Existence of The Asymptotically Periodic Solution to the System of Nonlinear Neutral Difference Equations

The system of nonlinear neutral difference equations with delays in the form { Δ(yi(n)+pi(n)yi(n−τi))=ai(n)fi(yi+1(n))+gi(n),Δ(ym(n)+pm(n)ym(n−τm))=am(n)fm(y1(n))+gm(n),\[\left\{ \begin{array}{l} \Delta ({y_i}(n) + {p_i}(n){y_i}(n - {\tau _i})) = {a_i}(n){f_i}({y_{i + 1}}(n)) + {g_i}(n),\\ \Delta ({y_m}(n) + {p_m}(n){y_m}(n - {\tau _m})) = {a_m}(n){f_m}({y_1}(n)) + {g_m}(n), \end{array} \right.\] for i = 1, . . . , m − 1, m ≥ 2, is studied. The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established. Here sequences (p_i(n)), i = 1,..., m, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.