Linearized oscillations in nonlinear delay difference equations
Consider the nonlinear difference equation \[ x_{n+1}-x_n=\sum^m_{j=1} p_jf_j(x_{n-k_j})=0,\quad n\geq n_0\geq 0\tag{*} \] and its linearized version \[ x_{n+1}-x_n+\sum^m_{i=1} p_jx_{n-k_i}=0,\quad n\geq n_0,\tag{**} \] where \(n_0\in\{0,1,2,\dots\}\). If \(p_j>0\), \(k_j=0,1,2,\dots\), \(f_j\in C(\mathbb{R},\mathbb{R}]\), \(u\cdot f_j(u)>0\) (\(u\neq 0\)), \(u\in(-\alpha,\alpha)\) and \(\lim u^{-1}f_j(u)=1\) for all \(j=1,2,\dots,m\), then every solution of (**) is oscillatory if and only if (*) oscillates.
scientific article; zbMATH DE number 1385074