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In this paper there are established necesssary and sufficient conditions for the second order quasilinear difference equation L(p,cp(iyn)) + f(n,y,,+ i) = 0 (n E No) to have various types of non-oscillatory solutions. In addition, in the case that the equation is either strongly superlinear or strongly sublinear, there are established necessary and sufficient conditions for all solutions to oscillate.

This paper deals with the following second order neutral differential equation of the formwhere z(t) = x(t) + a(t)x(t − τ ) + b(t)x(t + δ), andWe obtain some new oscillation criteria which extend some known results. Examples are presented to illustrate the main results.

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