A note on asymptotic stability condition for delay difference equations

Abstract

In this paper, we obtain the necessary and sufficient condition for the asymptotic stability of the linear delay difference equation $$ x_{n+1} - x_{n-1} + p \sum_{j=1}^{N} x_{n-k+(j-1)l} = 0 $$ where $n = 0, 1, 2, \dots,\ p$ is a real number, and $k$, $l$, and $N$ are positive integers such that $k \gt (N - 1)l$.