Lacunary Statistical $p$-Quasi Cauchy Sequences

Authors : Şebnem YILDIZ

Pages : 9-17

View : 15 | Download : 12

Publication Date : 2019-04-09

Article Type : Research Paper

Abstract :In this paper, we introduce a concept of lacunary statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $insert ignore into journalissuearticles values(\alpha_{k});$ is lacunary statistically $p$-quasi-Cauchy if $\lim_{r\rightarrow\infty}\frac{1}{h_{r}}|\{k\in I_{r}: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called lacunary statistically $p$-ward continuous on a subset $A$ of the set of real numbers $\mathbb{R}$ if it preserves lacunary statistically $p$-quasi-Cauchy sequences, i.e. the sequence $insert ignore into journalissuearticles values(finsert ignore into journalissuearticles values(\alpha_{n}););$ is lacunary statistically $p$-quasi-Cauchy whenever $\boldsymbol\alpha=insert ignore into journalissuearticles values(\alpha_{n});$ is a lacunary statistically $p$-quasi-Cauchy sequence of points in $A$. It turns out that a real valued function $f$ is uniformly continuous on a bounded subset $A$ of $\mathbb{R}$ if there exists a positive integer $p$ such that $f$ preserves lacunary statistically $p$-quasi-Cauchy sequences of points in $A$.
Keywords : Lacunary statistical convergence, Summability, Quasi Cauchy sequences, Continuity