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Assessment of rock mass quality significantly impacts the design and construction of underground and open-pit mines from the point of stability and economy. This study develops the novel Gromov-Hausdorff distance for rock quality (GHDQR) methodology for rock mass quality rating based on multi-criteria grey metric space. It usually presents the quality of surrounding rock by classes (metric spaces) with specified properties and adequate interval-grey numbers. Measuring the distance between surrounding rock sample characteristics and existing classes represents the core of this study. The Gromov-Hausdorff distance is an especially useful discriminant function, i.e. a classifier to calculate these distances, and assess the quality of the surrounding rock. The efficiency of the developed methodology is analyzed using the Mean Absolute Percentage Error (MAPE) technique. Seven existing methods, such as the Gaussian cloud method, Discriminant method, Mutation series method, Artificial neural network (ANN), Support vector machine (SVM), Grey wolf optimizer and Support vector classification method (GWO-SVC) and Rock mass rating method (RMR) are used for comparison with the proposed GHDQR method. The share of the highly accurate category of 85.71% clearly indicates compliance with actual values obtained by the compared methods. The results of comparisons showed that the model enables objective, efficient, and reliable assessment of rock mass quality. [ABSTRACT FROM AUTHOR]

We introduce a new type of mappings in metric spaces which are three-point analogue of the well-known Chatterjea type mappings, and call them generalized Chatterjea type mappings. It is shown that such mappings can be discontinuous as is the case of Chatterjea type mappings and this new class includes the class of Chatterjea type mappings. The fixed point theorem for generalized Chatterjea type mappings is proven. [ABSTRACT FROM AUTHOR]

In 2018, Erdal Karapinar introduced the concept of interpolative Kannan operators, a novel adaptation of the Kannan mapping originally defined in 1969 by Kannan. This new mapping condition is more lenient than the basic contraction condition. In this paper, we study the concept by introducing the IKC-iterated function/multi-function system using interpolative Kannan operators, including a broader area of mappings. Moreover, we establish the Collage Theorem endowed with the iterated function system (IFS) based on the IKC, and show the well-posedness of the IKC-IFS. Interpolative Kannan contractions are meaningful due to their applications in fractals, offering a more versatile framework for creating intricate geometric structures with potentially fewer constraints compared to classical approaches. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a new type of multi-valued operators and to present some basic problems of the fixed points and strict fixed points for them. Obtained results generalize, complement and extend classical results given by Ćirić (Mat. Vesnik 9 (24): 265-272, (1972)) or Nadler (Pacific J. Math. 30: 475-488 (1969)), as well as recent results given by Alecsa and PetruŞel (Anal. Univ. Vest Timisoara, LVII (1): 23-42 (2019)). [ABSTRACT FROM AUTHOR]

The Roe algebra C * (X) is a noncommutative C * -algebra reflecting metric properties of a space X, and it is interesting to understand the correlation between the Roe algebra of X and the (uniform) Roe algebra of its discretization. Here, we perform a minor step in this direction in the simplest non-trivial example, namely X = R , by constructing a continuous field of C * -algebras over [ 0 , 1 ] , with the fibers over non-zero points constituting the uniform C * -algebra of the integers, and the fibers over 0 constituting a C * -algebra related to R. [ABSTRACT FROM AUTHOR]

In the present study, we prove generalizations of Banach, Kannan, Chatterjea, Čirič-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 Pâles. 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. [ABSTRACT FROM AUTHOR]

This article is a revision and correction of the chapter book [S. Hadi Bonab, V. Parvaneh, Z. Bagheri, ηA-Admissible Mappings for Four Maps in C*-Algebra-Valued MP-Metric Spaces with an Application, In: P. Debnath, Delfim F. M. Torres, Yeol Je Cho, Advanced Mathematical Analysis and its Applications, CRC Press, 2023, 97-113.]. In this article, we first introduce the concept of η-admissible mapping in C* -algebra valued MP-metric spaces, which is a generalization and combination of ”modular metric spaces”, ”parametric metric spaces” and ”C* -algebra-valued metric spaces”. Then, for four mappings in these spaces, we prove several fixed-point theorems. We give an example and an application regarding the solvability of operator equations and integral equations, respectively, to support the new findings. [ABSTRACT FROM AUTHOR]

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.

Metric spaces form a subfamily of topological spaces, or some appropriate non-empty spaces. The fixed point can, almost always, be found using the Picard iteration (the most widely used fixed point iteration), as one can start from any initial point 0-x in space. Finding fixed points for mappings depends mainly on the settings of the studied spaces, which are defined using some intuitive axioms. Different classes of generalized spaces and several contractions will yield new dynamic research areas and thus different types of uniform fixed point conjectures at once. Most of the effort expended on fixed point theory has been directed at specifying a variety of applicable and easily verifiable sufficient conditions for fixed point problems. The paper consists of a variety of advanced discussions and contemporary topics on metric fixed point theory and its applications to present the feasibility of the results. The research framework undoubtedly contains pure theoretical mathematics. We explore the ideal combination of relaxed conditions to prove our new theorems and realization of the idea in full detail to prove all the results obtained using different, applicable and high-tech proof modes. The results of this research are theoretical and analytical in nature. This study follows the recent trend and latest development in the analysis of metric fixed point theory, and provides a very sound basic text on this theory. This research brings together selected chapters on modern topics of fixed point theory and its applications, each of which is divided into several sections, numbered gradually. [ABSTRACT FROM AUTHOR]

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. [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]

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. [ABSTRACT FROM AUTHOR]

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 work, the set of all functions that are Fourier transformable with regard to their structure both algebraic and topological is taken into account. Certain topological properties of the set of Fourier transformable functions with the help of a metric are described. Also determines the proofs of the statements that the set of all Fourier transformable functions is a commutative semigroup with respect to the convolution operation as well as Abelian group with respect to the operation of addition. Metric for two functions belonging to the set of all Fourier transformable functions is defined and the proof that the Fourier transformable functions space is complete with our metric is given. The separability theorem and that the Fourier transformable functions space is disconnected are also discussed. [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 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 the present study, we commenced by presenting a new class of maps, termed noncyclic $ (\varphi, \mathcal{R}^t) $-enriched quasi-contractions within metric spaces equipped with a transitive relation $ \mathcal{R}^t $. Subsequently, we identified the conditions for the existence of an optimal pair of fixed points pertaining to these mappings, thereby extending and refining a selection of contemporary findings documented in some articles. Specifically, our analysis will encompass the outcomes pertinent to reflexive and strictly convex Banach spaces.

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 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]

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]

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]

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 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]

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 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]

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]

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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 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.

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.

In 2018, Erdal Karapinar introduced the concept of interpolative Kannan operators, a novel adaptation of the Kannan mapping originally defined in 1969 by Kannan. This new mapping condition is more lenient than the basic contraction condition. In this paper, we study the concept by introducing the IKC-iterated function/multi-function system using interpolative Kannan operators, including a broader area of mappings. Moreover, we establish the Collage Theorem endowed with the iterated function system (IFS) based on the IKC, and show the well-posedness of the IKC-IFS. Interpolative Kannan contractions are meaningful due to their applications in fractals, offering a more versatile framework for creating intricate geometric structures with potentially fewer constraints compared to classical approaches.

The Roe algebra C*(X) is a noncommutative C*-algebra reflecting metric properties of a space X, and it is interesting to understand the correlation between the Roe algebra of X and the (uniform) Roe algebra of its discretization. Here, we perform a minor step in this direction in the simplest non-trivial example, namely X=R, by constructing a continuous field of C*-algebras over [0,1], with the fibers over non-zero points constituting the uniform C*-algebra of the integers, and the fibers over 0 constituting a C*-algebra related to R.