Necessary and sufficient conditions for oscillations of first order neutral delay difference equations with constant coefficients
Abstract
In this paper, we establish the necessary and sufficient conditions for oscillation of the following first order neutral delay difference equation\begin{equation*} \quad \quad \quad \quad \quad \quad \quad \quad \quad\quad \quad \quad \quad\Delta[x(n)+px(n-\tau)]+qx(n-\sigma)=0, \quad \quad n\geq n_0, \quad \quad \quad \quad \quad \quad {(*)} \end{equation*}
where \(\tau\) and \(\sigma\) are positive integers, \(p\neq 0\) is a real number and \(q\) is a positive real number. We proved that every solution of (*) oscillates if and only if its characteristic equation
\begin{equation*}\quad \quad \quad \quad\quad \quad \quad \quad\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (\lambda-1)(1+p\lambda^{-\tau})+q\lambda^{-\sigma}=0\quad \quad \quad \quad \quad \quad \quad \quad {(**)} \end{equation*}
has no positive roots.
About this article
Keywords
- Neutral
- delay difference equation
- oscillatory properties.
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