Consider the first order linear difference equation with general advanced argument and variable coefficients of the form $$\nabla x\left(n\right)-p\left(n\right)x\left(\tau \right(n\left)\right)=0,n⩾1,$$ where p(n) is a sequence of nonnegative real numbers, τ(n) is a sequence of positive integers such that $$\tau \left(n\right)⩾n+1,n⩾1,$$ and ▿ denotes the backward difference operator ▿x(n) = x(n) − x(n − 1). Sufficient conditions which guarantee that all solutions oscillate are established. Examples illustrating the results are given.

This paper is concerned with the oscillatory behavior of first-order nonlinear difference equations with variable deviating arguments. The corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.

The asymptotic and oscillatory behavior of solutions of mth order damped nonlinear difference equation of the form $$\Delta \left(a_n{\Delta }^{m-1}{y}_n\right)+p_n{\Delta }^{m-1}{y}_n+q_{n}f\left(y_{\sigma (n+m-1)}\right)=0$$ where $m$ is even, is studied. Examples are included to illustrate the results.

In this paper, sufficient conditions are obtained for oscillation of all solutions of third order difference equations of the form $$y_{n+3}+r_n{y}_{n+2}+q_n{y}_{n+1}+p_n{y}_n=0,n\ge 0.$$ These results are generalization of the results concerning difference equations with constant coefficients $$y_{n+3}+r{y}_{n+2}+q{y}_{n+1}+p{y}_n=0,n\ge 0.$$ Oscillation, nonoscillation and disconjugacy of a certain class of linear third order difference equations are discussed with help of a class of linear second order difference equations.

In this paper we discuss the existence of oscillatory and nonoscillatory solutions for first order impulsive dynamic inclusions on time scales. We shall rely of the nonlinear alternative of Leray-Schauder type combined with lower and upper solutions method.

We study oscillatory properties of the second order half-linear difference equation $$\Delta (r_k|\Delta y_k{|}^{\alpha -2}\Delta y_k)-p_k{\left|y_{k+1}\right|}^{\alpha -2}{y}_{k+1}=0,\alpha >1.\left(\mathrm{HL}\right)$$ It will be shown that the basic facts of oscillation theory for this equation are essentially the same as those for the linear equation $$\Delta \left(r_k\Delta y_k\right)-p_k{y}_{k+1}=0.$$ We present here the Picone type identity, Reid Roundabout Theorem and Sturmian theory for equation (HL). Some oscillation criteria are also given.