scholarly journals Completeness in Quasi-Pseudometric Spaces—A Survey

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1279
Author(s):  
Ştefan Cobzas
Keyword(s):  
Metric Space ◽  
London Math ◽  
Metric Spaces ◽  
Sequential Completeness ◽  
Definition Of ◽  
Directed Sets

The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included.

scholarly journals Fixed Point of Interpolative Rus–Reich–Ćirić Contraction Mapping on Rectangular Quasi-Partial b-Metric Space

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 32
Author(s):  
Pragati Gautam ◽  
Luis Manuel Sánchez Ruiz ◽  
Swapnil Verma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Real Number ◽  
Metric Space ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Rectangular Metric Space ◽  
New Type ◽  
Symmetry Requirement ◽  
Definition Of

The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the rectangular metric space defined by Branciari. Here, we obtain a fixed point theorem for interpolative Rus–Reich–Ćirić contraction mappings in the realm of rectangular quasi-partial b-metric spaces. Furthermore, an example is also illustrated to present the applicability of our result.

scholarly journals Fixed Points for a Pair of F-Dominated Contractive Mappings in Rectangular b-Metric Spaces with Graph

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 884 ◽  
Author(s):  
Tahair Rasham ◽  
Giuseppe Marino ◽  
Abdullah Shoaib
Keyword(s):  
Fixed Point ◽  
Metric Space ◽  
Metric Spaces ◽  
Partial Metric Space ◽  
Contractive Condition ◽  
Partial Metric ◽  
Contractive Mappings ◽  
Dislocated Metric Space ◽  
Definition Of ◽  
Dislocated Metric

Recently, George et al. (in Georgea, R.; Radenovicb, S.; Reshmac, K.P.; Shuklad, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 2015, 8, 1005–1013) furnished the notion of rectangular b-metric pace (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space. In this paper, we achieved fixed-point results for a pair of F-dominated mappings fulfilling a generalized rational F-dominated contractive condition in the better framework of complete rectangular b-metric spaces complete rectangular b-metric spaces. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. Some examples are given to illustrate our conclusions. New results in ordered spaces, partial b-metric space, dislocated metric space, dislocated b-metric space, partial metric space, b-metric space, rectangular metric spaces, and metric space can be obtained as corollaries of our results.

scholarly journals Approximations of Variational Problems in Terms of Variational Convergence

2017 ◽  
Vol 20 (K2) ◽  
pp. 107-116
Author(s):  
Diem Thi Hong Huynh
Keyword(s):  
Multiobjective Optimization ◽  
Metric Space ◽  
Metric Spaces ◽  
Equilibrium Problems ◽  
Variational Problems ◽  
Dimensional Case ◽  
Variational Convergence ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Definition Of

We show first the definition of variational convergence of unifunctions and their basic variational properties. In the next section, we extend this variational convergence definition in case the functions which are defined on product two sets (bifunctions or bicomponent functions). We present the definition of variational convergence of bifunctions, icluding epi/hypo convergence, minsuplop convergnece and maxinf-lop convergence, defined on metric spaces. Its variational properties are also considered. In this paper, we concern on the properties of epi/hypo convergence to apply these results on optimization proplems in two last sections. Next we move on to the main results that are approximations of typical and important optimization related problems on metric space in terms of the types of variational convergence are equilibrium problems, and multiobjective optimization. When we applied to the finite dimensional case, some of our results improve known one.

scholarly journals From Euclidean Spaces to Metric Spaces

2019 ◽  
Vol 8 (1) ◽  
Author(s):  
Ryan Joseph Rogers ◽  
Ning Zhong
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Definition Of

In this note, we provide the definition of a metric space and establish that, while all Euclidean spaces are metric spaces, not all metric spaces are Euclidean spaces. It is then natural and interesting to ask which theorems that hold in Euclidean spaces can be extended to general metric spaces and which ones cannot be extended. We survey this topic by considering six well-known theorems which hold in Euclidean spaces and rigorously exploring their validities in general metric spaces.

scholarly journals SPACES OF SMALL METRIC COTYPE

2010 ◽  
Vol 02 (04) ◽  
pp. 581-597 ◽  
Author(s):  
E. VEOMETT ◽  
K. WILDRICK
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Ultrametric Space ◽  
Partial Converse ◽  
Definition Of ◽  
Hausdorff Limits

Mendel and Naor's definition of metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz is equivalent to an ultrametric space having infimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov–Hausdorff limits, and use these facts to establish a partial converse of the main result.

scholarly journals Compactness Property of Fuzzy Soft Metric Space and Fuzzy Soft Continuous Function

2021 ◽  
pp. 3031-3038
Author(s):  
Raghad I. Sabri
Keyword(s):  
Functional Analysis ◽  
Continuous Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Compactness Property ◽  
Locally Compact ◽  
Fuzzy Metric Spaces ◽  
Fundamental Properties ◽  
Definition Of

      The theories of metric spaces and fuzzy metric spaces are crucial topics in mathematics.    Compactness is one of the most important and fundamental properties that have been widely used in Functional Analysis. In this paper, the definition of compact fuzzy soft metric space is introduced and some of its important theorems are investigated. Also, sequentially compact fuzzy soft metric space and locally compact fuzzy soft metric space are defined and the relationships between them are studied. Moreover, the relationships between each of the previous two concepts and several other known concepts are investigated separately. Besides, the compact fuzzy soft continuous functions are studied and some essential theorems are proved.

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.

Metric Spaces

2021 ◽  
pp. 103-125
Author(s):  
James Davidson
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Distance Measure ◽  
Function Spaces ◽  
Important Case ◽  
General Context ◽  
Compact Sets ◽  
Definition Of ◽  
Metric Distance

This chapter introduces and illustrates the concept of a metric (distance measure), and the definition of a metric space. Open, closed, and compact sets are discussed in a general context, and the concepts of separability and completeness introduced. It goes on to look at mappings on metric spaces, examines the important case of function spaces, and treats the Arzelà–Ascoli theorem.

scholarly journals Fixed Points of Eventually Δ-Restrictive and Δ(ϵ)-Restrictive Set-Valued Maps in Metric Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 127 ◽  
Author(s):  
Pradip Debnath ◽  
Manuel de La Sen
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Intrinsic Property ◽  
Point Theory ◽  
Symmetry Principles ◽  
Definition Of ◽  
Metric Function

The symmetry concept is an intrinsic property of metric spaces as the metric function generalizes the notion of distance between two points. There are several remarkable results in science in connection with symmetry principles that can be proved using fixed point arguments. Therefore, fixed point theory and symmetry principles bear significant correlation between them. In this paper, we introduce the new definition of the eventually Δ -restrictive set-valued map together with the concept of p-orbital continuity. Further, we introduce another new concept called the Δ ( ϵ ) -restrictive set-valued map. We establish several fixed point results related to these maps and proofs of these results also provide us with schemes to find a fixed point. In a couple of results, the stronger condition of compactness of the underlying metric space is assumed. Some results are illustrated with examples.

scholarly journals THE PROPERTIES OF OCCASIONALLY WEAKLY COMPATIBLE MAPPING ON FUZZY METRIC SPACES

EDUPEDIA ◽  
2018 ◽  
Vol 2 (1) ◽  
pp. 33
Author(s):  
Citra Rizki ◽  
Sumaji .
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fuzzy Metric Space ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Weakly Compatible ◽  
Compatible Mapping ◽  
Definition Of

In this paper, describe about the properties of occasionally weakly compatible mapping on fuzzy metric spaces in terms of fixed point theory. The discussion of this research is strarted from the concept of fuzzy set and metric space and they were expanded into the concept of fuzzy metric space using continuous norm-t. Furthermore, in investigating about occasionally weakly compatible mapping on fuzzy metric space, we start by given the definition of compatible mapping, weakly commuting, and weakly compatible. Moreover, we construct some theorems to investigating the properties of occasionally weakly compatible mapping on fuzzy metric spaces. 

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