Bounded solutions of $k$-dimensional system of nonlinear difference equations of neutral type

The $k$-dimensional system of neutral type nonlinear difference equations with delays in the following form \begin{equation*} \begin{cases} \Delta \Big(x_i(n)+p_i(n)\,x_i(n-\tau_i)\Big)=a_i(n)\,f_i(x_{i+1}(n-\sigma_i))+g_i(n),\\ \Delta \Big(x_k(n)+p_k(n)\,x_k(n-\tau_k)\Big)=a_k(n)\,f_k(x_1(n-\sigma_k))+g_k(n), \end{cases} \end{equation*} where $i=1,\dots,k-1$, is considered. The aim of this paper is to present sufficient conditions for the existence of nonoscillatory bounded solutions of the above system with various $(p_i(n))$, $i=1,\dots,k$, $k\geq 2$.