Linearized oscillation theorems for certain nonlinear delay partial difference equations
Consider the difference equation \[ A_{m+1,n} +A_{m,n+1} -A_{m,n} +p_{m,n} f(A_{m-k, n-1})= 0,m,n=0,1, \dots \tag{1} \] where \(f\in C(\mathbb{R}, \mathbb{R})\), \(p_{m,n}>0\), and corresponding with it the linear equation \[ A_{m+1,n} +A_{m,n+1} -A_{m,n} +pA_{m-k,n-1} =0,\;p>0. \tag{2} \] A solution \(\{A_{i,j}\}\) to (2) is said to be proper if there exist positive constants \(M, \alpha\) and \(\beta\) such that \(| A_{m,n} |\leq M\alpha^m \beta^n\) for all large \(m\) and \(n\). Let \(\lim\inf_{m,n \to\infty} p_{m,n} =p>0\), \((f(x))/x>0\) for \(x\neq 0\) and \(\lim_{x\to 0} (f(x))/x=1\), then if every proper solution to (2) is oscillatory the same holds for (1). A similar criterion is obtained for positivity of solutions.
nonlinear delay partial difference equations
scientific article; zbMATH DE number 1192707