This paper is concerned with the asymptotic behavior for nonoscillatory solutions of the second order neutral delay difference equations of the form ∆ 2 (x n + p n x n−τ − q n x n−σ) + r n f ν (x n−l) = 0, also oscillation of this equation is discussed using sublinear function. Examples are inserted to illustrate the results.

Some new oscillation criteria are obtained for the third-order quasilinear difference equation $\Delta^2\left(p_n\left(\Delta x_n\right)^\alpha\right)-$ $q_n\left(\Delta x_n\right)^\alpha+r_n f\left(x_n\right)=0, n=0,1,2, \ldots$, where $\alpha>0$ is the ratio of odd positive integers. The method uses techniques based on Schwarz's inequality. Example is inserted to illustrate the result.

The paper is concerned with asymptotic and stability behaviors of fixed solutions of the second order nonlinear delay difference equation of the form ?2(xn +pnxn-k -qnxn-l)+ f(xn) = 0, n = 0,1,2, · ··,. Examples are provided to illustrate the results.

No abstract

Abstract. In this paper some sufficient conditions for oscillation of all solutions of certain difference equations are obtained. Examples are given to illustrate the results.