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Necessary and sufficient conditions for oscillations of first order neutral delay difference equations with constant coefficients
Author(s) -
A. Murugesan,
P. Sowmiya
Publication year - 2015
Publication title -
international journal of advanced mathematical sciences
Language(s) - English
Resource type - Journals
ISSN - 2307-454X
DOI - 10.14419/ijams.v3i1.4462
Subject(s) - constant (computer programming) , order (exchange) , mathematics , mathematical analysis , computer science , economics , finance , programming language
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.

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