A New Approach to Statistically Quasi Cauchy Sequences

Abstract

A sequence $(\alpha _{k})$ of points in $\mathbb{R}$, the set of real numbers, is called $\rho$-statistically $p$ quasi Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{\rho _{n}}|\{k\leq n: |\Delta_{p}\alpha _{k} |\geq{\varepsilon}\}|=0 \] for each $\varepsilon>0$, where $\rho=(\rho_{n})$ is a non-decreasing sequence of positive real numbers tending to $\infty$ such that $\limsup _{n} \frac{\rho_{n}}{n}<\infty $, $\Delta \rho_{n}=O(1)$, and $\Delta_{p} \alpha _{k+p} =\alpha _{k+p}-\alpha _{k}$ for each positive integer $k$. A real-valued function defined on a subset of $\mathbb{R}$ is called $\rho$-statistically $p$-ward continuous if it preserves $\rho$-statistical $p$-quasi Cauchy sequences. $\rho$-statistical $p$-ward compactness is also introduced and investigated. We obtain results related to $\rho$-statistical $p$-ward continuity, $\rho$-statistical $p$-ward compactness, $p$-ward continuity, continuity, and uniform continuity.