Lacunary ward continuity in two normed spaces

Abstract

In this paper, we introduce lacunary statistically ward continuity in a $2$-normed space. A function $f$ defined on a subset $E$ of a $2$-normed space $X$ is lacunary statistically ward continuous if it preserves lacunary statistically quasi-Cauchy sequences of points in $E$ where a sequence $(x_{k})$ of points in $X$ is lacunary
statistically quasi-Cauchy if
\[
\lim_{r\rightarrow\infty}\frac{1}{h_{r}}|\{k\in I_{r}: ||x_{k+1}-x_{k},z||
\geq\varepsilon\}|=0
\]
for every positive real number $\varepsilon$ and $z\in X$, and $(k_r)$ is an increasing sequence of positive integers such that $k_0=0$ and $h_r=k_r-k_{r-1}\rightarrow\infty$ as $r\rightarrow\infty$, $I_r=(k_{r-1},k_r]$. We investigate not only lacunary statistically ward continuity, but also some other kinds of continuities in $2$-normed spaces.