Double Sequences and Selections

Djurcic, Dragan; Kočinac, Ljubiša D.R.; Žižović, Mališa

doi:10.1155/2012/497594

Cited by 32 publications

(27 citation statements)

References 16 publications

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“…However due to the change in settings, the definitions and methods of proofs will not always be analogous to those of the present work (for example see [1], [30], [41], and [47]). We also suggest to investigate statistically p-quasi-Cauchy double sequences of points in R (see [54], [53], [40], and [35] for the related definitions in the double case). For another further study, we suggest to investigate statistically p-quasi-Cauchy sequences in abstract metric spaces (see [28], [52], [36], [56], and [59]).…”

Section: Resultsmentioning

confidence: 99%

Variations on statistical quasi Cauchy sequences

Cakalli

2019

B Soc Paran Mat

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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 $A$.

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Section: Resultsmentioning

confidence: 99%

Variations on statistical quasi Cauchy sequences

Cakalli

2019

B Soc Paran Mat

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“…We suggest to investigate downward and upward quasi-Cauchy sequences of fuzzy points in asymmetric fuzzy spaces (see [16,33] for the definitions and related concepts in fuzzy setting). We also suggest to investigate downward and upward quasi-Cauchy double sequences (see for example [20,31,32,37] for the definitions and related concepts in the double sequences case). For another further study, we suggest to investigate upward quasi-Cauchy sequences of points in an asymmetric cone metric space since in a cone metric space the notion of an upward quasi-Cauchy sequence coincides with the notion of a downward quasi-Cauchy sequence, and therefore upward continuity coincides with downward continuity (see [21,23,35,41]).…”

Section: Resultsmentioning

confidence: 99%

Beyond cauchy and quasi-Cauchy sequences

Cakalli

2018

Filomat

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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 (x k) of points in R is called downward quasi-Cauchy if for every ε > 0 there exists an n 0 ∈ N such that x n+1 − x n < ε for n ≥ n 0 , and called upward quasi-Cauchy if for every ε > 0 there exists an n 1 ∈ N such that x n − x n+1 < ε for n ≥ n 1. 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.

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“…However, due to the change in settings, the definitions and methods of proofs will not always be analogous to those of the present work (see [18], and [38] for the definitions and related concepts in the fuzzy setting). For another further study, we suggest to investigate N α θ (p) quasi-Cauchy double sequence (see [22,30], and [41] for the definitions and related concepts in double case). One can introduce and give an investigation of N α θ (p) -quasi-Cauchy sequences in cone normed spaces as a further study (see [23,25,40], and [45] for basic concepts in cone setting).…”

Section: Resultsmentioning

confidence: 99%