Oscillation criteria for difference equations with non-monotone arguments

This paper is concerned with the oscillatory behavior of first-order retarded [advanced] difference equation of the form Δx(n)+p(n)x(τ(n))=0,n∈N0[∇x(n)−q(n)x(σ(n))=0,n∈N],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta x(n)+p(n)x\bigl(\tau (n)\bigr)=0, \quad n\in \mathbb{N} _{0} \qquad \bigl[\nabla x(n)-q(n)x\bigl(\sigma (n)\bigr)=0, \ n\in \mathbb{N} \bigr], $$\end{document}where (p(n))n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p(n))_{n\geq 0}$\end{document}[(q(n))n≥1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[(q(n))_{n\geq 1}]$\end{document} is a sequence of nonnegative real numbers and τ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau (n)$\end{document}[σ(n)]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\sigma (n)]$\end{document} is a non-monotone sequence of integers such that τ(n)≤n−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau (n)\leq n-1$\end{document}, for n∈N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\in \mathbb{N}_{0}$\end{document} and limn→∞τ(n)=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lim_{n\rightarrow \infty }\tau (n)=\infty $\end{document}[σ(n)≥n+1, for n∈N]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\sigma (n)\geq n+1,\mbox{ for }n\in \mathbb{N}]$\end{document}. Sufficient conditions, involving limsup, which guarantee the oscillation of all solutions are established. These conditions improve all previous well-known results in the literature. Also, using algorithms on MATLAB software, examples illustrating the significance of the results are given.