On Variations of Quasi-Cauchy Sequences in Cone Metric Spaces


Hüseyin Çakallı




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 $c\in\overset{\circ}{K}$ there exists an $n_0\in\mathbb N$ such that $\rho(x_{n+p},x_n)-c\in\overset{\circ}{K}$ for $n\geq n_0$, where $K$ is a proper, closed and convex pointed cone in a topological vector space $Y$ with $\overset{\circ}{K}\neq0$. 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 $X$.