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.

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.

The paper deals with the higher-order ordinary differential equations and the analogous higher-order difference equations and compares the corresponding fundamental concepts. Important dissimilarities appear for the moving frame method.

This paper presents two theorems for designing controllers to achieve directional partial generalized synchronization (PGS) of two independent (chaotic) differential equation systems or two independent (chaotic) discrete systems. Two numerical simulation examples are given to illustrate the effectiveness of the proposed theorems. It can be expected that these theorems provide new tools for understanding and studying PGS phenomena and information encryption.