Certain observations on statistical variations of bornological covers

We primarily make a general approach to the study of open covers and related selection principles using the idea of statistical convergence in metric space. In the process we are able to extend some results in (Caserta et al. 2012; Chandra et al. 2020) where bornological covers and related selection principles in metric spaces have been investigated using the idea of strong uniform convergence (Beer and Levi, 2009) on a bornology. We introduce the notion of statistical-Bs-cover, statistically-strong-B-Hurewicz and statistically-strong-B-groupable cover and study some of its properties mainly related to the selection principles and corresponding games. Also some properties like statistically-strictly Fr?chet Urysohn, statistically-Reznichenko property and countable fan tightness have also been investigated in C(X) with respect to the topology of strong uniform convergence ?sB.

Download Full-text

Statistical Convergence in Function Spaces

Abstract and Applied Analysis ◽

10.1155/2011/420419 ◽

2011 ◽

Vol 2011 ◽

pp. 1-11 ◽

Cited By ~ 35

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.

Download Full-text

Bornologies and bitopological function spaces

Filomat ◽

10.2298/fil1307345o ◽

2013 ◽

Vol 27 (7) ◽

pp. 1345-1349 ◽

Cited By ~ 3

Author(s):

Selma Özçağ

Keyword(s):

Uniform Convergence ◽

Metric Spaces ◽

Function Spaces ◽

Uniform Topology ◽

Selection Principles ◽

Strong Uniform Convergence

The aim of this paper is to study certain closure-type properties of function spaces over metric spaces endowed with two topologies: the topology of uniform convergence on a bornology and the topology of strong uniform convergence on a bornology. The study of function spaces with the strong uniform topology on a bornology was initiated by G. Beer and S. Levi in 2009, and then continued by several authors: A. Caserta, G. Di Maio and L'. Hol? in 2010, A. Caserta, G. Di Maio, Lj.D.R. Kocinac in 2012. Properties that we consider in this paper are defined in terms of selection principles.

Download Full-text

On statistical convergence in quasi-metric spaces

Demonstratio Mathematica ◽

10.1515/dema-2019-0019 ◽

2019 ◽

Vol 52 (1) ◽

pp. 225-236 ◽

Cited By ~ 2

Author(s):

Merve İlkhan ◽

Emrah Evren Kara

Keyword(s):

Functional Analysis ◽

Computer Science ◽

Distance Function ◽

Metric Space ◽

Statistical Convergence ◽

Triangle Inequality ◽

Metric Spaces ◽

Applied Mathematics ◽

Comprehensive Investigation ◽

Cauchy Sequences

AbstractA quasi-metric is a distance function which satisfies the triangle inequality but is not symmetric in general. Quasi-metrics are a subject of comprehensive investigation both in pure and applied mathematics in areas such as in functional analysis, topology and computer science. The main purpose of this paper is to extend the convergence and Cauchy conditions in a quasi-metric space by using the notion of asymptotic density. Furthermore, some results obtained are related to completeness, compactness and precompactness in this setting using statistically Cauchy sequences.

Download Full-text

Statistical convergence in probabilistic generalized metric spaces w.r.t. strong topology

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02669-w ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Rasoul Abazari

Keyword(s):

Metric Space ◽

Cauchy Sequence ◽

Statistical Convergence ◽

Metric Spaces ◽

Asymptotic Density ◽

Strong Topology ◽

Generalized Metric Spaces ◽

Generalized Metric ◽

Basic Facts ◽

Definition Of

AbstractIn this paper, the concept of probabilistic g-metric space with degree l, which is a generalization of probabilistic G-metric space, is introduced. Then, by endowing strong topology, the definition of l-dimensional asymptotic density of a subset A of $\mathbb{N}^{l}$ N l is used to introduce a statistically convergent and Cauchy sequence and to study some basic facts.

Download Full-text

On Some Further Generalizations of Strong Convergence in Probabilistic Metric Spaces Using Ideals

Abstract and Applied Analysis ◽

10.1155/2013/765060 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 4

Author(s):

Pratulananda Das ◽

Kaustubh Dutta ◽

Vatan Karakaya ◽

Sanjoy Ghosal

Keyword(s):

Strong Convergence ◽

Metric Space ◽

Statistical Convergence ◽

Metric Spaces ◽

Strong Topology ◽

Probabilistic Metric Space ◽

Probabilistic Metric Spaces ◽

New Approach ◽

Lacunary Statistical Convergence ◽

Probabilistic Metric

Following the line of (Das et al., 2011, Savas and Das, 2011), we make a new approach in this paper to extend the notion of strong convergence and more general strong statistical convergence (Şençimen and Pehlivan, 2008) using ideals and introduce the notion of strongℐ- andℐ*-statistical convergence and two related concepts, namely, strongℐ-lacunary statistical convergence and strongℐ-λ-statistical convergence in a probabilistic metric space endowed with strong topology. We mainly investigate their interrelationship and study some of their important properties.

Download Full-text

On the Lacunary Statistical Convergence of Triple Sequences in Fuzzy Metric Spaces

10.21203/rs.3.rs-185625/v1 ◽

2021 ◽

Author(s):

Mehmet Gürdal ◽

Ekrem Savaş

Keyword(s):

Metric Space ◽

Statistical Convergence ◽

Metric Spaces ◽

Research Paper ◽

Fuzzy Metric Space ◽

Fuzzy Metric Spaces ◽

Basic Properties ◽

Fuzzy Metric ◽

Lacunary Statistical Convergence

Abstract In this research paper, we analyze the lacunary statistical convergence and lacunary statistical Cauchy concepts of triple sequence in fuzzy metric space. We also introduce the concept of triple lacunary statistical completeness and prove some basic properties.

Download Full-text

Quantale-valued Cauchy tower spaces and completeness

Applied General Topology ◽

10.4995/agt.2021.15610 ◽

2021 ◽

Vol 22 (2) ◽

pp. 461

Author(s):

Gunther Jäger ◽

T. M. G. Ahsanullah

Keyword(s):

Uniform Convergence ◽

Metric Space ◽

Metric Spaces ◽

Probabilistic Metric Spaces ◽

Cauchy Space ◽

Cauchy Spaces ◽

Probabilistic Metric

Generalizing the concept of a probabilistic Cauchy space, we introduce quantale-valued Cauchy tower spaces. These spaces encompass quantale-valued metric spaces, quantale-valued uniform (convergence) tower spaces and quantale-valued convergence tower groups. For special choices of the quantale, classical and probabilistic metric spaces are covered and probabilistic and approach Cauchy spaces arise. We also study completeness and completion in this setting and establish a connection to the Cauchy completeness of a quantale-valued metric space.

Download Full-text

On Asymmetric Distances

Analysis and Geometry in Metric Spaces ◽

10.2478/agms-2013-0004 ◽

2013 ◽

Vol 1 ◽

pp. 200-231 ◽

Cited By ~ 6

Author(s):

Andrea C.G. Mennucci

Keyword(s):

Total Variation ◽

Calculus Of Variations ◽

Distance Function ◽

Metric Space ◽

Metric Spaces ◽

Geodesic Segment ◽

Intrinsic Metric ◽

Typical Application ◽

Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

Download Full-text

A new Ф-generalized quasi metric space with some fixed point results and applications

Filomat ◽

10.2298/fil1711157a ◽

2017 ◽

Vol 31 (11) ◽

pp. 3157-3172

Author(s):

Mujahid Abbas ◽

Bahru Leyew ◽

Safeer Khan

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Fixed Points ◽

Periodic Solutions ◽

Metric Space ◽

Delay Differential Equations ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Existence Of Periodic Solutions ◽

Delay Differential

In this paper, the concept of a new ?-generalized quasi metric space is introduced. A number of well-known quasi metric spaces are retrieved from ?-generalized quasi metric space. Some general fixed point theorems in a ?-generalized quasi metric spaces are proved, which generalize, modify and unify some existing fixed point theorems in the literature. We also give applications of our results to obtain fixed points for contraction mappings in the domain of words and to prove the existence of periodic solutions of delay differential equations.

Download Full-text

On Some New Contractive Conditions in Complete Metric Spaces

Mathematics ◽

10.3390/math9020118 ◽

2021 ◽

Vol 9 (2) ◽

pp. 118

Author(s):

Jelena Vujaković ◽

Eugen Ljajko ◽

Mirjana Pavlović ◽

Stojan Radenović

Keyword(s):

Current Literature ◽

Metric Space ◽

Metric Spaces ◽

Complete Metric Spaces ◽

Nonlinear Anal ◽

Increasing Mapping ◽

Strictly Increasing Mapping

One of the main goals of this paper is to obtain new contractive conditions using the method of a strictly increasing mapping F:(0,+∞)→(−∞,+∞). According to the recently obtained results, this was possible (Wardowski’s method) only if two more properties (F2) and (F3) were used instead of the aforementioned strictly increasing (F1). Using only the fact that the function F is strictly increasing, we came to new families of contractive conditions that have not been found in the existing literature so far. Assuming that α(u,v)=1 for every u and v from metric space Ξ, we obtain some contractive conditions that can be found in the research of Rhoades (Trans. Amer. Math. Soc. 1977, 222) and Collaco and Silva (Nonlinear Anal. TMA 1997). Results of the paper significantly improve, complement, unify, generalize and enrich several results known in the current literature. In addition, we give examples with results in line with the ones we obtained.

Download Full-text