(n)))+G(n,x(n), x (n−g(n))) . We first utilize the fundamental matrix solution of △x(n)=A(n)x (n) to transform the aforementioned system.  Subsequently, we construct suitable mappings and apply Krasnoselskii’s fixed point theorem to demonstrate the existence of periodic solutions for this nonlinear neutral difference system.   Moreover, by means of the contraction mapping 
principle, we study the uniqueness of periodic solutions.  Unlike linear neutral systems, our model introduces complete nonlinearity, presenting new analytical challenges.

Key words:  , Neutral difference equation, Krasnoselskii’s fixed point theorem, Contrac tion mapping principle, Fundamental matirx solution, Periodic solution