Necessary and sufficient conditions for oscillations of first order neutral delay difference equations with constant coefficients - Murugesan, A.; Sowmiya, P. (International Journal of Advanced Mathematical Sciences, 2015)
In this paper, we establish the necessary and sufficient conditions for oscillation of the following first order neutral delay difference equationÂ \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, \(p eq 0\) 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.