Oscillation of a class of partial difference equations with unbounded delay
Consider real double sequences \((a_{mn})_{mn}\), \((b_{mn})_{mn}\), \((p_{mn})_{mn}\), the set \(\mathbb{N}\) of natural numbers, the set \(\mathbb{Z}\) of integers, nondecreasing functions \(\sigma\) and \(\tau\) defined on \(\mathbb{N}\) and with values in \(\mathbb{Z}\) such that \(\sigma(n)<n\), \(\tau(n)<n\), \(n\in \mathbb{N}\), \(\lim_{n\to+\infty}\sigma(n)=\lim_{n\to+\infty}\tau(n)=+\infty\). Assume that there exist positive constants \(a\) and \(b\) such that for all large \(m\) and \(n\) we have \(a_{mn}\geq a\), \(b_{mn}\leq b\), \(p_{mn}\geq 0\). Consider the partial difference equation \[ A_{m+1,n}+a_{mn}A_{m,n+1} -b_{mn}A_{mn}+ p_{mn}A_{\sigma(m),\tau(n)}=0. \] A nontrivial double sequence \((A_{mn})_{mn}\) satisfying (1) for \(m\geq 0\), \(n\geq 0\), is called a solution of (1). A solution \((A_{mn})_{mn}\) of (1) is said to be eventually positive (or negative) if \(A_{mn}>0\) (or \(A_{mn}<0)\) for large \(m\) and \(n\). It is said to be oscillatory if it is neither eventually positive nor eventually negative, and nonoscillatory otherwise. The authors prove some oscillation criteria for the solutions of the equation (1). For example, if \[ \limsup_{m,n\to+\infty} p_{mn}b^{\sigma(m)-m-1}\left(\frac ab\right)^{n-\tau(n)} c^{n-\tau(n)}_{m-\sigma(m)+n-\tau(n)}>1 \] then every solution of (1) is oscillatory. The authors give some examples to illustrate the results obtained.
scientific article; zbMATH DE number 1755272