On statistical convergence of sequences of closed sets in metric spaces

2021 ◽  
Vol 71 (2) ◽  
pp. 409-422
Author(s):  
Dimitrios Georgiou ◽  
Athanasios Megaritis ◽  
Georgios Prinos ◽  
Fotini Sereti
Keyword(s):  
Statistical Convergence ◽  
Metric Spaces ◽  
Closed Sets ◽  
Convergence Of Sequences ◽  
The Relationship

Abstract In this paper, we do further investigations on the statistical inner and outer limits of sequences of closed sets in metric spaces, which were introduced by Nuray, Rhoades, and Talo, Sever, Başar, and generalize the conventional Painleve-Kuratowski inner and outer limits. Also, we provide criteria for checking statistical Wijsman and Hausdorff set convergences and we examine the relationship between Kuratowski and Wijsman statistical convergence. A closer look on the concept of statistical Cauchyness, with respect to the Hausdorff “extended” metric h, completes this research.

scholarly journals On Kuratowski I-convergence of sequences of closed sets

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 899-912
Author(s):  
Özer Talo ◽  
Yurdal Sever
Keyword(s):  
Statistical Convergence ◽  
Metric Spaces ◽  
Closed Sets ◽  
Convergence Of Sequences ◽  
Positive Integers ◽  
The Relationship ◽  
Convergent Sequences

In this paper we extend the concepts of statistical inner and outer limits (as introduced by Talo, Sever and Ba?ar) to I-inner and I-outer limits and give some I-analogue of properties of statistical inner and outer limits for sequences of closed sets in metric spaces, where I is an ideal of subsets of the set N of positive integers. We extend the concept of Kuratowski statistical convergence to Kuratowski I-convergence for a sequence of closed sets and get some properties for Kuratowski I-convergent sequences. Also, we examine the relationship between Kuratowski I-convergence and Hausdorff I-convergence.

scholarly journals On statistically convergent sequences of closed sets

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1497-1509 ◽  
Author(s):  
Özer Talo ◽  
Yurdal Sever ◽  
Feyzi Başarc
Keyword(s):  
Statistical Convergence ◽  
Metric Spaces ◽  
Closed Sets ◽  
The Relationship ◽  
Convergent Sequences

In this paper, we give the definitions of statistical inner and outer limits for sequences of closed sets in metric spaces. We investigate some properties of statistical inner and outer limits. For sequences of closed sets if its statistical outer and statistical inner limits coincide, we say that the sequence is Kuratowski statistically convergent. We prove some proporties for Kuratowski statistically convergent sequences. Also, we examine the relationship between Kuratowski statistical convergence and Hausdorff statistical convergence.

scholarly journals Statistical Convergence in Function Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Agata Caserta ◽  
Giuseppe Di Maio ◽  
Ljubiša D. R. Kočinac
Keyword(s):  
Uniform Convergence ◽  
Statistical Convergence ◽  
Statistical Approach ◽  
Metric Spaces ◽  
Function Spaces ◽  
Strong Uniform Convergence ◽  
Convergence Of Sequences

We study statistical versions of several classical kinds of convergence of sequences of functions between metric spaces (Dini, Arzelà, and Alexandroff) in different function spaces. Also, we discuss a statistical approach to recently introduced notions of strong uniform convergence and exhaustiveness.

scholarly journals I-lacunary statistical convergence of sequences of sets

Filomat ◽  
2014 ◽  
Vol 28 (8) ◽  
pp. 1567-1574 ◽  
Author(s):  
Uğur Ulusu ◽  
Erdinç Dündar
Keyword(s):  
Statistical Convergence ◽  
Lacunary Statistical Convergence ◽  
Convergence Of Sequences ◽  
The Relationship

In this paper we study the concepts of Wijsman I-statistical convergence, Wijsman I-lacunary statistical convergence and Wijsman strongly I-lacunary convergence of sequences of sets and investigate the relationship between them.

The Arzelà–Ascoli theorem by means of ideal convergence

2018 ◽  
Vol 25 (3) ◽  
pp. 475-479
Author(s):  
Emre Taş ◽  
Tugba Yurdakadim
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Statistical Convergence ◽  
Real Numbers ◽  
Ideal Convergence ◽  
Convergence Of Sequences ◽  
The Relationship

AbstractIn this paper, using the concept of ideal convergence, which extends the idea of ordinary convergence and statistical convergence, we are concerned with the I-uniform convergence and the I-pointwise convergence of sequences of functions defined on a set of real numbers D. We present the Arzelà–Ascoli theorem by means of ideal convergence and also the relationship between I-equicontinuity and I-continuity for a family of functions.

Some further studies on strong ℐλ-statistical convergence in probabilistic metric spaces

Analysis ◽  
2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Argha Ghosh ◽  
Samiran Das
Keyword(s):  
Statistical Convergence ◽  
Metric Spaces ◽  
Cluster Point ◽  
Probabilistic Metric Spaces ◽  
Basic Properties ◽  
Statistical Cluster ◽  
Cauchy Sequences ◽  
Convergence Of Sequences ◽  
Statistical Cluster Point ◽  
Probabilistic Metric

Abstract We prove some basic properties of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical convergence of sequences in probabilistic metric spaces and introduce the notion of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical cluster point. We also introduce the notion of strong ℐ λ {\mathcal{I}_{\lambda}} -statistical Cauchy sequences in probabilistic metric spaces. Further, we establish a connection between strong ℐ λ {\mathcal{I}_{\lambda}} -statistical convergence and strong ℐ λ {\mathcal{I}_{\lambda}} -statistical Cauchy sequences.

scholarly journals Riesz triple probabilisitic of almost lacunary ces$\acute{A}$ro $C_{111}$ statistical convergence of $\chi^{3}$ defined by a Musielak Orlicz function

2018 ◽  
Vol 36 (4) ◽  
pp. 23-32 ◽  
Author(s):  
Dr Vandana ◽  
_ Deepmala ◽  
N. Subramanian ◽  
Vishnu Narayan Mishra
Keyword(s):  
Functional Analysis ◽  
General Framework ◽  
Statistical Convergence ◽  
Metric Spaces ◽  
Orlicz Function ◽  
The Subject

In this paper we study the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{3}$ over probabilistic $p-$ metric spaces defined by Musielak Orlicz function. Since the study of convergence in PP-spaces is fundamental to probabilistic functional analysis, we feel that the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{2}$ over probabilistic $p-$ metric spaces defined by Musielak in a PP-space would provide a more general framework for the subject.

scholarly journals On Weak Contractive Cyclic Maps in Generalized Metric Spaces and Some Related Results on Best Proximity Points and Fixed Points

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
M. De la Sen
Keyword(s):  
Fixed Points ◽  
Metric Spaces ◽  
Nonempty Intersection ◽  
Generalized Metric Spaces ◽  
Best Proximity Points ◽  
Generalized Metric ◽  
Cyclic Maps ◽  
Convergence Of Sequences ◽  
Composite Maps ◽  
Decreasing Functions

This paper discusses the properties of convergence of sequences to limit cycles defined by best proximity points of adjacent subsets for two kinds of weak contractive cyclic maps defined by composite maps built with decreasing functions with either the so-calledr-weaker Meir-Keeler orr,r0-stronger Meir-Keeler functions in generalized metric spaces. Particular results about existence and uniqueness of fixed points are obtained for the case when the sets of the cyclic disposal have a nonempty intersection. Illustrative examples are discussed.

scholarly journals Extended partial b-metric spaces and some fixed point results

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2837-2850 ◽  
Author(s):  
V. Parvaneh ◽  
Z. Kadelburg
Keyword(s):  
Fixed Point ◽  
Integral Equations ◽  
Metric Space ◽  
Metric Spaces ◽  
Volterra Type ◽  
Fundamental Lemma ◽  
Contractive Mappings ◽  
Type Integral ◽  
Convergence Of Sequences ◽  
Weakly Contractive

In this paper, we introduce the concept of extended partial b-metric space. We demonstrate a fundamental lemma for the convergence of sequences in such spaces. Then we prove some fixed point results for weakly contractive mappings in the setup of ordered extended partial b-metric spaces. An example is given to verify the effectiveness and applicability of our main results. An application of these results to Volterra-type integral equations is provided at the end.

scholarly journals On the representation of metric spaces

1979 ◽  
Vol 20 (3) ◽  
pp. 367-375 ◽  
Author(s):  
G.J. Logan
Keyword(s):  
Algebraic Structure ◽  
Topological Structure ◽  
Dual Space ◽  
Metric Spaces ◽  
Closure Operator ◽  
The Family ◽  
Necessary And Sufficient ◽  
The Relationship

A closure algebra is a set X with a closure operator C defined on it. It is possible to construct a topology τ on MX, the family of maximal, proper, closed subsets of X, and then to examine the relationship between the algebraic structure of (X, C) and the topological structure of the dual space (MX τ) This paper describes the algebraic conditions which are necessary and sufficient for the dual space to be separable metric and metric respectively.

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