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

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
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.

scholarly journals Strongly $\lambda$-Statistically and Strongly Vallée-Poussin Pre-Cauchy Sequences in Probabilistic Metric Spaces

2021 ◽  
Vol 53 ◽  
Author(s):  
Argha Ghosh ◽  
Samiran Das
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Strong Topology ◽  
Probabilistic Metric Space ◽  
Probabilistic Metric Spaces ◽  
Cauchy Sequences ◽  
Probabilistic Metric ◽  
Convergent Sequences

We introduce the notions of strongly $\lambda$-statistically pre-Cauchy and strongly Vall´ee-Poussin pre-Cauchy sequences in probabilistic metric spaces endowed with strong topology. And we show that these two new notions are equivalent. Strongly $\lambda$-statistically convergent sequences are strongly $\lambda$-statistically pre-Cauchy sequences, and we give an example to show that there is a sequence in a probabilistic metric space which is strongly $\lambda$-statistically pre-Cauchy but not strongly $\lambda$-statistically convergent.

scholarly journals Existence of common fixed points for linear combinations of contractive maps in enhanced probabilistic metric spaces

2019 ◽  
Vol 24 (5) ◽  
Author(s):  
Shahnaz Jafari ◽  
Maryam Shams ◽  
Asier Ibeas ◽  
Manuel De La Sen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Linear Combination ◽  
Metric Space ◽  
Metric Spaces ◽  
Probabilistic Metric Space ◽  
Probabilistic Metric Spaces ◽  
Contractive Mappings ◽  
Probabilistic Metric ◽  
Infinite Linear

In this paper, we introduce the concept of enhanced probabilistic metric space (briefly EPM-space) as a type of probabilistic metric space. Also, we investigate the existence of fixed points for a (finite or infinite) linear combination of different types of contractive mappings in EPM-spaces. Furthermore, we investigate about the convergence of sequences (generated by a finite or infinite family of contractive mappings) to a common fixed point. The useful application of this research is the study of the stability of switched dynamic systems, where we study the conditions under which the iterative sequences generated by a (finite or infinite) linear combination of mappings (contractive or not), converge to the fixed point. Also, some examples are given to support the obtained results. In the end, a number of figures give us an overview of the examples.

scholarly journals Multivalued operator with respect generalized distance on Menger probabilistic metric spaces

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1675-1682
Author(s):  
Reza Saadati
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Multivalued Operator ◽  
Probabilistic Metric Space ◽  
Probabilistic Metric Spaces ◽  
Contractive Type ◽  
Probabilistic Metric ◽  
Menger Probabilistic Metric Space ◽  
Generalized Distance

In this paper, we recall the concept of r-distance on a Menger probabilistic metric space. Further we prove a fixed point theorem for contractive type multi-valued operators in terms of a r-distance on a complete Menger probabilistic metric space.

scholarly journals P-Cyclic C-Contraction Result in Menger Spaces Using a Control Function

2016 ◽  
Vol 49 (2) ◽  
Author(s):  
B. S. Choudhury ◽  
S. K. Bhandari
Keyword(s):  
Contraction Mapping ◽  
Metric Spaces ◽  
Optimization Problems ◽  
Control Function ◽  
Probabilistic Metric Space ◽  
Probabilistic Metric Spaces ◽  
Control Functions ◽  
Menger Spaces ◽  
Cyclic Contractions ◽  
Probabilistic Metric

AbstractThe intrinsic flexibility of probabilistic metric spaces makes it possible to extend the idea of contraction mapping in several inequivalent ways, one of which being the C-contraction. Cyclic contractions are another type of contractions used extensively in global optimization problems. We introduced here p-cyclic contractions which are probabilistic C-contraction types. It involves p numbers of subsets of the spaces and involves two control functions for its definitions. We show that such contractions have fixed points in a complete probabilistic metric space. The main result is supported with an example and extends several existing results.

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

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.

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

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.

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 Quantale-valued Cauchy tower spaces and completeness

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.

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