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.

We study the oscillatory properties of the solutions of the third-order nonlinear semi-noncanonical delay difference equation $$D_{3}y\left(n\right)+f\left(n\right)y^{\beta }\left(\sigma \left(n\right)\right)=0,$$ where $D_{3}y\left(n\right)=\Delta \left(b\left(n\right)\Delta \left(a\left(n\right){\left(\Delta y\left(n\right)\right)}^{\alpha }\right)\right)$ is studied. The main idea is to transform the semi-noncanonical operator into canonical form. Then we obtain new oscillation theorems for the studied equation. Examples are provided to illustrate the importance of the main results.