Abel Statistical Delta Quasi Cauchy Sequences of Real Numbers

Abstract

In this paper, we investigate the concept of Abel statistical delta quasi Cauchy sequences. A real function $f$ is called Abel statistically delta ward continuous it preserves Abel statistical delta quasi Cauchy sequences, where a sequence $(\alpha_{k})$ of points in $\mathbb{R}$ is called Abel statistically delta quasi Cauchy if $\lim_{x \to 1^{-}}(1-x)\sum_{k:|\Delta^{2} \alpha_{k}|\geq\varepsilon}^{}x^{k}=0$ for every $\varepsilon>0$, where $\Delta^{2} \alpha_{k}=\alpha_{k+2}-2\alpha_{k+1}+\alpha_{k}$ for every $k\in{\mathbb{N}}$. Some other types of continuities are also studied and interesting results are obtained.