Oscillatory behavior of quasilinear difference equations of second order (Q1585279)

The authors establish criteria for oscillation of all solutions to the equation \[ \Delta (a_n|\Delta y_n|^{\alpha -1}\Delta y_n) + F(n,y_{n-k},\Delta y_n)=0,\quad \alpha >1, \tag{*} \] where \(a_n>0\), \(k\) is a positive integer and the nonlinearity \(F\) satisfies certain restrictions. A typical result is the following statement. Theorem. Let \[ {\mathcal R}(n):=\sum_{j=n_0}^{n} {1\over a^{1/\alpha }}\to \infty,\quad \text{as}\quad n\to \infty \] and there exists \(\beta\) such that \(0<\beta\leq\alpha \) and \[ \sum_{n_0+k+1}^{\infty}{\mathcal R}^{\beta} {|F(n,v_n,v_{n-k},\Delta v_n)|\over |v_{n-k}|^{\beta }}=\infty \] for every positive nondecreasing or negative nonincreasing sequence \(\{v_n\}\). Then every solution of (*) is oscillatory. A prototype of equation (*) satisfying the assumptions of the Theorem is the equation \[ \Delta (a_n|\Delta y_n|^{\alpha -1}y_n)+q_n|y_{n-k}|^{\beta-1}y_{n-k}=0, \] where \(0<\beta<\alpha\), \(q_n>0\) and \[ \sum_{n_0+k+1}^{\infty}{\mathcal R}^{\beta}(n-k)q_n=\infty. \] Criteria for the existence of nonoscillatory solutions to equations similar to (*) are given as well.