Oscillations of advanced difference equations with variable arguments

Consider the first-order advanced difference equation of the form \begin{equation*} \nabla x(n)-p(n)x(\mu (n))=0\text{, }\ n\geq 1\, [\Delta x(n)-p(n)x(\nu (n))=0, n\geq 0], \end{equation*} where $\nabla$ denotes the backward difference operator $\nabla x(n)=x(n)-x(n-1)$, $\Delta$ denotes the forward difference operator $\Delta x(n)=x(n+1)-x(n)$, $\left\{ p(n)\right\}$ is a sequence of nonnegative real numbers, and $\left\{ \mu (n)\right\}$ $\ \left[ \left\{ \nu (n)\right\} \right]$ is a sequence of positive integers such that \begin{equation*} \mu (n)\geq n+1\ \text{ for all }n\geq 1\, \left[ \nu (n)\geq n+2 \ \text{ for all }n\geq 0\right] \text{.} \end{equation*} Sufficient conditions which guarantee that all solutions oscillate are established. The results obtained essentially improve known results in the literature. Examples illustrating the results are also given

Oscillations of equations caused by several deviating arguments

Linear delay or advanced differential equations with variable coefficients and several not necessarily monotone arguments are considered, and some new oscillation criteria are given. More precisely, sufficient conditions, involving $\lim\sup$ and $\lim\inf$, are obtained, which essentially improve several known criteria existing in the literature. Examples illustrating the results are also given, numerically solved in MATLAB

Existence of positive solutions of linear delay difference equations with continuous time

Consider the delay difference equation with continuous time of the form $$x(t)-x(t-1)+\sum_{i=1}^mP_i(t)x(t-k_i(t))=0,\qquad t\ge t_0,$$ where $P_i\colon[t_0,\infty)\mapsto\mathbb{R}$, $k_i\colon[t_0,\infty)\mapsto \{2,3,4,\dots\}$ and $\lim_{t\to\infty}(t-k_i(t))=\infty$, for $i=1,2,\dots,m$. We introduce the generalized characteristic equation and its importance in oscillation of all solutions of the considered difference equations. Some results for the existence of positive solutions of considered difference equations are presented as the application of the generalized characteristic equation

Oscillation properties of second-order quasilinear difference equations with unbounded delay and advanced neutral terms

summary:We obtain some new sufficient conditions for the oscillation of the solutions of the second-order quasilinear difference equations with delay and advanced neutral terms. The results established in this paper are applicable to equations whose neutral coefficients are unbounded. Thus, the results obtained here are new and complement some known results reported in the literature. Examples are also given to illustrate the applicability and strength of the obtained conditions over the known ones