Oscillation criteria for delay difference equations (Q2703365)

The authors obtain sufficient conditions to assure that all the solutions of the delay difference equation \(x_{n+1}-x_n+p_n x_{n-k}=0\), \(n=0,1, 2, \ldots\) are oscillatory. Here \(\{p_n\}\) is a sequence of nonnegative real numbers and \(k\) is a positive integer. NEWLINENEWLINENEWLINEIn the considered cases they do not assume one of the well known oscillation conditions NEWLINE\[NEWLINE\limsup_{n\to\infty}{\sum_{i=0}^{k}{p_{n-i} }} >1 \quad \text{ and } \quad \liminf_{n\to\infty}{\frac{1}{k}\sum_{i=1}^{k}{p_{n-i}}} > \frac{k^k}{(k+1)^{k+1}}.NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEPreviously, the authors proved some results in which necessary conditions to have a non oscillatory solution were given.