A sequence $(\alpha _{k})$ of points in $\mathbb{R}$, the set of real numbers, is called $\rho$-statistically $p$ quasi Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{\rho _{n}}|\{k\leq n: |\Delta_{p}\alpha _{k} |\geq{\varepsilon}\}|=0 \] for each $\varepsilon>0$, where $\rho=(\rho_{n})$ is a non-decreasing sequence of positive real numbers tending to $\infty$ such that $\limsup _{n} \frac{\rho_{n}}{n}<\infty $, $\Delta \rho_{n}=O(1)$, and $\Delta_{p} \alpha _{k+p} =\alpha _{k+p}-\alpha _{k}$ for each positive integer $k$. A real-valued function defined on a subset of $\mathbb{R}$ is called $\rho$-statistically $p$-ward continuous if it preserves $\rho$-statistical $p$-quasi Cauchy sequences. $\rho$-statistical $p$-ward compactness is also introduced and investigated. We obtain results related to $\rho$-statistical $p$-ward continuity, $\rho$-statistical $p$-ward compactness, $p$-ward continuity, continuity, and uniform continuity.

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A New Approach to Statistically Quasi Cauchy Sequences

In this paper, we introduce and study new kinds of continuities. It turns out
 that a function f defined on an interval is uniformly continuous if and only
 if there exists a positive integer p such that f preserves p-quasi-Cauchy
 sequences where a sequence (xn) is called p-quasi-Cauchy if the sequence of
 differences between p-successive terms tends to 0.

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Variations on quasi-Cauchy sequences

A sequence $(\alpha _{k})$ of points in $\mathbb {R}$, the set of real numbers, is called $\rho $-statistically convergent to an element $\ell $ of $\mathbb {R}$ if 

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\rho _{n}}|\{k\le n: |\alpha _{k}-\ell |\ge {\varepsilon }\}|=0 \end{aligned}$$

for each $\varepsilon >0$, where $\varvec{\rho }=(\rho _{n})$ is a non-decreasing sequence of positive real numbers tending to $\infty $ such that $\limsup _{n} \frac{\rho _{n}}{n}<\infty $, $\Delta \rho _{n}=O(1)$, and $\Delta \alpha _{n} =\alpha _{n+1}-\alpha _{n}$ for each positive integer n. A real-valued function defined on a subset of $\mathbb {R}$ is called $\rho $-statistically ward continuous if it preserves $\rho $-statistical quasi-Cauchy sequences where a sequence $(\alpha _{k})$ is defined to be $\rho $-statistically quasi-Cauchy if the sequence $(\Delta \alpha _{k})$ is $\rho $-statistically convergent to 0. We obtain results related to $\rho $-statistical ward continuity, $\rho $-statistical ward compactness, ward continuity, continuity, and uniform continuity. It turns out that the set of uniformly continuous functions coincides with the set of $\rho $-statistically ward continuous functions not only on a bounded subset of $\mathbb {R}$, but also on an interval.

A Variation on Statistical Ward Continuity

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.

/pdf/abel-statistical-delta-quasi-cauchy-sequences-of-real-4kuv1rrbac.pdf

Abel Statistical Delta Quasi Cauchy Sequences of Real Numbers

Recently, the first author has introduced a concept of G-sequential connectedness in the sense that a non-empty subset A of a Hausdorff topological group X is G-sequentially connected if the only subsets of A which are both G-sequentially open and G-sequentially closed are A and the empty set ∅. In this paper we investigate further properties of G-sequential connectedness and obtain some interesting results.

On connectedness via a sequential method

In this paper, we investigate slowly oscillating continuity in cone metric spaces. It turns out that the set of slowly oscillating continuous functions is equal to the set of uniformly continuous functions on a slowly oscillating compact subset of a topological vector space valued cone metric space.

/pdf/slowly-oscillating-continuity-in-abstract-metric-spaces-3q9yv0zrze.pdf

Ayșe Sӧnmez

Papers

Slowly oscillating continuity in abstract metric spaces