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

Abstract

Abstract The system of nonlinear neutral difference equations with delays in the form { Δ ( y i ( n ) + p i ( n ) y i ( n − τ i ) ) = a i ( n ) f i ( y i + 1 ( n ) ) + g i ( n ) , Δ ( y m ( n ) + p m ( n ) y m ( n − τ m ) ) = a m ( n ) f m ( y 1 ( n ) ) + g m ( 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 (pi (n)), i = 1,..., m, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.