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

Abstract. Consider the second order difference equation of the form ∆2(yn−1−pyn−1−k)+qnf(yn−ℓ) = 0, n = 1, 2, 3,... (E) where {qn} is a nonnegative real sequence, f: R → R is continuous such that uf(u)> 0 for u 6 = 0, 0 ≤ p < 1, k and ℓ are positive integers. We establish the necessary and/or sufficient conditions for the oscillation of all solutions of (E) when 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) = uγ, where γ is a odd positive integers. 1