Comparison and linearized oscillation theorems for a nonlinear partial difference equation (Q2725331)

The paper deals with the linear partial difference equation NEWLINE\[NEWLINEx_{m+1,n} + x_{m,n+1} - p x_{mn} + q x_{m-\sigma, n-\tau} = 0, \tag{1}NEWLINE\]NEWLINE and the nonlinear partial difference equation NEWLINE\[NEWLINEx_{m+1,n} + x_{m,n+1} - p x_{mn} + q_{mn} f(x_{m-\sigma, n-\tau}) = 0, \tag{2}NEWLINE\]NEWLINE where \(m, n = 0, 1,\dots\), \(p\) is a positive number and \(\sigma\) and \(\tau\) are positive integers, \(\{ q_{mn} \}\) a real double sequence, and \(f\) a real-valued function defined on \(\mathbb{R}.\) Using a comparison theorem, relations between existence of eventually positive solutions of equations (1) and (2) are established. The authors give conditions when every solution of (2) oscillates iff every solution of (1) oscillates.