Linearized oscillations of first-order nonlinear neutral delay difference equations (Q1827218)

The authors study oscillations of the first order nonlinear neutral delay equation \[ \Delta(x_n-px_{n-\tau})+\sum_{i=1}^m q_if_i(x_{n-\sigma_i})=0 \quad (n\geq n_0) \] and its corresponding linear equation \[ \Delta(x_n-px_{n-\tau})+\sum_{i=1}^m q_i x_{n-\sigma_i}=0 \quad (n\geq n_0), \] where \(p\in[0,1), f_i\in C({\mathbb R},{\mathbb R}), q_i\in(0,\infty), \tau,\sigma_i\in{\mathbb Z}_+, i\in\{1,\dots,m\}\), and \(\Delta x_n=x_{n+1}-x_n\). Let \[ F(\lambda):=(\lambda-1)(1-p\lambda^{-\tau})+\sum_{i=1}^m q_i\lambda^{-\sigma_i}. \] If there exists a \(\lambda_0\in(0,\infty)\) such that \(F(\lambda_0)=0\) and \(F(\lambda)>0\) for \(\lambda\in(0,\lambda_0)\cup(\lambda_0,\infty)\), then the above equations are said to be in the critical state. Otherwise, they are said to be in the noncritical state. The authors find conditions (both, for critical and noncritical states) under which every solution of the above nonlinear equation oscillates if and only if every solution of its corresponding linear equation oscillates.