Oscillation theorems for some nonlinear difference equations (Q1354153)

The authors consider the nonlinear difference equation \[ \Delta (r_n\Delta u_n) +q_nf(u_{n-k}) =0,\;n=0,1,2, \dots, \tag{1} \] where \(\Delta u_n =u_{n+1} -u_n\), \(k\) is a nonnegative integer, \(\{r_n\}\) is a sequence of positive real numbers, \(\{q_n\}\) is a sequence of nonnegative real numbers, \(f:\mathbb{R} \to\mathbb{R}\), \(uf(u) >0\) for \(u\neq 0\). A nontrivial solution \(\{u_n\}\) of (1) is said to be oscillatory if for every \(N>0\) there exists an \(n\geq N\) such that \(u_n u_{n+1} \leq 0\). Sufficient conditions for all solutions of (1) to be oscillatory are obtained.