Semimetric Spaces Research Articles

Certain Fixed Point Results via Contraction Mappings in Neutrosophic Semi-Metric Spaces

In this work, the authors introduce the concept of neutrosophic semi-metric spaces and prove several common fixed-point theorems for countable and uncountable family of mappings via an implicit relation of contractive and integral type by utilizing locally integrable functions. These results improve and generalize the several results in the existing literature. Further, the authors present some non-trivial examples to support our main results. Mathematics Subject Classification: 46S40, 47H10, 54H25.

On generalizations of some fixed point theorems in semimetric spaces with triangle functions

In the present study, we prove generalizations of Banach, Kannan, Chatterjea, Ćirić-Reich-Rus fixed point theorems, as well as of the fixed point theorem for mapping contracting perimeters of triangles. We consider corresponding mappings in semimetric spaces with triangle functions introduced by Bessenyei and Páles. Such an approach allows us to derive corollaries for various types of semimetric spaces such as metric spaces, ultrametric spaces, and b-metric spaces. The significance of these generalized theorems extends across multiple disciplines, such as optimization, mathematical modeling, and computer science. They may serve to establish stability conditions, demonstrate the existence of optimal solutions, and improve algorithm design.

Quasi-Contraction Maps in Subordinate Semimetric Spaces

Throughout this study, we discuss the subordinate Pompeiu–Hausdorff metric (SPHM) in subordinate semimetric spaces. Moreover, we present a well-behaved quasi-contraction (WBQC) to solve quasi-contraction (QC) problems in subordinate semimetric spaces under some local constraints. Furthermore, we provide examples to support our conclusion.

Coupled Fixed Point Theory in Subordinate Semimetric Spaces

The aim of this paper is to study the coupled fixed point of a class of mixed monotone operators in the setting of a subordinate semimetric space. Using the symmetry between the subordinate semimetric space and a JS-space, we generalize the results of Senapati and Dey on JS-spaces. In this paper, we obtain some coupled fixed point results and support them with some examples.

A note on generalized four-point inequality

In 2017, M. Bessenyei and Z. Páles [1] introduced a definition of a triangle function that generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept, we discuss already known inequalities in metric spaces that relate six distances determined by four points and introduce a definition of the generalized four-point inequality in semimetric spaces. Conditions under which quasisymmetric mappings and quasi-Möbius mappings between semimetric spaces preserve such an inequality are obtained.

A note on generalized four-point inequality

In 2017, M. Bessenyei and Z. Páles [1] introduced a definition of a triangle function that generates a concept of a generalized triangle inequality in semimetric spaces. Inspired by this concept, we discuss already known inequalities in metric spaces that relate six distances determined by four points and introduce a definition of the generalized four-point inequality in semimetric spaces. Conditions under which quasisymmetric mappings and quasi-Möbius mappings between semimetric spaces preserve such an inequality are obtained.

Uniqueness of best proximity pairs and rigidity of semimetric spaces

For arbitrary semimetric space \((X, d)\) and disjoint proximinal subsets \(A\), \(B\) of \(X\) we define the proximinal graph as a bipartite graph with parts \(A\) and \(B\) whose edges \(\{a, b\}\) satisfy the equality \(d(a, b) = {\text {dist}}(A, B)\). We characterize the semimetric spaces whose proximinal graphs have at most one edge and the semimetric spaces whose proximinal graphs have the vertices with degree at most \(1\) only. This allows us to describe the necessary and sufficient conditions for uniqueness of the best proximity pairs and best approximations.

Applications of ball spaces theory: fixed point theorems in semimetric spaces and ball convergence

In the paper, we apply some of the results from the theory of ball spaces in semimetric setting. This allows us to obtain fixed point theorems which we believe to be unknown to this day. As a byproduct, we obtain the equivalence of some different notions of completeness in semimetric spaces where the distance function is 1-continuous. In the second part of the article, we generalize the Caristi–Kirk results for b-metric spaces. Additionally, we obtain a characterization of semicompleteness for 1-continuous b-metric spaces via a fixed point theorem analogous to a result of Suzuki. In the epilogue, we introduce the concept of convergence in ball spaces, based on the idea that balls should resemble closed sets in topological sets. We prove several of its properties, compare it with convergence in semimetric spaces and pose several open questions connected with this notion.

Asymptotically Coupled Coincidence Points and Asymptotically Coupled Fixed Points in Fuzzy Semi-Metric Spaces

Asymptotically coupled coincidence points and asymptotically coupled fixed points in fuzzy semi-metric spaces are studied in this paper. The fuzzy semi-metric space is taken into account, which lacks symmetric conditions. In this case, the desired results are separately investigated based on four different types of triangle inequalities. The uniqueness of asymptotically coupled coincidence points cannot be guaranteed, and it can only be addressed in a weak sense of uniqueness. However, the uniqueness of asymptotically coupled fixed points can be guaranteed using different arguments.

Hutchinson’s theorem in semimetric spaces

One of the important consequences of the Banach fixed point theorem is Hutchinson’s theorem which states the existence and uniqueness of fractals in complete metric spaces. The aim of this paper is to extend this theorem for semimetric spaces using the results of Bessenyei and Páles published in 2017. In doing so, some properties of semimetric spaces as well as of the fractal space are investigated. We extend Hausdorff’s theorem to characterize compactness and Blaschke’s theorems to characterize the completeness of the fractal space. Based on these preliminaries, an analogue of Hutchinson’s theorem in the setting of semimetric spaces is proved, and finally, error estimates and stability of fractals are established as well.

A Solution to Two Old Problems by Menger Concerning Angle Spaces

Around 1930, Menger expressed his interest in the concept of abstract angle function. He introduced a general definition of this notion for metric and semi-metric spaces. He also proposed two problems concerning conformal embeddability of spaces endowed with an angle function into Euclidean spaces. These problems received attention in later years but only for some particular cases of metric spaces. In this article, we first update the definition of angle function to apply to the larger class of spaces with a notion of betweenness, which seem to us a more natural framework. In this new general setting, we solve the two problems proposed by Menger.

On quasisymmetric mappings in semimetric spaces

The class of quasisymmetric mappings on the real axis was first introduced by Beurling and Ahlfors in 1956. In 1980 Tukia and Väisälä considered these mappings between general metric spaces. In our paper we generalize the concept of a quasisymmetric mapping to the case of general semimetric spaces and study some properties of these mappings. In particular, conditions under which quasisymmetric mappings preserve triangle functions, Ptolemy's inequality and the relation "to lie between" are found. Considering quasisymmetric mappings between semimetric spaces with different triangle functions we give a new estimate for the ratio of diameters of two subsets, which are images of two bounded subsets. This result generalizes the well-known Tukia-Väisälä inequality. Moreover, we study connections between quasisymmetric mappings and weak similarities which form a special class of mappings between semimetric spaces.

Generalized fractals in semimetric spaces

The aim of the present paper is to extend the classical fractal theory using condensing maps and generalized contractions of semimetric spaces. Our method is independent from the approach of Hutchinson: It is based on the Kuratowski measure of noncompactness and avoids completely the Blaschke Completeness Theorem.

On Functions Preserving Products of Certain Classes of Semimetric Spaces

Abstract In the paper, we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. We extend their scope to a broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving b-metric inequality. We provide analogues of the results of Borsík and Doboš adjusted to the new broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.

Additive regression for non-Euclidean responses and predictors

Additive regression is studied in a very general setting where both the response and predictors are allowed to be non-Euclidean. The response takes values in a general separable Hilbert space, whereas the predictors take values in general semimetric spaces, which covers a very wide range of nonstandard response variables and predictors. A general framework of estimating additive models is presented for semimetric space-valued predictors. In particular, full details of implementation and the corresponding theory are given for predictors taking values in Hilbert spaces and/or Riemannian manifolds. The existence of the estimators, convergence of a backfitting algorithm, rates of convergence and asymptotic distributions of the estimators are discussed. The finite sample performance of the estimators is investigated by means of two simulation studies. Finally, three data sets covering several types of non-Euclidean data are analyzed to illustrate the usefulness of the proposed general approach.

Using the infimum form of auxiliary functions to study the common coupled coincidence points in fuzzy semi-metric spaces

The common coupled coincidence points and common coupled fixed points in fuzzy semi-metric spaces are investigated in this paper. In fuzzy semi-metric space, the symmetric condition is not necessarily assumed to be satisfied. In this case, regarding the non-symmetry of metric, there are four kinds of triangle inequalities that can be considered. In order to investigate the common coupled coincidence points and common coupled fixed points, the fuzzy semi-metric space is further assumed to satisfy the so-called canonical condition that is inspired from the intuitive observations. The sufficient conditions for guaranteeing the common coupled coincidence points and common coupled fixed points will be different for the four different kinds of triangle inequalities.

Using the Supremum Form of Auxiliary Functions to Study the Common Coupled Coincidence Points in Fuzzy Semi-Metric Spaces

This paper investigates the common coupled coincidence points and common coupled fixed points in fuzzy semi-metric spaces. The symmetric condition is not necessarily satisfied in fuzzy semi-metric space. Therefore, four kinds of triangle inequalities are taken into account in order to study the Cauchy sequences. Inspired by the intuitive observations, the concepts of rational condition and distance condition are proposed for the purpose of simplifying the discussions.

Constrained minimum Riesz energy problems for a condenser with intersecting plates

We study the constrained minimum energy problem with an external field relative to the α-Riesz kernel x−yα−n of order α ∈ (0, n) for a generalized condenser A = (Ai)i∈I in ℝn, n ⩾ 3, whose oppositely charged plates intersect each other over a set of zero capacity. Conditions sufficient for the existence of minimizers are found, and their uniqueness and vague compactness are studied. Conditions obtained are shown to be sharp. We also analyze continuity of the minimizers in the vague and strong topologies when the condenser and the constraint both vary, describe the weighted equilibrium vector potentials, and single out their characteristic properties. Our arguments are based particularly on the simultaneous use of the vague topology and a suitable semimetric structure on a set of vector measures associated with A, and the establishment of completeness theorems for proper semimetric spaces. The results remain valid for the logarithmic kernel on ℝ2 and A with compact Ai, i ∈ I. The study is illustrated by several examples.

Common fixed point results for non-compatible R-weakly commuting mappings in probabilistic semimetric spaces using control functions

In common fixed point problems in metric spaces several versions of weak commutativity have been considered. Mappings which are not compatible have also been discussed in common fixed point problems. Here we consider common fixed point problems of non-compatible and R-weakly commuting mappings in probabilistic semimetric spaces with the help of a control function. This work is in line with research in probabilistic fixed point theory using control functions. Further we support our results by examples.

Bipartite Sets of Spheres and Casey-Type Theorems

A bipartite set of spheres in $${\mathbb {R}}^n$$ is a set of colored spheres, where two colors are used, no sphere is contained in the closed ball bounded by another sphere in the set, and spheres of different colors are disjoint. For any two spheres in a bipartite set, the common-tangent distance between them is defined as the distance between two tangent points in a common tangent hyperplane to them, where an external common tangent hyperplane is taken if the two spheres are of the same color; otherwise, a common internal tangent hyperplane is taken. By this common-tangent distance, a bipartite set becomes a semi-metric space. It turns out that bipartite sets of spheres form an interesting family of semi-metric spaces. Casey’s theorem (a generalization of Ptolemy’s theorem) gives a condition for a bipartite set of four circles in $${\mathbb {R}}^2$$ to have a circle that is suitably tangent to all circles in the bipartite set. Ptolemy’s theorem is generalized to the n-dimensional situation via Cayley–Menger determinants. Among other results, we present a Casey-type theorem for a bipartite set of $$n+2$$ spheres in n dimensions as a generalization of the n-dimensional version of Ptolemy’s theorem, and we extend this further to a bipartite set with an arbitrary number of spheres in $${\mathbb {R}}^n$$ .