Upper and lower limit oscillation conditions for first-order difference equations

Abstract

AbstractIn this work, we consider the first-order difference equation with general argument \[ \varDelta x(n)+p(n)x\left( \tau (n)\right) =0,\quad n\geq 0, \]where $(p(n))$ is a sequence of non-negative real numbers, $(\tau (n))$ is a sequence of integers such that $\tau (n)< n$ for$\ n\in \mathbf {N},\,$and$\ \lim _{n\rightarrow \infty }\tau (n)=\infty.$ Under the assumption that the deviating argument is not necessarily monotone, we obtain some new oscillation conditions and improve the all known results for the above equation in the literature, involving only upper and only lower limit conditions. Two examples illustrating the results are also given.