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In this paper we formulate a setvalued fixed point problem by combining four prevalent trends of fixed point theory. We solve the problem by showing that the set of fixed points is nonempty. Further we have a data dependence result pertaining to the problem and also a stability result for the fixed point sets. The main result is extended to metric spaces with a graph. The results are obtained without the use of metric completeness assumption which is replaced by some other conditions suitable for solving the fixed point problem. There are some consequences of the main result. The main result is illustrated with an example.

In this paper, as a geometric approach to the fixed-point theory, we prove new fixed-figure results using the notion of k-ellipse on a metric space. For this purpose, we are inspired by the Caristi type mapping, Kannan type contraction, Chatterjea type contraction and Ćirić type contraction. After that, we give some existence and uniqueness theorems of a fixed k-ellipse. We also support our obtained results with illustrative examples. Finally, we present a new application to the S-Shaped Rectified Linear Activation Unit (SReLU) to show the importance of our theoretical results.

A complete metric space is a well-known concept. Kreyszig shows that every non-complete metric space can be developed into a complete metric space , referred to as completion of . We use the b-Cauchy sequence to form which “is the set of all b-Cauchy sequences equivalence classes”. After that, we prove to be a 2-normed space. Then, we construct an isometric by defining the function from to ; thus and are isometric, where is the subset of composed of the equivalence classes that contains constant b-Cauchy sequences. Finally, we prove that is dense in , is complete and the uniqueness of is up to isometrics

In this paper, the concepts of Wardowski-type set-valued contractions and Işik-type set-valued contractions are introduced and fixed point theorems for such contractions are established. A positive answer to the open Question is given. Examples to support main theorems and an application to integral inclusion are given.

In an outstanding article published in 2008, Suzuki obtained a nice generalization of the Banach contraction principle, from which a characterization of metric completeness was derived. Although Suzuki’s theorem has been successfully generalized and extended in several directions and contexts, we here show by means of a simple example that the problem of achieving, in an obvious way, its full extension to the framework of w-distances does not have an emphatic response. Motivated by this fact, we introduce the concept of presymmetric w-distance on metric spaces. We also give some properties and examples of this new structure and show that it provides a reasonable setting to obtain a real and hardly forced w-distance generalization of Suzuki’s theorem. This is realized in our main result, which consists of a fixed point theorem that involves presymmetric w-distances and certain contractions of Suzuki-type. We also discuss the relationship between our main result and the well-known w-distance full generalization of the Banach contraction principle, due to Suzuki and Takahashi. Connected to this approach, we prove another fixed point result that compares with our main result through some examples. Finally, we state a characterization of metric completeness by using our fixed point results.

In this paper, we prove a new coincidence and common fixed point theorem of hybrid mappings using the C-class function and T-weak commutativity. Finally, we give an example to illustrate our result. [ABSTRACT FROM AUTHOR]

The work is a continuation of the researches of presentation by values of the field „Mental and Behavioural Disorders in Epilepsy" (MBDE). Metric spaces for MBDE have been developed. These spaces were used in the diagnosis of MBDE and to make tables of distances between diagnoses. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider problems concerning the existence of solutions of the Beltrami equations and their convergence in the complex plane. We are mainly interested in the case when these solutions satisfy the so-called hydrodynamic normalization condition in the neighborhood of infinity. Under some conditions on dilatations of inverse mappings, we have established the existence of such solutions in the class of continuous Sobolev mappings. We have also obtained results on the locally uniform limit of a sequence of such solutions. [ABSTRACT FROM AUTHOR]

We calculate the Gromov–Hausdorff distance between vertex sets of regular polygons endowed with the round metric. We give a full answer for the case of - and -gons with divisible by . We also calculate all distances to -gons and -gons. [ABSTRACT FROM AUTHOR]

Recently, Jleli et. al. [9] introduced by the concept of a (H, ψ)-fixed point, and they establish some fixed point results for various classes of operators defined on a metric space (M, d). This study is devoted to investigate the problem whether the existence and uniqueness of integral type (H, ψ)F-contraction mappings on complete metric space. At the end, we give an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper we introduce multivalued modified F-contraction on a metric space. This is a multivalued mapping obtained by incorporating the idea of the recently introduced F-contraction which has attracted much attention in contemporary research. We explore the fixed point problem associated with the above contractive mapping. We also investigate the data dependence and stability properties of the fixed point sets associated with these multival- ued contractions. We discuss an illustration of the main result and present an application of the single valued version of the main theorem to a problem of an integral equation of Volterra type. The domain of the study is fixed point theory and set valued analysis. [ABSTRACT FROM AUTHOR]

Let (X, d) be a complete metric space, (Y, ρ) be a metric space and f, g : X → Y be two mappings. The problem is to give metric conditions which imply that, C(f,g) : = {x ∈∈ X \ f(x) = g(x)} ≠ 0. In this paper we give an abstract coincidence point result with respect to which some results such as of Peetre-Rus (I.A. Rus, Teoria punctului fix în analiza funcțională, Babeş-Bolyai Univ., Cluj-Napoca, 1973), A. Buică (A. Buică, Principii de coincidență şi aplicații, Presa Univ. Clujeană, Cluj-Napoca, 2001) and A.V. Arutyunov (A.V. Arutyunov, Covering mappings in metric spaces and fixed points, Dokl. Math., 76(2007), no.2, 665-668) appear as corollaries. In the case of multivalued mappings our result generalizes some results given by A.V. Arutyunov and by A. Petruşel (A. Petruşel, A generalization of Peetre-Rus theorem, Studia Univ. Babeş-Bolyai Math., 35(1990), 81-85). The impact on metric fixed point theory is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we present a new extension of Banach-Caristi type theorem for multivalued mappings. We show that our result is not a consequence of multivalued version of Banach contraction principle due to Nadler. We provide an application of our result to the solution of functional equations. [ABSTRACT FROM AUTHOR]

This work is inspired by some recent developments on the extension of Lipschitz real functions based on the minimization of the maximum value of the slopes of a reference set for this function. We propose a new method in which an integral p–average is optimized instead of its maximum value. We show that this is a particular case of a more general theoretical approach studied here, provided by measure-valued representations of the metric spaces involved, and a duality formula. For p = 2 , explicit formulas are proved, which are also shown to be a particular case of a more general class of measure-based extensions, which we call ellipsoidal measure extensions. The Lipschitz-type boundedness properties of such extensions are shown. Examples and concrete applications are also given. [ABSTRACT FROM AUTHOR]

We investigate the existence of metric spaces which, for any coloring with a fixed number of colors, contain monochromatic isomorphic copies of a fixed starting space K. In the main theorem we construct such a space of size 2 ℵ 0 for colorings with ℵ 0 colors and any metric space K of size ℵ 0 . We also give a slightly weaker theorem for countable ultrametric K where, however, the resulting space has size ℵ 1 . [ABSTRACT FROM AUTHOR]

A metric space (X, d) is called a subline if every 3-element subset T of X can be written as T = {x, y, z} for some points x, y, z such that d(x, z) = d(x, y)+d(y, z). By a classical result of Menger, every subline of cardinality ̸= 4 is isometric to a subspace of the real line. A subline (X, d) is called an n-subline for a natural number n if for every c ∈ X and positive real number r ∈ d[X2], the sphere S(c; r) := {x ∈ X: d(x, c) = r} contains at least n points. We prove that every 2-subline is isometric to some additive subgroup of the real line. Moreover, for every subgroup G ⊆ R, a metric space (X, d) is isometric to G if and only if X is a 2-subline with d[X2] = G+ := G ∩ [0,∞). A metric space (X, d) is called a ray if X is a 1-subline and X contains a point o ∈ X such that for every r ∈ d[X2] the sphere S(o; r) is a singleton. We prove that for a subgroup G ⊆ Q, a metric space (X, d) is isometric to the ray G+ if and only if X is a ray with d[X2] = G+. A metric space X is isometric to the ray R+ if and only if X is a complete ray such that Q+ ⊆ d[X2]. On the other hand, the real line contains a dense ray X ⊆ R such that d[X2] = R+. [ABSTRACT FROM AUTHOR]

Recently, Karapinar and Khojasteh1 proposed a new structure, namely the Super Metric Spaces, a revolutionary generalisation of metric spaces. In the framework of super metric spaces, we present a common fixed point result in this paper. To support this research, we consider an illustrative example. [ABSTRACT FROM AUTHOR]

The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion of ranking of observations which is absent in more than one dimension. Motivated by the rapid development of technology, in particular the advent of ‘Big Data’, we extend here that concept to general metric spaces, propose a natural depth measure and explore its properties as a statistical depth function. Working in a general metric space allows the depth to be tailored to the data at hand and to the ultimate goal of the analysis, a very desirable property given the polymorphic nature of modern data sets. This flexibility is thoroughly illustrated by several real data analyses. [ABSTRACT FROM AUTHOR]

Given Tychonoff spaces X and Y, Uspenskij proved in [15] that if X is pseudocompact and Cp(X) is uniformly homeomorphic to Cp(Y), then Y is also pseudocompact. In particular, if Cp(X) is linearly homeomorphic to Cp(Y), then X is pseudocompact if and only if so is Y. This easily implies Arhangel'skii's theorem [1] which states that, in the case when Cp(X) is linearly homeomorphic to Cp(Y the space X is compact if and only if Y is compact. We will establish that existence of a linear homeomorphism between the spaces Cp*(X) and Cp*(Y) implies that X is (pseudo)compact if and only if so is Y. We will also show that the methods of proof used by Arhangel'skii and Uspenskij do not work in our case. [ABSTRACT FROM AUTHOR]

The problem of continuation of a partially defined metric can be efficiently studied using graph theory. Let $G=G(V,E)$ be an undirected graph with the set of vertices $V$ and the set of edges $E$. A necessary and sufficient condition under which the weight $w\colon E\to\mathbb R^+$ on the graph $G$ has a unique continuation to a metric $d\colon V\times V\to\mathbb R^+$ is found.

This paper surveys regarding solutions of linear and nonlinear integral equations through fixed point theorem. Banach's contraction mapping principle is the most widely applied fixed point theorem in all of analysis with special applications to the theory of integral equations. Over a decade researchers have generalized the definition of a metric space and using the technique of fixed point have found solutions to many linear and nonlinear Volterra-Fredholm integral equations.

In this article we have introduced the sequence space $m(\phi,d)$ and $m(M,\phi,d)$ of W. L. C. Sargent type in a metric space $(X, d)$ on generalising the sequence space $m(\phi)$ and we have defined these sequence spaces using the Orlicz function $M$. We have investigated their different properties like solidness, symmetricity, monotone, sequence algebra, completeness etc. We have established some inclusion results involving the space $m(M,\phi,d)$ and some of the existing sequence spaces. We have provided suitable examples and discussed in detail, in order to justify the failure cases and the definitions we have introduced. The results established in this article generalized and unifies several existing results.

Through the textual analysis of eight poems from Variazioni belliche, the article intends to demonstrate how Amelia Rosselli’s appropriation of Inferno vmainly presents a metapoetic connotation. Dantean echoes in Poesie (1959), the collection's first section, embody a kind of poetry in which the lyrical I has a distinctive role – hence, the use of the free verse, even if already projected towards the spazio metrico, the fixed form which has the task to distinguish Rosselli’s poetry from the previous lyrical tradition. On the contrary, in Variazioni (1960-1961), the second section of the volume, the strong lyrical subject conveyed by Paolo and Francesca’s canto collides with the «pretesa di universalità» of Rosselli’s new poetic form. This conflict results in a deep fracture within the lyrical voice.

As in [6, 3, 4, 5], we discuss published assertions concerning fixed points in “digital metric spaces” - assertions that are incorrect or incorrectly proven, or reduce to triviality.

Using the technique of enriching contractive type mappings, we introduce a more general concept of enriched Ćirić-Reich-Rus contraction than the one studied in [Berinde, V.; Păcurar, M. Fixed point theorems for enriched Ćirić-Reich-Rus contractions in Banach spaces and convex metric spaces. Carpathian J. Math. 37 (2021), no. 2, 173–184.] and provide convergence results for the Krasnoselskij iterative algorithm used to approximate their fixed points. Examples to illustrate the effectiveness of the new results as well as comparison to other classes of contractive type mappings existing in literature are also presented. In this context, we also conclude that Ćirić-Reich-Rus contractions form a class of unsaturated mappings.

We study mappings defined in the domain of a metric space that distort the modulus of families of paths by the type of the inverse Poletsky inequality. It is proved that such mappings have a continuous extension to the boundary of the domain in terms of prime ends. Under some additional conditions, the families of such mappings are equicontinuous in the closure of the domain with respect to the space of prime ends. [ABSTRACT FROM AUTHOR]

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The notion of convergence wins its own important part in the branch of Topology. Convergences in metric spaces, topological spaces, fuzzy topological spaces, fuzzy metric spaces, partially ordered sets (in short, posets), and fuzzy ordered sets (in short, fosets) develop significant chapters that attract the interest of many studies. In particular, statistical and ideal convergences play their own important role in all these areas. A lot of studies have been devoted to these special convergences, and many results have been proven. As a consequence, the necessity to produce and extend new results arises. Since there are many results on different kinds of convergences in different areas, we present a review paper on this research topic in order to collect the most essential results, which leads us to provide open questions for further investigation. More precisely, we present and gather definitions and results which have been proven for different kinds of convergences, mainly on statistical/ideal convergences, in metric spaces, topological spaces, fuzzy topological spaces, fuzzy metric spaces, posets, and fosets. Based on this presentation, we provide new open problems for further investigation on related topics. The study of these problems will create new "roads", enriching the branch of convergences in the field of Topology. [ABSTRACT FROM AUTHOR]

We establish a unique best proximity point theorem for generalized set-valued contractions on metric spaces without involving the Hausdorff distance. This result subsumes and generalizes few important fixed point and best proximity point results for set-valued mappings. In particular, our result enables us to derive the Mizoguchi-Takahashi's fixed point result for closed valued map rather than closed and bounded valued. Moreover, we obtain a best proximity point result for maps satisfying Mizoguchi-Takahashi contractions uniformly locally. [ABSTRACT FROM AUTHOR]

In this paper we will study existence, uniqueness and data dependence of the fixed points of multi-valued operators on a set endowed with two metrics. The case of multi-valued graph contractions is considered. Then, an extension to a more general contraction type condition is also given. [ABSTRACT FROM AUTHOR]

The classical Cantor's intersection theorem states that in a complete metric space X, intersection of every decreasing sequence of nonempty closed bounded subsets, with diameter approaches zero, has exactly one point. In this article, we deal with decreasing sequences {Kn} of nonempty closed bounded subsets of a metric space X, for which the Hausdorff distance H(Kn,Kn+1) tends to 0, as well as for which the excess of Kn over X \ Kn tends to 0. We achieve nonempty intersection properties in metric spaces. The obtained results also provide partial generalizations of Cantor's theorem. [ABSTRACT FROM AUTHOR]

A series of such metric spaces is constructed (subgeometric sequences of real numbers), for which the multiplication of the metric by any positive number not equal to one, gives a space at an infinite Gromov-Hausdorff distance from the original space. [ABSTRACT FROM AUTHOR]

The Hofmann-Mislove theorem says that the ordered set of open filters of the open-set lattice of a sober topological space is isomorphic to the ordered set of compact saturated sets (ordered by reverse inclusion) of that space. This paper concerns a metric analogy of this result. To this end, the notion of compact functions of approach spaces is introduced. Such functions are an analog of compact subsets in the enriched context. It is shown that for a sober approach space X, the metric space of proper open [0;∞]-filters of the metric space of upper regular functions of X is isomorphic to the opposite of the metric space of inhabited and saturated compact functions of X, establishing a Hofmann-Mislove theorem for approach spaces. [ABSTRACT FROM AUTHOR]

In this work, we study the interplay of ordered relation and the structure of the convex metric space on which it is defined: First, we show when a convex metric space is isomorphic with a convex subset of some vector space. Second, we obtain conditions for a representation of a preference relation on a convex metric space. [ABSTRACT FROM AUTHOR]

In the context of a complete metric space, the most important fixed-point result for multi-valued operators was given in 1969. Many extensions of this fixed-point principle for multi-valued operators were proved by different authors. Based on some of the above-mentioned results, we introduce the notion of the multi-valued Feng–Liu-type operator and we construct an abstract fixed-point theory for this general class of multi-valued operators. Our results extend and complement some theorems in metric fixed-point theory for multi-valued operators. An application to a Cauchy problem related to a first-order differential inclusion is also given. In this case, our theorem improves several previous theorems on this subject by relaxing the contraction-type condition (with respect to its second argument) on the multi-valued right-hand side.

This article introduces the novel concept of an extended parametric Sb-metric space, which is a generalization of both Sb-metric spaces and parametric Sb-metric spaces. Within this extended framework, we first establish an analog version of the Banach fixed-point theorem for self-maps. We then prove an improved version of the Banach contraction principle for symmetric extended parametric Sb-metric spaces, using an auxiliary function to establish the desired result. Finally, we provide illustrative examples and an application for determining solutions to Fredholm integral equations, demonstrating the practical implications of our work.

The purpose of this study is to present fixed-point results for Suzuki-type multi-valued maps using relation theory. We examine a range of implications that arise from our primary discovery. Furthermore, we present two substantial cases that illustrate the importance of our main theorem. In addition, we examine the stability of fixed-point sets for multi-valued maps and the concept of well-posedness. We present an application to a specific functional equation which arises in dynamic programming.

In this work we show that if iterates of a nonexpansive self-mapping of a complete metric with a graph converge uniformly on a subset of the space, then this convergence is stable under the presence of small computational errors.

The purpose of this paper is to prove Boyd-Wong and Matkowski type fixed point theorems in orthogonal metric space which was defined by M.E. Gordji in 2017 and is an extension of the metric space. Some examples are established in support of our main results. Finally, we apply our results to establish the existence of a unique solution of a periodic boundary value problem.

Recently, we improved our long-standing Metatheorem in Fixed Point Theory. In this paper, as its applications, some well-known order theoretic fixed point theorems are equivalently formulated to existence theorems on maximal elements, common fixed points, common stationary points, and others. Such theorems are the ones due to Banach, Nadler, Browder-Gohde-Kirk, Caristi-Kirk, Caristi, Brøndsted, and possibly many others.