General Metric Spaces Research Articles

The Study of Fixed Point Theorem over Modular F-metric Spaces

One of the most significant discoveries in the annals of mathematical history is Schauder's fixed-point theorem, which is generally recognized to be among the most important discoveries. The fact that the Brüwer fixed point hypothesis cannot be used in dimensions of space that are infinitely large is one of the most important discoveries in the history of mathematics. The facts that have been supplied make it feasible to claim that the great majority of them are of a topological nature. In 2019, N. Manav and D. Turkoglu introduced a new class of generalized metric space called modular F metric space as a generalization of metric space. It is generally agreed upon that this is one of the most significant discoveries that has been made in the subject of mathematics that has ever been produced. In this article, we study the basic structure of modular F-metric spaces and the concept of an equivalent relationship between modular F metric space and modular F metric bounded space. Moreover, we study the fixed point theorem (New version of Banach Contraction Principle) over modular F metric spaces. This is something that one would be able to predict occurring given the circumstances that are present. These results extend, broaden, and integrate many previously published results.

I-Symmetric Spaces and the π-Images of Metric Spaces

Symmetric spaces and sn-symmetric spaces, as a generalization of metric spaces, have many important properties and have been widely discussed. We consider characterizations and mapping properties of sn-symmetric spaces under ideal convergence. I-symmetric spaces and I-sn-symmetric spaces are defined and studied. These not only generalize some classical results on symmetric spaces but also provide new directions to study generalized metric spaces using the notion of ideal convergence. As an application of I-sn-symmetric spaces, some relevant properties of statistical convergence are obtained. Some unanswered questions in this field are raised.

On Neutrosophic Fuzzy Metric Space and Its Topological Properties

The present research introduces a novel concept termed “neutrosophic fuzzy metric space”, which extends the traditional metric space framework by incorporating the notion of neutrosophic fuzzy sets. A thorough investigation of various structural and topological properties within this newly proposed generalization of metric space has been conducted. Additionally, counterparts of well-known theorems such as the Uniform Convergence Theorem and the Baire Category Theorem have been established for this generalized metric space. Through rigorous analysis, a detailed understanding of its fundamental characteristics has been attained, illuminating its potential applications and theoretical significance.

Original new fixed point theorems in nth order G− metric spaces

Many generalizations of the traditional metric space have been introduced in the literature, such as 2−, D−, G−, S− and b−metric spaces. When the studies on these generalized metric spaces are examined, it is seen that the main motivation of the researchers is to develop and generalize the famous Banach fixed point theorem. Although introduced with a similar motivation, its ability to measure the distance between n points simultaneously distinguishes the n th order G−metric space from other generalized metric spaces. In this study, we will give new and original fixed point theorems that reveal the importance of G−metric techniques since they cannot be reduced to the framework of quasi and conventional metric spaces.

A (Discrete) Homotopy Theory for Geometric Spaces

We define the concepts of homotopy and fundamental group for geometric spaces as a generalization of metric spaces, digital spaces, and graphs; then, we compare them with corresponding concepts in these spaces. Also, we state some properties of the fundamental group of geometric spaces and some theorems to calculate them.

EXPANSIVE FIXED POINT RESULTS IN SUPER METRIC SPACES

Super Metric Spaces are a ground-breaking generalization of metric spaces that were recently developed by Karapinar and Khojasteh (Filomat, in press). In this paper, we initiate the study of expansive fixed points in the context of the supermetric spaces. Our results may open the door to more expansive fixed point results in a different direction.

Diversity on \(\wp_{fin}\) (X)

In tight-span theory, a diversity is a generalization of metric space where the metric is defined over the set \(\wp_{fin}\) (X) which is composed of finite subsets of X. In this paper we are going to generalize the results of D. Silvestru and C. Gosa to derive some sharp inequalities for the diameter diversity. This sharp inequality can be used to study models with diversity in a collective manner.

A New Generalization of Metric Spaces Satisfying the T2-Separation Axiom and Some Related Fixed Point Results

A New Generalization of Metric Spaces Satisfying the T2-Separation Axiom and Some Related Fixed Point Results

Fixed point of weak contraction mappings on suprametric spaces

Suprametric spaces are a very recent generalization of metric spaces. In this study, various fixed point results are given in suprametric spaces. We prove the existence and uniqueness of fixed point for ? ? ?-weak contractive mapping. Our results generalize those corresponding in the literature.

Fixed point results in α, β partial b-metric spaces using C-contraction type mapping and its generalization

Banach contraction mapping has main role in nonlinear analysis courses and has been modified to get new kind of generalizations in some abstract spaces to produce many fixed point theory. Fixed point theory has been proved in partial metric spaces and b-metric spaces as generalizations of metric spaces to obtain new theorems. In addition, using modified of contraction mapping we get some fixed point that have been used to solve differential equations or integral equations, and have many applications. Therefore, this area is actively studied by many researchers. The goal of this article is present and prove some fixed point theorems for extension of contraction mapping in α, β partial b-metric spaces. In this research, we learn about notions of b-metric spaces and partial metric that are combined to generated partial b-metric spaces from many literatures. Afterwards, generalizations are made to get α, β partial b-metric spaces. Using the properties of convergence, Cauchy sequences, and notions of completeness in α, β partial b-metric spaces, we prove some fixed point theorem. Fixed point theory that we generated used C-contraction mapping and its generalizations with some conditions. Existence and uniqueness of fixed point raised for some restrictions of α, β conditions. Some corollaries of main results are also proved. Our main theorems extend and increase some existence in the previous results.©2022 JNSMR UIN Walisongo. All rights reserved.

Partial b-Rectangular Metric Space with Some Results

A new generalization of metric space called partial b-rectangular metric space is introduced. Also, the relation between this generalization and the other generalizations for example a b-rectangular metric space is given. Moreover, we have proved Banach theorem and Kannan theorem of fixed Point in partial b-rectangular metric space. Furthermore, some definitions and results dealing with partial b-rectangular metric space are discussed

SOME COMMON FIXED POINTS THEOREMS FOR OCCASIONALLY WEAKLY COMPATIBLE MAPPINGS IN MENGER SPACE

The fixed point theory as a part of the non-linear analysis is the study of function in metric or non-metric settings. K. Menger in1942 introduced the notion of probabilistic metric space (or statistical space or Menger Space) which is an important generalization of metric space and the study of this space was expanded rapidly with the pioneering work of B. Schweizer and A. Skalar [21] in 1960 and the work of V.M. Sehgal and A.T. Bharucha Reid [23] in 1972. This space broadens in weakly compatible in Menger space by Singh and Jain [24], B.D. Pant et. al [16] and this notion also extend to occasionally weakly compatible by AI. Thapagi and Shahzad [27]. The purpose of this paper is to establish a common fixed point result in Menger space in two pairs and three pairs of mappings by using occasionally weakly compatible mappings. Our first theorem generalizes the theorem of Sharma and Shahu [28] and B. Fisher et.al [29] and both theorems deduce some similar results in the literature.

Generalization of metric spaces and some fixed point theorems

The primary purpose of this paper is to adduce the concept of extended metric spaces, inspired by the concepts of metric space and an extended metric space. We also create some fixed-point theorems arising in this space.

F-Bipolar metric spaces and fixed point theorems with applications

In this paper, we propose a new generalization of metric spaces by the unification of two novel notions, namely \(\mathfrak{F}\)-metric spaces and bipolar metric spaces, under the name \(\mathfrak{F}\)-bipolar metric spaces. Further, in this newly generalized notion we provide a binary topology and prove some fixed point results. As applications of our result, we prove the existence and uniqueness of solution of integral equation and the existence of a unique solution in homotopy theory. We also give some non-trivial examples to vindicate our claims. Our fixed point results extend several results in the existing literature.

Function weighted $mathcal{G}$-metric spaces and Hausdorff $Delta$-distances; an application to fixed point theory

In this paper, we introduce a new space which is a generalization of function weighted metric space introduced by Jleli and Samet [On a new generalization of metric spaces, J. Fixed Point Theory Appl. 2018, 20:128] where namely function weighted $mathcal{G}$-metric space. Also, a Hausdorff $Delta$-distance is introduced in these spaces. Then several fixed point results for both single-valued and multi-valued mappings in such spaces are proved. We also construct some examples for the validity of the given results and present an application to the existence of a solution of the Volterra-type integral equation.

EXISTENCE AND UNIQUENESS OF COMMON FIXED POINT FOR TWO MAPPINGS IN RECTANGULAR METRIC SPACES AND RECTANGULAR b METRIC SPACES

The conception of rectangular b-metric space is introduced as a generalization of metric space, b-metric space and rectangular metric space. In this article we present existence and uniqueness of some fixed point results for new contractions in rectangular metric spaces and rectangular b-metric spaces. Some appropriate and innovative examples also displayed to support and validate these new outcomes.

S^{JS}- metric and topological spaces

We introduce the idea of S^{JS}- metric spaces which is a generalization of metric spaces. We next study the properties of S^{JS}- metric spaces and prove several theorems. We also deal with abstract S^{JS}- topological spaces induced by S^{JS}- metric and obtain several classical results including Cantor's intersection theorem in this setting. DOI: 10.1735/jme.2021.01.01134

A note on $${\mathcal {F}}$$-metric spaces

A new generalization of the metric space notion, named $${\mathcal {F}}$$ -metric space, was given in [M. Jleli, B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl. 20 (2018), no. 3, Art. 128, 20 pp.]. In this paper, we investigate some properties of $${\mathcal {F}}$$ -metric spaces. A simple proof is given to show that the natural topology induced by an $${\mathcal {F}}$$ -metric is metrizable. We present a method to construct s- $$\hbox {relaxed}_{{p}}$$ spaces and, therefore, $${\mathcal {F}}$$ -metric spaces from bounded metric spaces. We give some results that reveal differences between metric and $${\mathcal {F}}$$ -metric spaces. In particular, we show that the ordinary open and closed balls in $${\mathcal {F}}$$ -metric spaces are not necessarily topological open and closed, respectively. This answers a question posed implicitly in the quoted paper. We also show that $${\mathcal {F}}$$ -metrics are not necessarily jointly continuous functions. Despite some topological differences between metrics and $${\mathcal {F}}$$ -metrics, we show that the Nadler fixed point theorem and, therefore, the Banach contraction principle in the frame of $${\mathcal {F}}$$ -metric spaces can be reduced to their original metric versions. This reduction even happens when the Schauder fixed point theorem is investigated in $${\mathcal {F}}$$ -normed spaces structure. By applying the given technique in this paper, it turns out that some nonlinear $${\mathcal {F}}$$ -metric contractions and, therefore, the related $${\mathcal {F}}$$ -metric fixed point results can naturally be reduced to their metric versions. In addition, the same happens for some topological fixed point results.

Ultradiversities and their spherical completeness

Abstract Diversities are a generalization of metric spaces which associate a positive real number to every finite subset of the space. In this paper, we introduce ultradiversities which are themselves simultaneously diversities and a sort of generalization of ultrametric spaces. We also give the notion of spherical completeness for ultradiversities based on the balls defined in such spaces. In particular, with the help of nonexpansive mappings defined between ultradiversities, we show that an ultradiversity is spherically complete if and only if it is injective.

The topology of a quantale valued metric space

The ‘the’ in the title hides a subtlety. A metric space induces not one but four topologies - by means of open sets, closed sets, closure, and interior - they just so happen to coincide. The agreement between these four structures arising from a metric function d:X×X→[0,∞] is due to a combination of the metric axioms and the lattice structure of [0,∞]. Further motivation materializes from Lawvere's observation from 1973 to the effect that a (slightly generalized) metric space is a category enriched in [0,∞]. Metric spaces taking values in structures other than [0,∞] are relevant for generalizations of metric spaces and find a natural home in Lawvere's categorical setting. In particular, in recent years quantales emerged as structures occupying an important niche in between [0,∞] and arbitrary monoidal categories. Since a category enriched in a quantale Q is the same thing as a metric space taking values in Q one may ask whether such a thing belongs to algebra or geometry. Further, does the quadruplet of topologies associated to a Q-valued space/category still consist of identical siblings? We propose a litmus test for the geometricity of Q-valued spaces as we investigate these issues.