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In this manuscript, utilizing interpolative contractions with fractional forms, some unique fixed-point results were studied in the context of extended $ b $-metric spaces. For the validity of the presented results some examples are given. In the last section an existence theorem is provided to study the existence of a solution for the Fredholm integral equation.

This paper presents some existence and uniqueness results for a system of mappings on the finite product of metric spaces. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math. (Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Song-il Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results.

This paper continues a series discussing flaws in published assertions concerning fixed points in digital images.

Recently, Karapınar introduced a new Hardy-Rogers type contractive mapping using the concept of interpolation and proved a fixed point theorem in complete metric space. This new type of mapping, called "interpolative Hardy-Rogers type contractive mapping" is a generalization of Hardy-Rogers's fixed point theorem. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for cyclic mappings on complete metric spaces. Moreover, an example is given to illustrate the usability of the obtained results.

We introduce a weak Gurov–Reshetnyak class and discuss its connections to a weak Muckenhoupt 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. [ABSTRACT FROM AUTHOR]

In this article, we present two resilient algorithms, the improved alternative hard c‐means (IAHCM) and the improved alternative fuzzy c‐means (IAFCM). We implement the Gaussian distance‐dependent function proposed by Zhang and Chen (D.‐Q. Zhang and Chen, 2004). In some cases, Zhang and Chen's metric distance does not account for the clustering centroid effect predicted by the large value. R* is employed in IAHCM and IAFCM to discover robust results while minimizing its sensitivity. Experiments are conducted using two‐and three‐dimensional data, including Diamond and Iris real‐world data. The results are based on demonstrating the robust simplicity and applicability of the offered algorithms. Similarly, computational complexity is assessed. [ABSTRACT FROM AUTHOR]

In this work, we analyse the class of strictly pseudocontractive mappings in general metric spaces by providing a comprehensive and appropriate definition of a strictly pseudocontractive mapping, which serves as a natural extension of the existing notion. Moreover, we establish its various characterizations and explore several significant properties of these mappings in relation to fixed point theory in CAT(0) spaces. Specifically, we establish that these mappings are Lipschitz continuous, satisfying the demiclosedness-type property, and possessing a closed convex fixed point set. Furthermore, we show that the fixed points of the mappings can be effectively approximated using an iterative scheme for fixed points of nonexpansive mappings. The results in this work contribute to a deeper understanding of strictly pseudocontractive mappings and their applicability in the context of fixed point theory in metric spaces. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the memory of the esteemed Serbian mathematician Slaviša B. Prešić (1933–2008). The primary aim of this survey paper is to compile articles on Prešić-type mappings published since 1965. Additionally, it introduces a novel class of symmetric contractions known as Prešić–Menger and Prešić–Ćirić–Menger contractions, thereby enriching the literature on Prešić-type mappings. The paper endeavors to furnish young researchers with a comprehensive resource in functional and nonlinear analysis. The relevance of Prešić's method, which generalizes Banach's theorem from 1922, remains significant in metric fixed point theory, as evidenced by recent publications. The overview article addresses the growing importance of Prešić's approach, coupled with new ideas, reflecting the ongoing advancements in the field. Additionally, the paper establishes the existence and uniqueness of fixed points in Menger spaces, contributing to the filling of gaps in the existing literature on Prešić's works while providing valuable insights into this specialized domain. [ABSTRACT FROM AUTHOR]

The analogue of polar coordinates in the Euclidean space, a polar decomposition in a metric space, if well-defined, can be very useful in dealing with integrals with respect to a sufficiently regular measure. In this note we handle the technical details associated with such polar decompositions. Supplementary Information: The online version contains the Armenian language version of the article available at https://doi.org/10.1007/s10958-023-06674-w. [ABSTRACT FROM AUTHOR]

This thesis consists of four parts. In the first chapter, the basic definitions and theorems that will be used throughout the thesis are given. In the second part, we obtain some of the best periodic proximity point results in metric spaces for p-contraction type mappings. Thus, we generalize and improve the similar results existing in the literature. In the third section, we demonstrate some of the best periodic proximity point results in metric spaces for nonunique contraction mappings. Some generalizations of both the fixed point and best proximity point results have been obtained for nonunique contraction mappings. [ABSTRACT FROM AUTHOR]

The study deals with plastic and non-plastic sub-spaces A of the real-line E with the usual Euclidean metric d. It investigates non-expansive bijections, proves properties of such maps, and demonstrates their relevance by hands of examples. Finally, it is shown that the plasticity property of a sub-space A contains at least two complementary questions, a purely geometric and a topological one. Both contribute essential aspects to the plasticity property and get more critical in higher dimensions and more abstract metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, a general idea of Presic type L-fuzzy fixed point results using some weakly contractive conditions in the setting of metric space is initiated. Stability of L-fuzzy mappings and associated new concepts are proposed herein to complement their corresponding notions related to crisp multi-valued and single-valued mappings. Illustrative nontrivial examples are provided to support the assertions of our main results. [ABSTRACT FROM AUTHOR]

Let (X, d) be a metric space, f: X → X be a mapping and G(., f(.)) be an admissible perturbation of f. In this paper we study the following problems: In which conditions imposed on f and G we have the following: (DDE) data dependence estimate for the mapping f perturbation; (UH) Ulam-Hyers stability for the equation, x = f(x); (WP) well-posedness of the fixed point problem for f; (OP) Ostrowski property of the mapping f. Some research directions are suggested. [ABSTRACT FROM AUTHOR]

G. Klir proposed to describe the behavior of complex systems using behavior functions (BFs) - invariant constraints on the set of system states. BFs are one of the most productive tools for studying the functioning of systems. To study systems, it is necessary to have a metric for measuring of the difference between two BFs. To describe BFs modern researchers do not use distributions other than probability or possibility. But these distributions can be considered as special cases of Sugeno fuzzy measures, the use of which greatly expands the possibilities in the study of systems. However, metrics to measure the difference between fuzzy measures have not been developed. Therefore, in this article, the authors proposed a new metric and an algorithm for its calculation for the case when BFs are described by Sugeno fuzzy measures. This metric is based on the Cartesian product of fuzzy measures and the use of our proposed concentration function. The metric makes it possible to compare the behavior of systems in the case of describing BFs by Sugeno fuzzy measures with different modalities, as well as to ensure the priority of taking into account the set of the most significant states of the system.

In this study, we explored a novel type of contraction, known as paired contraction (PC), to establish fixed points in metric spaces. It has been demonstrated that mappings possessing the PC property are continuous. We have also provided proofs for the existence of fixed points for such mappings with the classical Banach fixed point theorem emerging as a corollary. Furthermore, we presented examples of mappings that do not satisfy the standard contraction condition, but do exhibit the PC property.

The study deals with plastic and non-plastic sub-spaces A of the real-line ℝ with the usual Euclidean metric d. It investigates non-expansive bijections, proves properties of such maps, and demonstrates their relevance by hands of examples. Finally, it is shown that the plasticity property of a sub-space A contains at least two complementary questions, a purely geometric and a topological one. Both contribute essential aspects to the plasticity property and get more critical in higher dimensions and more abstract metric spaces.

Recent studies have highlighted fixed point theorems in the context of bicomplex valued metric spaces, utilizing rational type contractions with coefficients defined by two-variable control functions. In our research, we extend these findings by proposing new theorems for identifying common fixed points within bicomplex valued metric spaces, employing rational type contractions characterized by three-variable control functions as coefficients. We have refined the contraction conditions presented in numerous existing theorems by substituting constants with a limited number of control functions for more versatility in bicomplex valued metric spaces. This advancement broadens the scope of several significant findings in the literature. To demonstrate the efficacy of our results, we offer compelling examples that validate our theorems. Furthermore, we apply our primary findings to effectively address the Urysohn integral equation system, showcasing the practical application of our research.

In this work, a new class of general contractive mappings, with the name Jaggi-Suzuki-type hybrid (K-α-ϕ)-contractive mapping is discussed in metric space equipped with a graph and new criteria for which the mapping is a Picard operator are studied. The superiority of this type of contractive mapping lies in the fact that its contractive inequality can be fixed in different ways, depending on the specified constants. Substantial illustrations are furnished to validate the axioms of our obtained ideas and to show their difference from the existing concepts. Supplementarily, some corollaries which collapse our obtained notion to recently propounded results in the literature are brought out and analysed.

The process of sorting or categorizing objects or information about these objects into clusters according to certain criteria is a fundamental procedure in data analysis. Where it is feasible to determine the distance metric for any pair of objects, the significance and reliability of the separation can be evaluated by calculating the separation/segregation power (SP) index proposed herein. The latter index is the ratio of the average inter distance to the average intra distance, independent of the scale parameter. Here, the calculated SP value is compared to its statistical distribution obtained by a simulation study for a given partition under the homogeneity null hypothesis to draw a conclusion using standard statistical procedures. The proposed concept is illustrated using three examples representing different types of objects under study. Some general considerations are given regarding the nature of the SP distribution under the null hypothesis and its dependence on the number of divisions and the amount of data within them. A detailed modus operandi (working method) for analyzing a metric data partition is also offered. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents some existence and uniqueness results for a solution of a system of equations. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math. (Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results. [ABSTRACT FROM AUTHOR]

Recently, Karapinar introduced a new Hardy-Rogers type contractive mapping using the concept of interpolation and proved a fixed point theorem in complete metric space. This new type of mapping, called "interpolative Hardy-Rogers type contractive mapping" is a generalization of Hardy-Rogers's fixed point theorem. Following this direction of research, in this paper, we will present some fixed point results of Hardy-Rogers-type for cyclic mappings on complete metric spaces. Moreover, an example is given to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]

This paper continues a series discussing flaws in published assertions concerning fixed points in digital images. [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. [ABSTRACT FROM AUTHOR]

This article introduces the novel concept of an extended parametric S b -metric space, which is a generalization of both S b -metric spaces and parametric S b -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 S b -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. [ABSTRACT FROM AUTHOR]

In this paper, we use an inequality involving a coupled multivalued mapping and a singlevalued mapping to obtain a coupled coincidence point theorem. We discuss special conditions under which coupled common fixed point theorems are obtained. The result combines several ideas prevalent in fixed point theory studies. There are several corollaries and illustrative examples. The Hausdorff-Pompeiu metric between sets is used. The work is in the context of metric spaces and is a part of set-valued analysis with the singlevalued consequences. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a pseudometric on the family of isometry classes of (extended) metric spaces. Using it, we obtain a natural compactification of the Gromov–Hausdorff space, which is compatible with ultralimit. [ABSTRACT FROM AUTHOR]

In our recent research we studied a feasibility problem with infinitely many sets in a metric space, presented a novel algorithm and analyzed its convergence. In the present paper we study our algorithm introduced in that recent work for common fixed problem taking into account computational errors. We show that our algorithm generates a good approximate solution, if computational errors are bounded from above by a small positive constant. [ABSTRACT FROM AUTHOR]

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In this study, we explored a novel type of contraction, known as paired contraction (PC), to establish fixed points in metric spaces. It has been demonstrated that mappings possessing the PC property are continuous. We have also provided proofs for the existence of fixed points for such mappings with the classical Banach fixed point theorem emerging as a corollary. Furthermore, we presented examples of mappings that do not satisfy the standard contraction condition, but do exhibit the PC property. [ABSTRACT FROM AUTHOR]

We show that limsup sets generated by a sequence of open sets in compact Ahlfors s-regular (0 < s < ∞) space (X , 풝 , μ , ρ) belong to the classes of sets with large intersections with index λ, denoted by G λ (X) , under some conditions. In particular, this provides a lower bound on Hausdorff dimension of such sets. These results are applied to obtain that limsup random fractals with indices γ2 and δ belong to G s − δ − γ 2 (X) almost surely, and random covering sets with exponentially mixing property belong to G s 0 (X) almost surely, where s0 equals to the corresponding Hausdorff dimension of covering sets almost surely. We also investigate the large intersection property of limsup sets generated by rectangles in metric space. [ABSTRACT FROM AUTHOR]

Outlier detection is an important data mining technique. In this article, the triangle inequality of distances is leveraged to design a pre-cutoff value (PCV) algorithm that calculates the outlier degree pre-threshold without additional distance computations. This algorithm is suitable for accelerating various metric space outlier detection algorithms. Experimental results on multiple real datasets demonstrate that the PCV algorithm reduces the runtime and number of distance computations for the iORCA algorithm by 14.59% and 15.73%, respectively. Even compared to the new high-performance algorithm ADPOD, the PCV algorithm achieves 1.41% and 0.45% reductions. Notably, the non-outlier exclusion for the first data block in the dataset is significantly improved, with an exclusion rate of up to 36.5%, leading to a 23.54% reduction in detection time for that data block. While demonstrating excellent results, the PCV algorithm maintains the data type generality of metric space algorithms. [ABSTRACT FROM AUTHOR]

In this article, we introduce the fundamentals of the theory of demicontractive mappings in metric spaces and expose the key concepts and tools for building a constructive approach to approximating the fixed points of demicontractive mappings in this setting. By using an appropriate geodesic averaged perturbation technique, we obtained strong convergence and Δ-convergence theorems for a Krasnoselskij-Mann type iterative algorithm to approximate the fixed points of quasinonexpansive mappings within the framework of CAT(0) spaces. The main results obtained in this nonlinear setting are natural extensions of the classical results from linear settings (Hilbert and Banach spaces) for both quasi-nonexpansive mappings and demicontractive mappings. We applied our results to solving an equilibrium problem in CAT(0) spaces and showed how we can approximate the equilibrium points by using our fixed point results. In this context we also provided a numerical example in the case of a demicontractive mapping that is not a quasi-nonexpansive mapping and highlighted the convergence pattern of the algorithm in Table 1. It is important to note that the numerical example is set in non-Hilbert CAT(0) spaces. [ABSTRACT FROM AUTHOR]

The present paper is on fixed point theory in set valued analysis. Here putting several concepts together, we define a set valued contraction and establish that such contractions have fixed points in a complete metric space. We have two main theorems in one of which we use α -regularity condition of the space as a substitute of the continuity requirement on the multivalued contraction. There are some illustrative examples and corollaries. We also perform stability analysis of fixed point sets. We establish that the fixed point sets associated with the mappings we consider in our theorems are stable. In the last section we discuss some consequences of the main results of this paper. [ABSTRACT FROM AUTHOR]

In this article, we introduce the fundamentals of the theory of demicontractive mappings in metric spaces and expose the key concepts and tools for building a constructive approach to approximating the fixed points of demicontractive mappings in this setting. By using an appropriate geodesic averaged perturbation technique, we obtained strong convergence and $ \Delta $-convergence theorems for a Krasnoselskij-Mann type iterative algorithm to approximate the fixed points of quasi-nonexpansive mappings within the framework of CAT(0) spaces. The main results obtained in this nonlinear setting are natural extensions of the classical results from linear settings (Hilbert and Banach spaces) for both quasi-nonexpansive mappings and demicontractive mappings. We applied our results to solving an equilibrium problem in CAT(0) spaces and showed how we can approximate the equilibrium points by using our fixed point results. In this context we also provided a numerical example in the case of a demicontractive mapping that is not a quasi-nonexpansive mapping and highlighted the convergence pattern of the algorithm in Table 1. It is important to note that the numerical example is set in non-Hilbert CAT(0) spaces.

This paper continues a series discussing flaws in published assertions concerning fixed points in digital metric spaces.

In this paper, we introduce four new types of contractions called in this order Kannan-type cyclic contraction via interpolation, interpolative Ćirić-Reich-Rus type cyclic contraction, and we prove the existence and uniqueness for a fixed point for each situation.

The well-known Caristi fixed point theorem has numerous generalizations and modifications. Recently there have appeared its equivalent dual forms and generalizations based on new concept of lower semicontinuity from above by several authors. In the present article, we give new proofs of such new versions and their equivalent formulations by applying our Metatheorem in the ordered fixed point theory.

Let (X,d) be a quasi-metric space. A Rus-Hicks-Rhoades (RHR) map f:X→X is the one satisfying d(fx,f2x)≤αd(x,fx) for every x∈X, where α∈[0,1). In our previous work [37], we collected various fixed-point theorems closely related to RHR maps. In the present article, we collect almost all the things we know about RHR maps and their examples. Moreover, we derive new classes of generalized RHR maps and fixed point theorems on them. Consequently, many of the known results in metric fixed point theory are improved and reproved in an easy way.

Any two points are close together in a

In order to study Maia type fixed point results for several well-known contractions, we suggest two novel contractions called $ \mathcal{A} $-contraction and $ \mathcal{A}' $-contraction. The majority of the Maia type fixed point results for various contractions can now be unified through these, which eliminate the need to manage various contractions individually. The advantage of including such contractions in the study of Maia type fixed point results has been demonstrated in suitable examples. We present an application of one of our established results towards the conclusion of the paper.

This paper deals with the existence and uniqueness of common fixed points in metric and dislocated metric spaces. In addition, an application and some illustrative examples are given in order to justify the validity and credibility of our results, also, their superiority over some similar theorems existing in unique common fixed points theory’s domain.

Topology is branch of modern mathematics that plays an important role in applications of biology. The aim of this paper is to study DNA sequence mutations using multisets, relations, metric functions, topology and association indices. Moreover, we use association indices to study the similarity between DNA sequences. These different ways of identifying a mutation help biologists to make a decision. A decision of mutation that depends on metrics between two sequences of genes and the topological structure produced by their relationship is presented.

This paper is dedicated to the memory of the esteemed Serbian mathematician Slaviša B. Prešić (1933–2008). The primary aim of this survey paper is to compile articles on Prešić-type mappings published since 1965. Additionally, it introduces a novel class of symmetric contractions known as Prešić–Menger and Prešić–Ćirić–Menger contractions, thereby enriching the literature on Prešić-type mappings. The paper endeavors to furnish young researchers with a comprehensive resource in functional and nonlinear analysis. The relevance of Prešić’s method, which generalizes Banach’s theorem from 1922, remains significant in metric fixed point theory, as evidenced by recent publications. The overview article addresses the growing importance of Prešić’s approach, coupled with new ideas, reflecting the ongoing advancements in the field. Additionally, the paper establishes the existence and uniqueness of fixed points in Menger spaces, contributing to the filling of gaps in the existing literature on Prešić’s works while providing valuable insights into this specialized domain.

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. [ABSTRACT FROM AUTHOR]

We establish a stability theorem for geometric separating flows on metric spaces, which may or may not be compact. Notably, this result extends Thomas' stability theorem (Proc London Math Soc. 45:479–505 1982). To prove our theorem, we employ set-valued analysis (Aubin and Frankowska, 1990). We give some applications of our result. [ABSTRACT FROM AUTHOR]

What is the analogous notion of Gromov–Hausdorff convergence for sequences of spacetimes? Since a Lorentzian manifold is not inherently a metric space, one cannot simply use the traditional definition. One approach offered by Sormani and Vega (Class Quant Gravity, 33:085001, 2016) is to define a metric space structure on a spacetime by means of the null distance. Then one can define convergence of spacetimes using the usual definition of Gromov–Hausdorff convergence. In this paper we explore this approach by giving many examples of sequences of warped product spacetimes with the null distance converging in the Gromov–Hausdorff sense. In addition, we give an optimal convergence theorem which shows that under natural geometric hypotheses a sequence of warped product spacetimes converge to a specific limiting warped product spacetime. The examples given further serve to show that the hypotheses of this convergence theorem are optimal. [ABSTRACT FROM AUTHOR]