Setting Of Metric Spaces Research Articles

COUPLED AND COMMON COUPLED FIXED POINT THEOREMS UNDER NEW COUPLED IMPLICIT RELATION IN PARTIAL METRIC SPACES

The purpose of this paper is to study existence and uniqueness of coupledand common coupled fixed point theorems for self-mappings satisfying a new coupledimplicit relation in the setting of partial metric spaces and give some corollaries ofTheorem 3.1. Furthermore, we prove well-posedness of a coupled fixed point problem.We also provide some applications of our result to a mapping with a contraction ofintegral type. The results of findings in this paper extend and generalize several resultsfrom the existing literature.

Solving Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions Using Covariant JS-Contractions

This paper investigates the existence, uniqueness, and symmetry of solutions for Φ–Atangana–Baleanu fractional differential equations of order μ∈(1,2] under mixed nonlocal boundary conditions. This is achieved through the use of covariant and contravariant JS-contractions within a generalized framework of a sequential extended bipolar parametric metric space. As a consequence, we obtain the results on covariant and contravariant Ćirić, Chatterjea, Kannan, and Reich contractions as corollaries. Additionally, we substantiate our fixed-point findings with specific examples and derive similar results in the setting of sequential extended fuzzy bipolar metric space.

The Application of Geogebra in Teaching Open and Inner Sets in Metric Spaces

Topology of metric spaces is a standard and important course at the college level mathematics studied at University of Mumbai. As the course is abstract, students feel it as a difficult course and hence tend to develop a fear. In this paper, we will see how to use the open-source software GeoGebra to teach the concepts of this course effectively. We will concentrate on the topics of open set and interior set. GeoGebra applets are created for various examples in different metric spaces to teach these concepts efficiently.

A note on rearrangement Poincaré inequalities and the doubling condition

We introduce Poincaré-type inequalities based on rearrangement invariant spaces in the setting of metric measure spaces and analyze when they imply the doubling condition on the underline measure.

Best Proximity Point Results via Simulation Function with Application to Fuzzy Fractional Differential Equations

In this study, we prove the existence and uniqueness of a best proximity point in the setting of non-Archimedean modular metric spaces via the concept of simulation functions. A non-Archimedean metric modular is shaped as a parameterized family of classical metrics; therefore, for each value of the parameter, the positivity, the symmetry, the triangle inequality, or the continuity is ensured. Also, we demonstrate how analogous theorems in modular metric spaces may be used to generate the best proximity point results in triangular fuzzy metric spaces. The utility of our findings is further demonstrated by certain examples, illustrated consequences, and an application to fuzzy fractional differential equations.

Generic families of finite metric spaces with identical or trivial 1-dimensional persistence

Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change of persistence in the bottleneck distance under perturbations of points, but without giving a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing finite metric spaces, which is evident for non-isometric point sets with identical persistence. We describe generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish finite point sets up to isometry.

Wijsman Ideal Convergence Sequence in Neutrosophic Metric Spaces

Objectives: An examination of convergence for a sequence of closed sets in Neutrosophic Metric Spaces is the main goal of this research. Particularly, the focus of this study is directed at Wijsman -convergence ( and Wijsman -convergence . Methods: The various properties of Neutrosophic Metric Spaces are investigated in this study. Using exact mathematical techniques, this study comprehends the features of convergent sequences in these domains. Furthermore, this study expresses particular views on Wijsman -Cauchy and Wijsman -Cauchy sequences. Findings: The convergence behaviour of closed sets in Neutrosophic Metric Spaces is a result of this research. it presents findings and remarks about the sequences of Wijsman - convergence, Wijsman - convergence, Wijsman -Cauchy, and Wijsman -Cauchy. These findings improve the current understanding of convergence in relation to NMS. Novelty: What makes this work unique is that it focuses on convergence notions in Neutrosophic Metric Spaces, specifically on Wijsman and Wijsman - convergence enriches the current body of information in this domain. Keywords: Neutrosophic Metric Spaces, Wijsman I ­convergent and Wijsman I ¿ ­ convergent sequences, Wijsman I ¿ ­Cauchy and Wijman I ­Cauchy sequences

Classical multidimensional scaling on metric measure spaces

Abstract We study a generalization of the classical multidimensional scaling procedure (cMDS) which is applicable in the setting of metric measure spaces. Metric measure spaces can be seen as natural ‘continuous limits’ of finite data sets. Given a metric measure space ${\mathcal{X}} = (X,d_{X},\mu _{X})$, the generalized cMDS procedure involves studying an operator which may have infinite rank, a possibility which leads to studying its traceability. We establish that several continuous exemplar metric measure spaces such as spheres and tori (both with their respective geodesic metrics) induce traceable cMDS operators, a fact which allows us to obtain the complete characterization of the metrics induced by their resulting cMDS embeddings. To complement this, we also exhibit a metric measure space whose associated cMDS operator is not traceable. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov–Wasserstein distance.

Weak Gurov–Reshetnyak class in metric measure spaces

We introduce a weak Gurov–Reshetnyak class and discuss its connections to a weak Muckenhoupt A ∞ A_\infty condition and a weak reverse Hölder inequality in the setting of metric measure spaces with a doubling measure. A John–Nirenberg type lemma is shown for the weak Gurov–Reshetnyak class which gives a specific decay estimate for the oscillation of a function. It implies that a function in the weak Gurov–Reshetnyak class satisfies the weak reverse Hölder inequality. This comes with an upper bound for the reverse Hölder exponent depending on the Gurov–Reshetnyak parameter which allows the study of the asymptotic behavior of the exponent.

Coupled coincidence best proximity point results for generalized Ciric contractions

In this paper, we define a generalized Ćiric contraction with respect to ℸ and prove some coupled coincidence best proximity point results in the setting of complete metric spaces. We also obtain some coupled best proximity point results for such contractions. We provide an example for the validity of our main result. Our results are the generalization of many results existing in the literature. As an application, we find the solution of the differential equation.

Results in Cone Metric Spaces and Related Fixed Point Theorems for Contractive Type Mappings With Application

In this article, we prove some new fixed point theorems for contractive type mappings in the setting of complete cone metric spaces and provide examples to illustrate the concepts and results developed in the article. We consider some consequences of our results to establish fixed point theorems in the context of cone metric spaces. As an application of our results, periodic point results for the contractive type mappings are proved.

Locating network trees by a bilevel scheme

AbstractIn this paper we investigate how to choose an optimal position of a specific facility that is constrained to a network tree connecting some given demand points in a given area. A bilevel formulation is provided and existence results are given together with some properties when a density describes the construction cost of the networks in the area. This includes the presence of an obstacle or of free regions. To prove existence of a solution of the bilevel problem, that is framed in Euclidean spaces, a lower semicontinuity property is required. This is obtained proving an extension of Goła̧b’s theorem in the general setting of metric spaces, which allows for considering a density function.

Fixed point theorem using contraction in complete multiplicative metric space

In our paper, we prove a fixed point theorem which extends results of Gupta and Garg and broadens a number of findings in the setting of multiplicative metric space.

Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

A modified implicit relation and ?-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided. As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation X = B + Pk i=1A* i T(X)Ai. An example is given, using matrices that are randomly generated, as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.

RATIONAL TYPE INTUITIONISTIC FUZZY CONTRACTION RESULTS IN INTUITIONISTIC FUZZY METRIC SPACES WITH AN APPLICATION

The purpose of this study is to propose a new idea of rational type intuitionistic fuzzy contraction mappings in intuitionistic fuzzy metric spaces. We show several fixed point results in intuitionistic fuzzy metric spaces under rational type intuitionistic fuzzy contraction conditions, with detailed examples to back up our claims. This revolutionary idea will be crucial in the theory of intuitionistic fuzzy fixed point outcomes and can be applied to other contractive type mappings in the setting of intuitionistic fuzzy metric spaces. Furthermore, to support our work, we offer an application of a nonlinear integral type problem to obtain the current result for a unique solution.

On a class of Hausdorff measure of cartesian product sets in metric spaces

In this paper we study, in a separable metric space, a class of Hausdorff measures ${\mathcal H}_\mu^{q, \xi}$ defined using a measure $\mu$ and a premeasure $\xi$. We discuss a Hausdorff structure of product sets. Weighted Hausdorff measures ${\mathcal W}_\mu^{q, \xi}$ appeare as an important tool when studying the product sets. When $\mu$ and $\xi$ satisfy the doubling condition, we prove that ${\mathcal H}_\mu^{q, \xi} = {\mathcal W}_\mu^{q, \xi}$. As an application, the case where $\xi$ is defined as the Hausdorff function is considered.

Proximity Point Results for Generalized p-Cyclic Reich Contractions: An Application to Solving Integral Equations

This article studies new classes of contractions called the p-cyclic Reich contraction and p-cyclic Reich contraction pair and develops certain best proximity point results for such contractions in the setting of partial metric spaces. Furthermore, the best proximity point results for p-proximal cyclic Reich contractions of the first and second types are also discussed.

Solving a Nonlinear Fractional Differential Equation Using Fixed Point Results in Orthogonal Metric Spaces

This research article aims to solve a nonlinear fractional differential equation by fixed point theorems in orthogonal metric spaces. To achieve our goal, we define an orthogonal Θ-contraction and orthogonal (α,Θ)-contraction in the setting of complete orthogonal metric spaces and prove fixed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in fixed point theory. We provide a non-trivial example to show the legitimacy of the established results.

Weighted ensemble clustering with multivariate randomness and random walk strategy

Ensemble clustering algorithms have made significant progress in recent years due to their excellent performance. However, most of these algorithms face two challenges: one is to focus on the selection of subspaces since there is limited discussion on how to construct a potential metric space, the other is to treat basic clustering equally without fully considering the local connection between clusters when constructing the cooperative association matrix. To solve these issues, we propose a weighted ensemble clustering algorithm with multiple randomness and random walk strategy. We define the free exponential similarity kernel to create a diverse set of random metric spaces coupled with random subspaces and use spectral clustering to generate base clustering. Moreover, we use random walk strategy to discover the local connection between clusters and weight the collaborative association matrix. Finally, the collaborative association matrix uses consensus functions based on hierarchical clustering and meta clustering to obtain clustering results. On this basis, two specific ensemble clustering algorithms WECMR-HC and WECMR-MC are proposed. Theoretical analysis and experimental results demonstrate that our proposed algorithm outperforms existing ensemble algorithms.

Certain Interpolative Proximal Contractions, Best Proximity Point Theorems in Bipolar Metric Spaces with Applications

In this manuscript, we present several types of interpolative proximal contraction mappings including Reich–Rus–Ciric-type interpolative-type contractions and Kannan-type interpolative-type contractions in the setting of bipolar metric spaces. Further, taking into account the aforementioned mappings, we prove best proximity point results. These results are an extension and generalization of existing ones in the literature. Furthermore, we provide several nontrivial examples, an application to find the solution of an integral equation, and a nonlinear fractional differential equation to show the validity of the main results.