Variations on strong lacunary quasi-Cauchy sequences

2012

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 (xn) is called p-quasi-Cauchy if (x n+p − xn) ∞ n=1 is a null sequence. Recently, in [1], a concept of quasi-Cauchy continuity, and a concept of quasi-Cauchy compactness have been introduced in the senses that a real function is called quasi-Cauchy continuous if lim n→∞ ∆f (x n) = 0 whenever lim n→∞ ∆x n = 0, and a subset E of R is called quasi-Cauchy compact if whenever (x n) is a sequence of points in E there is a subsequence (y k) = (x n k) of (x n) with lim k→∞ ∆y k = 0 where ∆y k = y k+1 − y k. We note that forward continuity and forward compactness

Mathematical and Computer Modelling, 2011

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 ( 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 ( 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.

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 conti...

Annals of the University of Craiova - Mathematics and Computer Science Series, 2018

A real function is lacunary ideal ward continuous if it preserves lacunary ideal quasi Cauchy sequences where a sequence (x_{n}) is said to be lacunary ideal quasi Cauchy (or I_{θ}-quasi Cauchy) when (Δx_{n})=(x_{n+1}-x_{n}) is lacunary ideal convergent to 0. i.e. a sequence (x_{n}) of points in R is called lacunary ideal quasi Cauchy (or I_{θ}-quasi Cauchy) for every e>0 if {r∈N:(1/(h_{r}))∑_{n∈J_{r}}|x_{n+1}-x_{n}|≥e}∈I. Also we introduce the concept of lacunary ideal ward compactness and obtain results related to lacunary ideal ward continuity, lacunary ideal ward compactness, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary continuity, δ-ward continuity, and slowly oscillating continuity. Finally we introduce the concept of ideal Cauchy continuous function in metric space and prove some results related to this notion.

Math Comput Modelling, 2011

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 (xn) is called p-quasi-Cauchy if (x n+p − xn) ∞ n=1 is a null sequence. Recently, in [1], a concept of quasi-Cauchy continuity, and a concept of quasi-Cauchy compactness have been introduced in the senses that a real function is called quasi-Cauchy continuous if lim n→∞ ∆f (x n) = 0 whenever lim n→∞ ∆x n = 0, and a subset E of R is called quasi-Cauchy compact if whenever (x n) is a sequence of points in E there is a subsequence (y k) = (x n k) of (x n) with lim k→∞ ∆y k = 0 where ∆y k = y k+1 − y k. We note that forward continuity and forward compactness

For a fixed positive i nteger p, a sequence (x n) in a metric space X is c alled p-quasi-Cauchy if (Δ p x n) is a null sequence where Δ p x n = d(x n+p , x n) for each positive integer n. A subset E of X is called p-ward compact if any sequence (x n) 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 compact. A function f from a subset E of X into a metric space Y is called p-ward continuous if it preserves p-quasi Cauchy sequences, i.e. (f (x n)) is a p-quasi Cauchy sequence in Y whenever (x n) is a p-quasi Cauchy sequence of points of E. A function f from a totally bounded subset of X into Y preserves p-quasi Cauchy sequences if and only if it is uniformly continuous. If a function is uniformly continuous on a subset E of X into Y, then (f (x n) is p-quasi Cauchy in Y whenever (x n) is a quasi cauchy sequence of points in E.

Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2017

In this paper, the concept of a lacunary statistically-quasi-Cauchy sequence is investigated. In this investigation, we proved interesting theorems related to lacunary statistically-ward continuity, and some other kinds of continuities. A real valued function f de…ned on a subset A of R, the set of real numbers, is called lacunary statistically ward continuous on A if it preserves lacunary statistically delta quasi-Cauchy sequences of points in A, i.e. (f (k)) is a lacunary statistically delta quasi-Cauchy sequence whenever (k) is a lacunary statistically delta quasi-Cauchy sequence of points in A, where a sequence (k) is called lacunary statistically delta quasi-Cauchy if (k) is a lacunary statistically quasi-Cauchy sequence. It turns out that the set of lacunary statistically ward continuous functions is a closed subset of the set of continuous functions.

2011

In this paper we call a real-valued function N θ -ward continuous if it preserves N θ -quasi-Cauchy sequences where a sequence α = (α k ) is defined to be N θ -quasi-Cauchy when the sequence ∆α is in N 0 θ . We prove not only inclusion and compactness type theorems, but also continuity type theorems.

2021

The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in [7]) that has been used to define several new concepts in recent articles [9, 10]. We first introduce a new notion of precompactness based on the idea of quasi-Cauchy sequences and establish several results including a new characterization of compactness in metric spaces. Next we consider associated idea of continuity, namely, ward continuous functions [8], as this class of functions strictly lies between the classes of continuous and uniformly continuous functions and mainly establish certain coincidence results. Finally a new class of Lipschitz functions called “quasi-Cauchy Lipschitz functions” is introduced following the line of investigations in [3, 4, 5, 12] and again several coincidence results are proved along with a very interesting observation that every real valued ward continuous function defined...

Journal of Inequalities and Applications, 2012

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. A sequence (xn) of real numbers is said to be I-convergent to a real number L, if for each ε > 0 the set {n : |xn − L| ≥ ε} belongs to I. We introduce I-ward compactness of a subset of R, the set of real numbers, and I-ward continuity of a real function in the senses that a subset E of R is I-ward compact if any sequence (xn) 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 (xn) is called to be I-quasi-Cauchy when (∆xn) is I-convergent to 0. We obtain results related to I-ward continuity, I-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, δward continuity, and slowly oscillating continuity.

arXiv: Functional Analysis, 2017

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-Cauchy sequences of points in $A$, i.e. $(f( \alpha_{k}))$ is an $N_{\theta}^{2}$ quasi-Cauchy sequence whenever $(\alpha_{k})$ is an $N_{\theta}^{2}$ quasi-Cauchy sequences of points in $A$, where a sequence $(\alpha_{k})$ is called $N_{\theta}^{2}$ quasi-Cauchy if $(\Delta^{2} \alpha_{k})$ is an $N_{\theta}$ quasi-Cauchy sequence where $\Delta^{2}\alpha_{k}=\alpha_{k+2}-2\alpha_{k+1}+\alpha_{k}$ for each positive integer $k$.

Filomat

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 sequences, where a sequence (xk) of points in R is called downward quasi-Cauchy if for every ? > 0 there exists an n0 ? N such that xn+1 - xn < ? for n ? n0, and called upward quasi-Cauchy if for every ? > 0 there exists an n1 ? N such that xn - xn+1 < ? for n ? n1. We investigate the notions of downward compactness and upward compactness and prove that downward compactness coincides with above boundedness. It turns out that not only the set of downward continuous functions, but also the set of upward continuous functions is a proper subset of the set of continuous functions.

Boletim da Sociedade Paranaense de Matemática

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 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 $f(\textbf{x})=(f(\alpha_{n}))$ is statistically $p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ 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 $A$.

2011

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 $E\subset{\textbf{R}}$ is uniformly continuous.

Cornell University - arXiv, 2010

Recall that a subset E of a metric space (X, d) is called bounded if δ(A) = sup{d(a, b) : a, b ∈ E} ≤ M where M is a positive real constant number, X is a non empty set, and d : X 2 → R satisfies (M1) d(x, y) = 0 if and only if x = y, (M2) d(x, y) = d(y, x), and (M 3) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z ∈ X. A metric space (X, d) is said to be an ε-net if X = a∈E B(a, ε). The metric space (X, d) is called totally bounded if it has a finite ε-net for each ε > 0. A subspace (E, d E) of (X, d) is said to be totally bounded if it is totally bounded as a metric space in its own right. A subset E of a metric space (X, d) is said to be totally bounded if it is totally bounded as a metric subspace. The