Necessary and sufficient conditions for oscillation of second order neutral difference equations

Abstract

Consider the second order difference equation of the form

$\Delta^2(y\n-py_{n-1-k})+q_nf(y_{n-\ell})=0,\quad n=1,2,3,\ldots  \hskip 1.9cm\hbox{(E)}$

where $ \{q_n\}$ is a nonnegative real sequence, $ f:{\Bbb R}\rightarrow {\Bbb R}$ is continuous such that $ uf(u)>0$ for $ u\not= 0$, $ 0\le p<1$, $ k$ and $ \ell$ are positive integers. We establish the necessary and/or sufficient conditions for the oscillation of all solutions of (E) when $ \int$ is linear, superlinear or sublinear and the results reduce to the well known theorems of Hooker and Patula in the special case when $ f(u)=u^\gamma$, where $ \gamma$ is a odd positive integers.