A Final Result on the Oscillation of Solutions of the Linear Discrete DelayedEquation Δx(n)=−p(n)x(n−k) with a Positive Coefficient

Abstract

A linear $(k+1)$ th-order discrete delayed equation $\Delta x\left(n\right)=-p\left(n\right)x(n-k)$ where $p\left(n\right)$ a positive sequence is considered for $n\to \infty$. This equation is known to have a positive solution if the sequence $p\left(n\right)$ satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for $p\left(n\right)$ , all solutions of the equation considered are oscillating for $n\to \infty$ .