A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation $$\Delta \left(a_n{\left|\Delta x_n\right|}^{\alpha }sgn\Delta x_n\right)+b_n{\left|x_{n+1}\right|}^{\beta }sgn{x}_{n+1}=0$$ is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.

We have established sufficient conditions for oscillation of a class of first order neutral impulsive difference equations with deviating arguments and fixed moments of impulsive effect.

In this work, oscillatory behaviour of solutions of a class of fourth-order neutral functional difference equations of the form $${\Delta }^2\left(r\left(n\right){\Delta }^2(y\left(n\right)+p\left(n\right)y(n-m))\right)+q\left(n\right)G\left(y(n-k)\right)=0$$ is studied under the assumption $$\sum _{n=0}^{\infty }\frac{n}{r\left(n\right)}<\infty .$$ New oscillation criteria have been established which generalize some of the existing results in the literature.

This paper is divided in two parts. In the first part we study a convergent $q$-analog of the divergent Euler series, with $q\in (0,1)$, and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding $q$-difference equation. In the second part, we work under the assumption $q\in (1,+\infty )$. In this case, at least four different $q$-Borel sums of a divergent power series solution of an irregular singular...

We study the solutions and attractivity of the difference equation $x_{n+1}=x_{n-3}/(-1+x_n{x}_{n-1}{x}_{n-2}{x}_{n-3})$ for $n=0,1,2,\cdots$ where $x_{-3},x_{-2},x_{-1}$ and $x_0$ are real numbers such that $x_0{x}_{-1}{x}_{-2}{x}_{-3}\ne 1.$

In this paper we establish some new nonlinear difference inequalities. We also present an application of one inequality to certain nonlinear sum-difference equation.