Existence of the asymptotically periodic solution to the system of nonlinear neutral difference equations

Abstract

\noindent The system of nonlinear neutral difference equations with delays  in the  form
\[
\left
\{\begin{array}{l}
\Delta \big(y_i(n)+p_i(n)\,y_i(n-\tau_i)\big)=a_i(n)\,f_i(y_{i+1}(n))+g_i(n),
\\
\Delta \big(y_m(n)+p_m(n)\,y_m(n-\tau_m)\big)=a_m(n)\,f_m(y_1(n))+g_m(n),
\end{array}
\right.
\]
for $i=1,\dots,m-1$,  $m\geq 2$, is studied.
The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established.
Here sequences $(p_i(n))$, $i=1,\dots,m$, are bounded away from -1.
The presented results are illustrated by theoretical and numerical examples.