AbstractIn this article, sufficient conditions are obtained so that every solution of the neutral difference equationΔm(yn−pnL(yn−s))+qnG(yn−k)=0,$$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0, \end{equation*}$$or every unbounded solution ofΔm(yn−pnL(yn−s))+qnG(yn−k)−unH(yα(n))=0,n≥n0,$$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})-u_nH(y_{\alpha(n)})=0,\quad n\geq n_0, \end{equation*}$$oscillates, wherem=2 is any integer, Δ is the forward difference operator given by Δyn=yn+1−yn; Δmyn= Δ(Δm−1yn) and other parameters have their usual meaning. The non linear functionL∈C(ℝ, ℝ) inside the operator Δmincludes the caseL(x) =x. Different types of super linear and sub linear conditions are imposed onGto prevent the solution approaching zero or ±∞. Further, all the three possible cases,pn≥ 0,pn≤ 0 andpnchanging sign, are considered. The results of this paper generalize and extend some known results.