A new approach to Abel statistical convergence in metric spaces

A new approach to Abel statistical convergence

Abstract

In this study, we first consider the sequences in the sense of Abel statistical together with the functions preserving the convergence of this kind of sequences called Abel statistical continuous functions in a metric space $X$. Then we relate this kind of continuity with some others. A function $f$ is Abel statistically continuous on a subset $E$ of a metric space $X$, if it preserves Abel statistical convergent sequences, i.e. $(f(p_{k}))$ is Abel statistically convergent whenever $(p_{k})$ is an Abel statistical convergent sequence of points in $E$, where a sequence $(p_{k})$ of points in $X$ is called Abel statistically convergent to a point $L$ in $X$ if $\lim_{x \to 1^{-}}(1-x)\sum_{k\in{\N}:d(p_{k}, L)\geq\varepsilon}^{}x^{k}=0$ for every $\varepsilon>0$. Some other types of continuities are also studied and interesting results are obtained.