Lacunary statistical ward continuity in metric spaces

Abstract

In this paper, we introduce a concept of lacunary statistically $p$-quasi-Cauchyness of a sequence in a metriic space in the sense that a sequence $(x_{k})$ is lacunary statistically $p$-quasi-Cauchy if $\lim_{r\rightarrow\infty}\frac{1}{h_{r}}|\{k\in I_{r}: d(x_{k+p}, x_{k})\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$ where
$I_{r}=(k_{r-1},k_{r}]$ and $k_{0}=1$,
$h_{r}=k_{r}-k_{r-1}\rightarrow \infty$ as $r\rightarrow\infty$
and $\theta=(k_{r})$ is an increasing sequence of positive integers. A function $f$ is called lacunary statistically $p$-ward continuous on a subset $A$ of $X$ if it preserves lacunary statistically $p$-quasi-Cauchy sequences, i.e. the sequence $(f(x_{n}))$ is lacunary statistically $p$-quasi-Cauchy whenever $\textbf{x}=(x_{n})$ is a lacunary statistically $p$-quasi-Cauchy sequence of points in $A$. It turns out that a function $f$ is uniformly continuous on a totally bounded subset $A$ of $X$ if there exists a positive integer $p$ such that $f$ preserves lacunary statistically $p$-quasi-Cauchy sequences of points in $A$.
}