Oscillatory behavior of solutions of certain third order mixed neutral difference equations

The objective of this paper is to study the oscillatory and asymptotic properties of the mixed type third order neutral difference equation of the form Δ(anΔ2(xn+bnxn−r1+cnxn+r2))+qnxn+1−σ1β+pnxb+1+σ2β=0,where {an}, {bn}, {cn}, {qn} and {pn} are positive real sequences, β is a ratio of odd positive integers, τ1, τ2, σ1 and σ2 are positive integers. We establish some sufficient conditions which ensure that all solutions are either oscillatory or converges to zero. Some examples are presented to illustrate the main results.