Oscillations of second-order nonlinear partial difference equations (Q1880828)

The paper deals with the partial difference equation \[ T(\Delta_1,\Delta_2)[c_{mn}T(\Delta_1,\Delta_2)(y_{mn})]+ p_{mn}(y_{m+1,n}+y_{m,n+1})^\nu=0, \] where \(T(\Delta_1,\Delta_2)=\Delta_1+\Delta_2+I\), \(\Delta_1y_{mn}=y_{m+1,n}-y_{mn}\), \(\Delta_2y_{mn}=y_{m,n+1}-y_{mn}\), \(Iy_{mn}=y_{mn}\); \(c_{mn}>0\), \(p_{mn}>0\), \(m\geq0\), \(n\geq0\); \(\nu>0\) is a quotient of odd positive integers. Denote \(r_{mn}=\sum_{i=m}^\infty\sum_{j=n}^\infty1/c_{ij}\) and suppose that \(r_{00}<\infty\). The authors establish the following result: All solutions of the above equation are oscillatory if \(\nu\neq1\) and \(\sum_{i=0}^\infty\sum_{j=0}^\infty(1/2)^{i+j}p_{ij} r_{i+1,j}^\theta=\infty\), where \(\theta=\max\{1,\nu\}\geq1\).