Ward Continuous Research Articles

A variation on strongly lacunary delta ward continuity in 2-normed spaces

A sequence $(x_{k})$ of points in a subset E of a 2-normed space $X$ is called strongly lacunary $\delta$-quasi-Cauchy, or $N_\theta$-$\delta$-quasi-Cauchy if $(\Delta x_k)$ is $N_\theta$-convergent to 0, that is $\lim_{r\rightarrow\infty}\frac{1}{h_r}\sum_{k\in I_r}||\Delta^2 x_k, z||=0$ for every fixed $z\in X$. A function defined on a subset $E$ of $X$ is called strongly lacunary $\delta$-ward continuous if it preserves $N_{\theta}$-$\delta$-quasi-Cauchy sequences, i.e. $(f(x_{k}))$ is an $N_{\theta}$-$\delta$-quasi-Cauchy sequence whenever $(x_{k})$ is. In this study we obtain some theorems related to strongly lacunary $\delta$-quasi-Cauchy sequences.

İstatistiksel boşluklu delta 2 quasi Cauchy dizileri

The notion of a lacunary statistical d 2 -quasi-Cauchyness of sequence of real numbers is introduce and investigated. In this work, we present interesting theorems related to lacunary statistically d 2 -ward continuity. A function f, whose domain is included in R, and whose range included in R is called lacunary statistical d 2 ward continuous if it preserves lacunary statistical d 2 quasi-Cauchy sequences, i.e. (f(x k )) is a lacunary statistically d 2 quasi-Cauchy sequence whenever (x k ) is a lacunary statistically d 2 quasi-Cauchy sequence, where a sequence (x k ) is called lacunary statistically d 2 quasi-Cauchy if ( D 2 x k ) is a lacunary statistically quasi-Cauchy sequence. We find out that the set of lacunary statistical d 2 ward continuous functions is closed as a subset of the set of continuous functions.

A Variation on Statistical Ward Continuity

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.

ϕ -statistically quasi Cauchy sequences

Abstract Let P denote the space whose elements are finite sets of distinct positive integers. Given any element σ of P, we denote by p(σ) the sequence {pn(σ)} such that p n ( σ ) = 1 for n ∈ σ and p n ( σ ) = 0 otherwise. Further P s = { σ ∈ P : ∑ n = 1 ∞ p n ( σ ) ≤ s } , i.e. Ps is the set of those σ whose support has cardinality at most s. Let (ϕn) be a non-decreasing sequence of positive integers such that n ϕ n + 1 ≤ ( n + 1 ) ϕ n for all n ∈ N and the class of all sequences (ϕn) is denoted by Φ. Let E ⊆ N . The number δ ϕ ( E ) = lim s → ∞ 1 ϕ s | { k ∈ σ , σ ∈ P s : k ∈ E } | is said to be the ϕ-density of E. A sequence (xn) of points in R is ϕ-statistically convergent (or Sϕ-convergent) to a real number l for every e > 0 if the set { n ∈ N : | x n − l | ≥ ɛ } has ϕ-density zero. We introduce ϕ-statistically ward continuity of a real function. A real function is ϕ-statistically ward continuous if it preserves ϕ-statistically quasi Cauchy sequences where a sequence (xn) is called to be ϕ-statistically quasi Cauchy (or Sϕ-quasi Cauchy) when ( Δ x n ) = ( x n + 1 − x n ) is ϕ-statistically convergent to 0. i.e. a sequence (xn) of points in R is called ϕ-statistically quasi Cauchy (or Sϕ-quasi Cauchy) for every e > 0 if { n ∈ N : | x n + 1 − x n | ≥ ɛ } has ϕ-density zero. Also we introduce the concept of ϕ-statistically ward compactness and obtain results related to ϕ-statistically ward continuity, ϕ-statistically ward compactness, statistically ward continuity, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary continuity, δ-ward continuity, and slowly oscillating continuity.

λ-Statistically Ward Continuity

Abstract 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.

Lacunary ward continuity in 2-normed spaces

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/hr |{k?Ir : ||xk+1 - xk, z||? ?}| = 0 for every positive real number ? and z ? X, and (kr) is an increasing sequence of positive integers such that k0 = 0 and hr = kr - kr-1 ? ? as r ? ?, Ir = (kr-1, kr]. We investigate not only lacunary statistical ward continuity, but also some other kinds of continuities in 2-normed spaces.

Strongly lacunary ward continuity in 2-normed spaces.

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

A Study on -Quasi-Cauchy Sequences

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 continuous if and only if (()) is -quasi-Cauchy whenever () is a quasi-Cauchy sequence of points in .

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

Statistical ward continuity

Recently, it has been proved that a real-valued function defined on an interval A of R, the set of real numbers, is uniformly continuous on A if and only if it is defined on A and preserves quasi-Cauchy sequences of points in A . In this paper we call a real-valued function statistically ward continuous if it preserves statistical quasi-Cauchy sequences where a sequence ( α k ) is defined to be statistically quasi-Cauchy if the sequence ( Δ α k ) is statistically convergent to 0. It turns out that any statistically ward continuous function on a statistically ward compact subset A of R is uniformly continuous on A . We prove theorems related to statistical ward compactness, statistical compactness, continuity, statistical continuity, ward continuity, and uniform continuity.

Formula omitted]-quasi-Cauchy sequences

Recently, it has been proved that a real-valued function defined on a subset E of R , the set of real numbers, is uniformly continuous on E if and only if it is defined on E and preserves quasi-Cauchy sequences of points in E where a sequence is called quasi-Cauchy if ( Δ x n ) is a null sequence. In this paper we call a real-valued function defined on a subset E of R δ -ward continuous if it preserves δ -quasi-Cauchy sequences where a sequence x = ( x n ) is defined to be δ -quasi-Cauchy if the sequence ( Δ x n ) is quasi-Cauchy. It turns out that δ -ward continuity implies uniform continuity, but there are uniformly continuous functions which are not δ -ward continuous. A new type of compactness in terms of δ -quasi-Cauchy sequences, namely δ -ward compactness is also introduced, and some theorems related to δ -ward continuity and δ -ward compactness are obtained.