On Variations of Quasi-Cauchy Sequences in Cone Metric Spaces

Abstract

A sequence $(x_{n})$ of points in a topological vector space valued cone metric space $(X, \rho)$ is called $p$-quasi-Cauchy if for each $c\in \ici K$ there exists an $n_0\in \N$ such that $\rho (x_{n+p},x_{n})-c\in{\ici{K}}$ for $n\geq n_0$, where $K$ is a proper, closed and convex pointed cone in a topological vector space, $Y$ with $\ici K\neq \emptyset$. We investigate $p$-ward continuity in topological vector space valued cone metric spaces.
It turns out that $p$-ward continuity coincides with uniform continuity not only on a totally bounded subset but also on a connected subset of $X$.