Variations on the Strongly Lacunary Quasi Cauchy Sequences

Abstract

In this paper, we introduce a concept of a strongly lacunary $p$-quasi-Cauchy sequence in the sense that a sequence $(\alpha_{k})$ is strongly lacunary $p$-quasi-Cauchy if $\lim_{r\rightarrow\infty}\frac{1}{h_{r}}\sum^{}_{k\in{I_{r}}} |\alpha_{k+p}-\alpha_{k}|=0$. A function $f$ is called strongly lacunary $p$-ward continuous on a subset $A$ of the set of real numbers $\mathbb{R}$ if it preserves strongly lacunary $p$-quasi-Cauchy sequences, i.e. the sequence $f(\textbf{x})=(f(\alpha_{n}))$ is strongly lacunary $p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ is a strongly lacunary $p$-quasi-Cauchy sequence of points in $A$. It turns out that a real valued function $f$ is strongly lacunary ward continuous on a subset $A$ of $\mathbb{R}$ if it preserves strongly lacunary $p$-quasi-Cauchy sequences of points in $A$.