arXiv subjects

Huseyin Cakalli

Publications and source records attributed to Huseyin Cakalli.

At least 19 recordsLinked to original sources

arXiv2018

A study on downward half Cauchy sequences

In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence $(f(α_{n}))$ is downward half Cauchy whenever $(α_{n})$ is a downward half Cauchy sequence of points in $E$, where a sequence $(α_{ k})$ of points in $\R$ is called downward half Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\N}$ such that $α_{m}-α_{n} <\varepsilon$ for $m \geq n \geq n_0$. It turns out that the set of down continuous functions is a proper subset of the set of continuous functions.

math.FA↗

arXiv2017

A new variation on statistical ward continuity

A real valued function defined on a subset $E$ of $\mathbb{R}$, the set of real numbers, is $ρ$-statistically downward continuous if it preserves $ρ$-statistical downward quasi-Cauchy sequences of points in $E$, where a sequence $(α_{k})$ of real numbers is called $ρ$-statistically downward quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{ρ_{n} }|\{k\leq n: Δα_{k} \geq \varepsilon\}|=0 $ for every $\varepsilon>0$, in which $(ρ_{n})$ is a non-decreasing sequence of positive real numbers tending to $\infty$ such that $\limsup _{n} \frac{ρ_{n}}{n}<\infty $, $Δρ_{n}=O(1)$, and $Δα_{k} =α_{k+1} - α_{k}$ for each positive integer $k$. It turns out that a function is uniformly continuous if it is $ρ$-statistical downward continuous on an above bounded set.

math.FA↗

arXiv2017

On Abel statistical convergence

In this paper, we introduce and investigate a concept of Abel statistical continuity. A real valued function $f$ is Abel statistically continuous on a subset $E$ of $\R$, the set of real numbers, 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 point in $\R$ is called Abel statistically convergent to a real number $L$ if Abel density of the set $\{k\in{\N}: |p_{k}-L|\geq\varepsilon \}$ is $0$ for every $\varepsilon>0$. Some other types of continuities are also studied and interesting results are obtained.

math.FA↗

arXiv2017

On Variations of statistical ward continuity

In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $(α_{k})$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: |α_{k+p}-α_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called statistically $p$-ward continuous on a subset $A$ of the set of real umbers $\mathbb{R}$ if it preserves statistically $p$-quasi-Cauchy sequences, i.e. the sequence $f(\textbf{x})=(f(α_{n}))$ is statistically $p$-quasi-Cauchy whenever $\boldsymbolα=(α_{n})$ is a 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 statistically $p$-quasi-Cauchy sequences of points in $A$.

math.FA↗

arXiv2017

A new study on the strongly lacunary quasi Cauchyness

In this paper, the concept of an $N_θ^{2}$ quasi-Cauchy sequence is introduced. We proved interesting theorems related to $N_θ^{2}$-quasi-Cauchy sequences. A real valued function $f$ defined on a subset $A$ of $\mathbb{R}$, the set of real numbers, is $N_θ^{2}$ ward continuous on $A$ if it preserves $N_θ^{2}$ quasi-Cauchy sequences of points in $A$, i.e. $(f( α_{k}))$ is an $N_θ^{2}$ quasi-Cauchy sequence whenever $(α_{k})$ is an $N_θ^{2}$ quasi-Cauchy sequences of points in $A$, where a sequence $(α_{k})$ is called $N_θ^{2}$ quasi-Cauchy if $(Δ^{2} α_{k})$ is an $N_θ$ quasi-Cauchy sequence where $Δ^{2}α_{k}=α_{k+2}-2α_{k+1}+α_{k}$ for each positive integer $k$.

math.FA↗

arXiv2014

Statistical quasi Cauchyness in two normed spaces

A function $f$ defined on a subset $E$ of a two normed space $X$ is statistically ward continuous if it preserves statistically quasi-Cauchy sequences of points in $E$ where a sequence $(x_n)$ is statistically quasi-Cauchy if $(Δx_{n})$ is a statistically null sequence. A subset $E$ of $X$ is statistically ward compact if any sequence of points in $E$ has a statistically quasi-Cauchy subsequence. In this paper, new kinds of continuities are investigated in two normed spaces. It turns out that uniform limit of statistically ward continuous functions is again statistically ward continuous.

math.FA↗

arXiv2013

Functions preserving slowly oscillating double sequences

A double sequence $\textbf{x}=\{x_{k,l}\}$ of points in $\textbf{R}$ is slowly oscillating if for any given $\varepsilon>0$, there exist $α=α(\varepsilon)>0$, $δ=δ(\varepsilon) >0$, and $N=N(\varepsilon)$ such that $|x_{k,l}-x_{s,t}|<\varepsilon$ whenever $k,l\geq N(\varepsilon)$ and $k\leq s \leq (1+α)k$, $l\leq t \leq (1+δ)l$. We study continuity type properties of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of $\textbf{R}^{2}$ into $\textbf{R}$.

math.GM↗

arXiv2013

On Double Sequences

A double sequence $\{x_{k,l}\}$ is quasi-Cauchy if given an $ε> 0$ there exists an $N \in {\bf N}$ such that $$\max_{r,s= 1\mbox{ and/or} 0} \left \{|x_{k,l} - x_{k+r,l+s}|< ε\right \} .$$ We study continuity type properties of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$, and obtain interesting results related to uniform continuity, sequential continuity, continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset $A\times A$ of ${\bf R}^{2}$ into $\textbf{R}$.

math.GM↗

arXiv2013

A study on ideal ward continuity

In this paper, we prove that any ideal ward continuous function is uniformly continuous either on an interval or on an ideal ward compact subset of $\textbf{R}$. A characterization of uniform continuity is also given via ideal quasi-Cauchy sequences.

math.GM↗

arXiv2013

On sequences in 2-normed spaces

A function $f$ defined on a 2-normed space $ (X,||.,.||)$ is ward continuous if it preserves quasi-Cauchy sequences where a sequence $(x_n)$ of points in $X$ is called quasi-Cauchy if $lim_{n\rightarrow\infty}||Δx_{n},z||=0$ for every $z\in X$. Some other kinds of continuties are also introduced via quasi-Cauchy sequences in 2-normed spaces. It turns out that uniform limit of ward continuous functions is again ward continuous.

math.FA↗

arXiv2013

Upward and downward statistical continuities

A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is statistically upward continuous if it preserves statistically upward half quasi-Cauchy sequences, is statistically downward continuous if it preserves statistically downward half quasi-Cauchy sequences; and a subset $E$ of $\textbf{R}$, is statistically upward compact if any sequence of points in $E$ has a statistically upward half quasi-Cauchy subsequence, is statistically downward compact if any sequence of points in $E$ has a statistically downward half quasi-Cauchy subsequence where a sequence $(x_{n})$ of points in $\textbf{R}$ is called statistically upward half quasi-Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k}-x_{k+1}\geq \varepsilon\}|=0 \] is statistically downward half quasi-Cauchy if \[ \lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: x_{k+1}-x_{k}\geq \varepsilon\}|=0 \] for every $\varepsilon>0$. We investigate statistically upward continuity, statistically downward continuity, statistically upward half compactness, statistically downward half compactness and prove interesting theorems. It turns out that uniform limit of a sequence of statistically upward continuous functions is statistically upward continuous, and uniform limit of a sequence of statistically downward continuous functions is statistically downward continuous.

math.GM↗

arXiv2013

$λ$-statistically quasi-Cauchy sequences

The main object of this paper is to investigate $λ$-statistically quasi-Cauchy sequences. A real valued function $f$ defined on a subset $E$ of $\textbf{R}$, the set of real numbers, is called $λ$-statistically ward continuous on $E$ if it preserves $λ$-statistically quasi-Cauchy sequences of points in $E$. It turns out that uniform continuity coincides with $λ$-statistically ward continuity on $λ$-statistically ward compact subsets.

math.GM↗

arXiv2012

Half quasi-Cauchy sequences

A real function $f$ is ward continuous if $f$ preserves quasi-Cauchyness, i.e. $(f(x_{n}))$ is a quasi-Cauchy sequence whenever $(x_{n})$ is quasi-Cauchy; and a subset $E$ of $\textbf{R}$ is quasi-Cauchy compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a quasi-Cauchy subsequence where $\textbf{R}$ is the set of real numbers. These known results suggest to us introducing a concept of upward (respectively, downward) half quasi-Cauchy continuity in the sense that a function $f$ is upward (respectively, downward) half quasi-Cauchy continuous if it preserves upward (respectively, downward) half quasi-Cauchy sequences, and a concept of upward (respectively, downward) half quasi-Cauchy compactness in the sense that a subset $E$ of $\textbf{R}$ is upward (respectively, downward) half quasi-Cauchy compact if any sequence of points in $E$ has an upward (respectively, downward) half quasi-Cauchy subsequence. We investigate upward(respectively, downward) half quasi-Cauchy continuity and upward (respectively, downward) half quasi-Cauchy compactness, and prove related theorems.

math.GM↗

arXiv2012

P-Quasi-Cauchy Sequences

In this paper we generalize the concept of a quasi-Cauchy sequence to a concept of a $p$-quasi-Cauchy sequence for any fixed positive integer $p$. For $p=1$ we obtain some earlier existing results as a special case. We obtain some interesting theorems related to $p$-quasi-Cauchy continuity, $G$-sequential continuity, slowly oscillating continuity, and uniform continuity. 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 $(x_{n})$ is called $p$-quasi-Cauchy if $(x_{n+p}-x_{n})_{n=1}^{\infty}$ is a null sequence.

math.GM↗

arXiv2012

Ideal-quasi-Cauchy sequences

An ideal $I$ is a family of subsets of positive integers $\textbf{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_n)$ of real numbers is said to be $I$-convergent to a real number $L$, if for each \;$ \varepsilon> 0$ the set $\{n:|x_{n}-L|\geq \varepsilon\}$ belongs to $I$. We introduce $I$-ward compactness of a subset of $\textbf{R}$, the set of real numbers, and $I$-ward continuity of a real function in the senses that a subset $E$ of $\textbf{R}$ is $I$-ward compact if any sequence $(x_{n})$ of points in $E$ has an $I$-quasi-Cauchy subsequence, and a real function is $I$-ward continuous if it preserves $I$-quasi-Cauchy sequences where a sequence $(x_{n})$ is called to be $I$-quasi-Cauchy when $(Δx_{n})$ is $I$-convergent to 0. We obtain results related to $I$-ward continuity, $I$-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, $δ$-ward continuity, and slowly oscillating continuity.

math.GM↗

arXiv2012

Sequential Definitions of Connectedness

A topological group $X$ is called connected if the only subsets which are both open and closed are the whole space $X$ and the null set $\emptyset$. A subset of a topological group is connected if the subspace is connected. We say that a subset $A$ of $X$ is $G$-sequentially connected if the only subsets of $A$ which are both $G$-sequentially open and $G$-sequentially closed, with respect to the relative $G$-sequentially open and $G$-sequentially closed subsets of $A$, are open and closed subsets of $A$ are $A$ and the null set, $\emptyset$. We investigate the impact of changing the definition of convergence of sequences on the structure of sequential connectedness of subsets of $X$ via sequential closure of sets in the sense of $G$-sequential closure. Sequential connectedness for topological groups is a special case of this generalization when G = lim.

math.GN↗

arXiv2011

Metrizability of Topological Vector Space Valued Cone Metric Spaces

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. This is applied by Du (2010) [A note on cone metric fixed point theory and its equivalence, Nonlinear Analysis 72 2259-2261] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable.

math.GN↗

arXiv2011

On $Δ$-quasi-slowly oscillating sequences

A sequence $(x_{n})$ of points in a topological group is called $Δ$-quasi-slowly oscillating if $(Δx_{n})$ is quasi-slowly oscillating, and is called quasi-slowly oscillating if $(Δx_{n})$ is slowly oscillating. A function $f$ defined on a subset of a topological group is quasi-slowly (respectively, $Δ$-quasi-slowly) oscillating continuous if it preserves quasi-slowly (respectively, $Δ$-quasi-slowly) oscillating sequences, i.e. $(f(x_{n}))$ is quasi-slowly (respectively, $Δ$-quasi-slowly) oscillating whenever $(x_{n})$ is. We study these kinds of continuities, and investigate relations with statistical continuity, lacunary statistical continuity, and some other types of continuities in metrizable topological groups.

math.FA↗