Huseyin Cakalli

• Professor
• Professor (Full) at Maltepe University

In this paper, we introduce the concept of neutrosophic $G$-sequential continuity as a new tool to further studies presenting the definitions of neutrosophic soft sequence, neutrosophic soft quasi-coincidence, neutrosophic soft $q$-neighborhood, neutrosophic soft cluster point, neutrosophic soft boundary point, neutrosophic soft sequential closure,...

In this study, we introduce and examine the concepts of Δm−weighted statistical convergence and Δm−weighted (N¯,pn)−summability. Also some relations between Δm−weighted statistical convergence and Δm−weighted (N¯,pn)−summability are given.

In this paper, we introduce the concept of strong ρ−convergence of order β ( or Npβ (ρ) −convergence ) of sequence of real numbers and give some inclusion relations between the set of all ρ−statistical convergence of order β and strong Npβ (ρ)− convergence.

In this extended abstract, we introduce the concept of delta quasi Cauchy sequences in metric spaces. A function f defined on a subset of a metric space X to X is called delta ward continuous if it preserves delta quasi Cauchy sequences, where a sequence (xk) of points in X is called delta quasi Cauchy if limn→∞[d(xk+2,xk+1)−d(xk+1,xk)]=0. A new ty...

In this paper we study the domain of generalized Riesz difference matrix RqΔ(α) of fractional order α in the classical sequence spaces c0 and c and introduced the sequence spaces r0q(Δ(α)) and rcq(Δ(α)). We obtain the α−, β− and γ−duals of these spaces and using Hausdorff measure of noncompactness, we characterize certain classes of compact operato...

In this paper, we investigate the concept of Abel statistical ward continuity in 2-normed spaces. A function f defined on a 2-normed space X into X is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (xk) of points in X is called Abel statistically quasi Cauchy if limx→1−(1−x)∑k:||Δ...

Scientists have always adopted the concept of sequential continuity as an indispensable subject, not only in Topology but also in some other branches of Mathematics. Connor and Grosse-Erdmann gave this concept for real functions by using an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequences instead...

In this paper, we investigate the concept of Abel statistical delta ward compactness and Abel statistical delta ward continuity in metric spaces. A function f defined on a metric space X into X is called Abel statistically delta ward continuous it preserves Abel statistical delta quasi Cauchy sequences, where a sequence (xk) of points in X is calle...

Continuity, in particular sequential continuity, is an important subject of investigation not only in topology but also in some other branches of mathematics. Connor and Grosse‐Erdmann remodelled its definition for real functions by replacing lim with an arbitrary linear functional G defined on a linear subspace of the vector space of all real sequ...

It is well known that for a Hausdorff topological group $X$, the limits of convergent sequences in $X$ define a function denoted by $\lim$ from the set of all convergent sequences in $X$ to $X$. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing $\lim$ with an arbitrary linear functional $G$ defined on a line...

A sequence (αk) of points in ℝ, the set of real numbers, is called ρ-statistically convergent of order β to an element ℓ of ℝ if limn→∞ℓρnβ|{k≤n:|αk−ℓ|≥ε}|=0 for each ε > 0 and 0 < β ≤ 1, where ρ = (ρn) is a non-decreasing sequence of positive real numbers tending to ∞ such that lim supnρnβn<∞, Δρnβ=O(1), and Δαn = αn+1 − αn for each positive inte...

In this study, using a lacunary sequence we introduce the concepts of lacunary d−statistically convergent sequences and lacunary d−statistically bounded sequences in general metric spaces.

In this paper, the concept of deferred statistical convergence of order α is generalized to topological groups, and some inclusion relations between the set of all statistically convergent sequences of order α in topological groups and the set of all deferred statistically convergent sequences of order α in topological groups are given.

We adapt strong θ-precontinuity into fuzzy soft topology and investigate its properties. Also, the relations with the other types of continuities in fuzzy soft topological spaces are analized. Moreover, we give some new definitions.

In this extended abstract, we introduce a concept of statistically quasi-Cauchyness of a sequence in X in the sense that a sequence (xk) is statistically quasi-Cauchy in X if limn→∞1n |{k ≤ n : d(xk+1, xk) − c ∈ P}| for each c ∈P̈ where (X, d) is a cone metric space, and P̈ denotes interior of a cone P of X. It turns out that a function f from a to...

In this paper, the definitions of lacunary strong A−convergence of order (α, β) with respect to a modulus and lacunary A–statistical convergence of order (α, β) are given. We study some connections between lacunary strong A-convergence of order (α, β) with respect to a modulus and lacunary A-statistical convergence of order (α, β).

In this study, we investigate the concepts of Abel statistical convergence and Abel statistical quasi Cauchy sequences. A function f from a subset E of a metric space X into X is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchyness, where a sequence (xk) of point in E is called Abel statistically quasi Cauchy...

Çakalli extended the concept of G-sequential compactness to a fuzzy topological group and introduced the notion of G-fuzzy sequential compactness, where G is a function from a suitable subset of the set of all sequences of fuzzy points in a fuzzy first countable topological space X. The aim of this paper is to investigate whether an idea like the G...

The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of order (alpha,beta) of sequences of real numbers. Strongly ideal lacunary ward continuity of order (alpha,beta) is also investigated. Interesting results are obtained.

In this paper we introduce the concepts of Wijsman $% \left( f,I\right) -$lacunary statistical{\Large \ }convergence of order $% \alpha $ and Wijsman strongly $\left( f,I\right) -$lacunary statistical% {\Large \ }convergence of order $\alpha ,$ and investigated between their relationship.

The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of order (α, β) of sequences of real numbers. Strongly ideal lacunary ward continuity of order (α, β) is also investigated. Interesting results are obtained.

For a fixed positive i nteger p, a sequence (xn) in a metric space X is c alled p-quasi-Cauchy if (Δpxn) is a null sequence where Δpxn=d(xn+p, xn) for each positive integer n. A subset E of X is called p-ward compact if any sequence (xn) of points in E has a p-quasi-Cauchy subsequence. A subset of X is totally bounded if and only if it is p-ward co...

A sequence (αk) of points in ℝ, the set of real numbers, is called ρ-statistically p quasi Cauchy if limn→∞1ρn|{k≤n:|Δpαk|≥ε}|=0 for each ε > 0, where ρ = (ρn) is a non-decreasing sequence of positive real numbers tending to ∞ such that lim supnρnn<∞, Δρn = O(1) and Δpαk+p = αk+p – αk for each positive integer k, p is a fixed positive integer. A r...

In this paper we introduce the concepts of Wijsman (f, I) −lacunary statistical convergence of order α and Wijsman strongly (f, I) −lacunary statistical convergence of order α, and investigated between their relationship.

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 (αk) of point in ℝ is called Abel statistically delta quasi Cauchy if limx→1−(1−x)∑k:|Δ2αk|≥εxk=0 for every ε >...

In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $(\alpha_{k})$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called statistically $p$-ward co...

In this paper, we investigate the concept of Abel statistical quasi Cauchy sequences. A real function f is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (?k) of point in R is called Abel statistically quasi Cauchy if limx?1-(1-x) ?k:|??k|?? xk = 0 for every ? > 0, where ??k = ?k+...

A sequence (αk) of real numbers is called λ-statistically upward quasi-Cauchy if for every ε > 0 limn→∞1λn|{k∈In:αk−αk+1≥ε}|=0, where (λn) is a non-decreasing sequence of positive numbers tending to ∞ such that λn+1 ≤ λn+1, λ1=1, and In=[n − λn+1, n] for any positive integer n. A real valued function f defined on a subset of R, the set of real numb...

In this paper, the concept of a strongly lacunary δ² quasi-Cauchy sequence is introduced. We proved interesting theorems related to strongly lacunary δ²-quasi-Cauchy sequences. A real valued function f defined on a subset A of the set of real numbers, is strongly lacunary δ² ward continuous on A if it preserves strongly lacunary δ² quasi-Cauchy seq...

In this paper, we give a generalization of absolutely almost convergence, and prove interesting results.

The main purpose of this paper is to introduce the concept of strongly ideal lacunary quasi-Cauchyness of sequences of real numbers. Strongly ideal lacunary ward continuity is also investigated. Interesting results are obtained.

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 $(f(\alpha_{n}))$ is downward half Cauchy whenever $(\alpha_{n})$ is a downward half Cauchy...

In this paper, we investigate the concepts of downward continuity and upward continuity. A real valued function on a subset E of ℝ, the set of real numbers, is downward continuous if it preserves downward quasi-Cauchy sequences; and is upward continuous if it preserves upward quasi-Cauchy se- quences, where a sequence (xk) of points in R is called...

It is a well known fact that for a Hausdorff topological group X, the limits of convergent sequences in X define a function denoted by lim from the set of all convergent sequences in X to X. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing lim with an arbitrary linear functional G defined on a linear subspa...

A sequence (α k ) of real numbers is called λ-statistically upward quasi-Cauchy if for every (forumala presented)., where (λ n ) is a non-decreasing sequence of positive numbers tending to ∞ such ⁿ that λ n+1 ≤ λ n + 1, λ 1 = 1, and I n = [n − λ n + 1, n] for any positive integer n. A real valued function f defined on a subset of R, the set of real...

A real valued function defined on a subset $E$ of $\mathbb{R}$, the set of real numbers, is $\rho$-statistically downward continuous if it preserves $\rho$-statistical downward quasi-Cauchy sequences of points in $E$, where a sequence $(\alpha_{k})$ of real numbers is called ${\rho}$-statistically downward quasi-Cauchy if $\lim_{n\rightarrow\infty}...

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 conver...

In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real sequence in the sense that a sequence $(\alpha_{k})$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called statistically $p$-ward co...

In this paper, the concept of an $N_{\theta}^{2}$ quasi-Cauchy sequence is introduced. We proved interesting theorems related to $N_{\theta}^{2}$-quasi-Cauchy sequences. A real valued function $f$ defined on a subset $A$ of $\mathbb{R}$, the set of real numbers, is $N_{\theta}^{2}$ ward continuous on $A$ if it preserves $N_{\theta}^{2}$ quasi-Cauch...

A sequence (xk) of points in R, the set of real numbers, is called arithmetically convergent if for each ϵ > 0 there is an integer n such that for every integer m we have /xm - x<m,n>/ < ϵ, where k/n means that k divides n or n is a multiple of k, and the symbol < m, n > denotes the greatest common divisor of the integers m and n. We prove that a s...

The first sentence in the abstract should be replaced with the sentence "A sequence $(x_{k})$ is called arithmetically convergent if for each $\varepsilon > 0$ there is an integer $n_{0}$ such that $|x_{m} - x_{<m,n>}|<\varepsilon$ for every integers $m, n$ satisfying $<m, n> \geq n_{0}$, where the symbol $< m, n >$ denotes the greatest common divi...

In this paper, we introduce a concept of a soft matrix on a soft multiset, and investigate how to use soft matrices to solve decision making problems. An algorithm for a multiple choose selection problem is also provided. Finally, we demonstrate an illustrative example to show the decision making steps.

It is a well-known fact that for a Hausdorff topological group X, the limits of convergent sequences in X define a function denoted by lim from the set of all convergent sequences in X to X. This notion has been modified by Connor and Grosse-Erdmann for real functions by replacing lim with an arbitrary linear functional G defined on a linear subspa...

In the present paper, we introduce a concept of ideal lacunary statistical quasi-Cauchy sequence of order α of real numbers in the sense that a sequence (x k ) of points in R is called I−lacunary statistically quasi-Cauchy of order α, if for each ε > 0 and for each δ > 0, where an ideal I is a family of subsets of positive integers N which is close...

We introduce a new function space, namely the space of N θ (p)-ward continuous functions, which turns out to be a closed subspace of the space of continuous functions for each positive integer p. -ward continuity is also introduced and investigated for any fixed 0 < α ≤ 1, and for any fixed positive integer p. A real valued function f defined on a...

In this paper, we investigate the concept of upward continuity. A real valued function on a subset E of R, the set of real numbers is upward continuous if it preserves upward quasi Cauchy sequences in E, where a sequence (x k ) of points in R is called upward quasi Cauchy if for every ε > 0 there exists a positive integer n 0 such that x n − x n+1...

An ideal I is a family of subsets of N, the set of positive integers which is closed under taking finite unions and subsets of its elements. A sequence (x k ) of real numbers is said to be S(I)-statistically convergent to a real number L, if for each ε > 0 and for each δ > 0 the set belongs to I. We introduce S(I)-statistically ward compactness of...

We introduce a new function space, namely the space of Nαθ(p)-ward continuous functions, which turns out to be a closed subspace of the space of continuous functions. A real valued function f defined on a subset A of ℝ, the set of real numbers, is Nαθ(p)-ward continuous if it preserves Nαθ(p)-quasi-Cauchy sequences, that is, (f(xn)) is an Nαθ(p)-qu...

A sequence (xn) of points in a topological vector space valued cone metric space (X;ρ) is called p-quasi-Cauchy if for each c Є K there exists an n0 Є ℕ such that ρ(xn+p, xn) - c Є K for n ≥ n0, where K is a proper, closed and convex pointed cone in a topological vector space γ with K ≠ θ. We investigate p-ward continuity in topological vector spac...

A sequence (xn) of points in a 2-normed space X is statistically quasi-Cauchy if the sequence of difference between successive terms statistically converges to 0. In this paper we mainly study statistical ward continuity, where a function f defined on a subset E of X is statistically ward continuous if it preserves statistically quasi-Cauchy sequen...

A function f defined on a subset A of a cone normed space X is strongly lacunary ward continuous if it preserves strongly lacunary quasi-Cauchy sequences of points in A; that is, (f (xk)) is a strongly lacunary quasi-Cauchy sequence whenever (xk) is strongly lacunary quasi-Cauchy. In this paper, not only strongly lacunary ward continuity, but also...

In this paper, we introduce lacunary statistical ward continuity in a 2-normed space. A function f defined on a subset E of a 2-normed space X is lacunary statistically ward continuous if it preserves lacunary statistically quasi-Cauchy sequences of points in E where a sequence (xk) of points in X is lacunary statistically quasi-Cauchy if limr?1 1/...

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 \(\va...

The main object of this paper is to investigate lacunary statistically ward continuity. We obtain some relations between this kind of continuity and some other kinds of continuities. It turns out that any lacunary statistically ward continuous real valued function on a lacunary statistically ward compact subset E ⊂ R is uniformly continuous.

In this paper, the concepts of a lacunary statistically δ-quasi-Cauchy sequence and a strongly lacunary δ-quasi-Cauchy sequence are introduced, and investigated. In this investigation, we proved interesting theorems related to some newly defined continuities here, mainly, lacunary statistically δ-ward continuity, and strongly lacunary δ-ward contin...

A real valued function defined on a subset E of R, the set of real numbers, is lacunary statistically upward continuous if it preserves lacunary statistically upward half quasi-Cauchy sequences where a sequence (xn) of points in R is called lacunary statistically upward half quasi-Cauchy if limr→∞1hr|{k∈Ir:xk−xk+1≥ε}|=0 for every ε > 0, and θ = (k...

In this paper, we introduce and investigate ideal strong lacunary ward continuity in 2-normed spaces. A function f on a subset A of a 2-normed space X into X is ideal strongly lacunary ward continuous if it preserves ideal strong lacunary quasi-Cauchy sequences of points in A. We also studied some other kinds of continuities.

In this paper, we introduce and investigate the concept of Abel ward continuity. A real function f is Abel ward continuous if it preserves Abel quasi Cauchy sequences, where a sequence (p k ) of point in R is called Abel quasi-Cauchy if the series Σ k = 0 ∞ Δ p k ⋅ x k is convergent for 0 ≤ x < 1 and lim x → 1 − ( 1 − x ) Σ k = 0 ∞...

A real valued function f defined on a subset of R, the set of real numbers is called Abel statistically ward continuous if it preserves Abel statistical quasi Cauchy sequences, where a sequence (α k ) of point in R is called Abel statistically quasi-Cauchy if Abel density of the set {k ∈ N : |Δα k | ≥ ε} is 0 for every ε > 0. In this paper, we gi...

In this paper, we introduce and investigate the concept of ward continuity 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 (xn) of points in X is called quasi-Cauchy if (Formula presented) for every z ∈ X. Some other kinds of continuities are also...

The main object of this paper is to investigate λ-statistically ward continuity. We obtain some relations between this kind of continuity and some other kinds of continuities. It turns out that any λ-statistically ward continuous real valued function on a λ-statistically ward compact subset E ⊂ R is uniformly continuous.

A double sequence {xk,l} is quasi-Cauchy if given an Ɛ > 0 there exists an N ∈ N such that We study continuity type properties of factorable double functions defined on a double subset A x A of R2 into R, and obtain interesting results related to uniform continuity, sequential continuity, continuity, and a newly introduced type of continuity of fa...

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...

A function defined on a subset of a 2-normed space is strongly lacunary ward continuous if it preserves strongly lacunary quasi-Cauchy sequences of points in ; that is, is a strongly lacunary quasi-Cauchy sequence whenever () is strongly lacunary quasi-Cauchy. In this paper, not only strongly lacunary ward continuity, but also some other kinds of c...

We investigate the concept of Abel continuity. A function f defined on a subset of ℝ, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences. Some other types of continuities are also studied and interesting result is obtained. It turned out that uniform limit of a sequence of Abel continuous functions is Abel continu...

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 $(\Delta x_{n})$ is a statistically null sequence. A subset $E$ of $X$ is statistically ward compact if any sequence of poin...

In this paper, we introduce a concept of a soft matrix on a soft multiset, and investigate how to use soft matrices to solve decision making problems. An algorithm for a multiple choose selection problem is also provided. Finally, we demonstrate an illustrative example to show the decision making steps.

In this paper, we study (Pn, s)-absolutely almost convergence where npn = 0(Pn) as n → ∞,Pn = Σnv=0 pv and s > 0. We also give necessary and sufficient conditions for a summability matrix to be (λn; s)-absolutely almost conservative, and obtain some inclusion results.

A double sequence $\textbf{x}=\{x_{k,l}\}$ of points in $\textbf{R}$ is slowly oscillating if for any given $\varepsilon>0$, there exist $\alpha=\alpha(\varepsilon)>0$, $\delta=\delta (\varepsilon) >0$, and $N=N(\varepsilon)$ such that $|x_{k,l}-x_{s,t}|<\varepsilon$ whenever $k,l\geq N(\varepsilon)$ and $k\leq s \leq (1+\alpha)k$, $l\leq t \leq (1...

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 empty set. In this paper we investigate further properti...

A double sequence $\{x_{k,l}\}$ is quasi-Cauchy if given an $\epsilon > 0$ there exists an $N \in {\bf N}$ such that $$\max_{r,s= 1\mbox{ and/or} 0} \left \{|x_{k,l} - x_{k+r,l+s}|< \epsilon\right \} .$$ 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...

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...

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 $lim_{n\rightarrow\infty}||\Delta x_{n},z||=0$ for every $z\in X$. Some other kinds of continuties are also introduced via quasi-Cauchy sequences in 2-normed spaces....

In this paper, we introduce soft continuous mappings which are defined over an initial universe set with a fixed set of parameters. Later we study soft open and soft closed mappings, soft homeomorphism and investigate some properties of these concepts.

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.

The concept of I-convergence is an important generalization of statistical convergence which depends on the notion of an ideal I of subsets of the set N of positive integers. In this paper we introduce the ideas of I-Cauchy and I * -Cauchy sequences in cone metric spaces and study their properties. We also investigate the relation between this new...

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

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.

Recently, the concept of -ward continuity was introduced and studied. In this paper, we prove that the uniform limit of -ward continuous functions is -ward continuous, and the set of all -ward continuous functions is a closed subset of the set of all continuous functions. We also obtain that a real function defined on an interval is uniformly conti...

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 R. A characterization of uniform continuity is also given via ideal quasi-Cauchy sequences.

A function f is continuous if and only if f preserves convergent sequences; that is, ( f ( α n ) ) is a convergent sequence whenever ( α n ) is convergent. The concept of N θ -ward continuity is defined in the sense that a function f is N θ -ward continuous if it preserves N θ -quasi-Cauchy sequences; that is, ( f ( α n ) ) is an N θ -quasi-Cauchy...

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...

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,...

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 compactnes...

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. Recently this has been applied by Du (2010) [14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in genera...

We investigate the impact of changing the definition of convergence of sequences on the structure of the set of connected subsets of a topological group, XX. A non-empty subset AA of XX is called GG-sequentially connected if there are no non-empty and disjoint GG-sequentially closed subsets UU and VV, both meeting A, such that A⊆U⋃VA⊆U⋃V. Sequentia...

Recently, Cakalli 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 there are no non-empty, disjoint $G$-sequentially closed subsets $U$ and $V$ meeting $A$ such that $A\subseteq U\bigcup V$. In this paper we investigate further pro...

A sequence (xn) of points in a topological group is called Δ-quasi-slowly oscillating if (Δxn) is quasi-slowly oscillating, and is called quasi-slowly oscillating if (Δxn) 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...

Recently, it has been proved that a real-valued function defined on an interval AA of R, the set of real numbers, is uniformly continuous on AA if and only if it is defined on AA and preserves quasi-Cauchy sequences of points in AA. In this paper we call a real-valued function statistically ward continuous if it preserves statistical quasi-Cauchy s...

A subset E of a metric space (X,d) is totally bounded if and only if any sequence of points in E has a Cauchy subsequence. We call a sequence (x"n) statistically quasi-Cauchy if st-lim"n"->"~d(x"n"+"1,x"n)=0, and lacunary statistically quasi-Cauchy if S"@q-lim"n"->"~d(x"n"+"1,x"n)=0. We prove that a subset E of a metric space is totally bounded if...

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...

In this paper we call a real-valued function $N_{\theta}$-ward continuous if it preserves $N_{\theta}$-quasi-Cauchy sequences where a sequence $\boldsymbol{\alpha}=(\alpha_{k})$ is defined to be $N_{\theta}$-quasi-Cauchy when the sequence $\Delta \boldsymbol{\alpha}$ is in $N^{0}_{\theta}$. We prove not only inclusion and compactness type theorems,...

The main purpose of the paper is to introduce the notion of summability in abstract Hausdorff topological spaces. We give a characterization of such summability methods when the space allows a countable base. We also provide several Tauberian theorems in topological structures. Some open problems are discussed.

A sequence $(x_{n})$ of points in a topological group is called $\Delta$-quasi-slowly oscillating if $(\Delta x_{n})$ is quasi-slowly oscillating, and is called quasi-slowly oscillating if $(\Delta x_{n})$ is slowly oscillating. A function $f$ defined on a subset of a topological group is quasi-slowly (respectively, $\Delta$-quasi-slowly) oscillati...