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Rough weighted $\mathcal{I}$-limit points and weighted $\mathcal{I}$-cluster points in $θ$-metric space

In: Mathematica Slovaca, vol. 70, no. 3
Sanjoy Ghosal - Avishek Ghosh

Details:

Year, pages: 2020, 667 - 680
Keywords:
θ-metric space, rough weighted I-convergence, rough weighted I-limit set, weighted I-boundedness, weighted I-cluster point
About article:
In 2018, Das et al. [\textit{Characterization of rough weighted statistical statistical limit set}, Math. Slovaca \textbf{68}(4) (2018), 881–896] (or, Ghosal et al. [\textit{Effects on rough $\mathcal{I}$-lacunary statistical convergence to induce the weighted sequence}, Filomat \textbf{32}(10) (2018), 3557–3568]) established the result: The diameter of rough weighted statistical limit set (or, rough weighted $\mathcal{I}$-lacunary limit set) of a sequence $x=\{xn\}_{n\in \mathbb{N}}$ is $\frac{2r}{{\liminfn\in A} tn}$ if the weighted sequence $\{tn\}_{n\in \mathbb{N}}$ is statistically bounded (or, self weighted $\mathcal{I}$-lacunary statistically bounded), where $A=\{k\in \mathbb{N}: tk< M\}$ and $M$ is a positive real number such that natural density (or, self weighted $\mathcal{I}$-lacunary density) of $A$ is $1$ respectively. Generally this set has no smaller bound other than $\frac{2r}{{\liminfn\in A} tn}.$ We concentrate on investigation that whether in a $θ$-metric space above mentioned result is satisfied for rough weighted $\mathcal{I}$-limit set or not? Answer is no. In this paper we establish infinite as well as unbounded $θ$-metric space (which has not been done so far) by utilizing some non-trivial examples. In addition we introduce and investigate some problems concerning the sets of rough weighted $\mathcal{I}$-limit points and weighted $\mathcal{I}$-cluster points in $θ$-metric space and formalize how these sets could deviate from the existing basic results.
How to cite:
ISO 690:
Ghosal, S., Ghosh, A. 2020. Rough weighted $\mathcal{I}$-limit points and weighted $\mathcal{I}$-cluster points in $θ$-metric space. In Mathematica Slovaca, vol. 70, no.3, pp. 667-680. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0380

APA:
Ghosal, S., Ghosh, A. (2020). Rough weighted $\mathcal{I}$-limit points and weighted $\mathcal{I}$-cluster points in $θ$-metric space. Mathematica Slovaca, 70(3), 667-680. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0380
About edition:
Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava
Published: 23. 5. 2020