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The aim of this paper is to construct a fractal of generalized Θ-Hutchinson Operator with the help of a finite family of Θ-contraction mappings, a class of mappings more general than contractions, defined on a complete metric space. Our results unify, generalize and extend various results in the existing literature. [ABSTRACT FROM AUTHOR]

In many fields, e.g., data mining and machine learning, distance-based outlier detection (DOD) is widely employed to remove noises and find abnormal phenomena, because DOD is unsupervised, can be employed in any metric spaces, and does not have any assumptions of data distributions. Nowadays, data mining and machine learning applications face the challenge of dealing with large datasets, which requires efficient DOD algorithms. We address the DOD problem with two different definitions. Our new idea, which solves the problems, is to exploit an in-memory proximity graph. For each problem, we propose a new algorithm that exploits a proximity graph and analyze an appropriate type of proximity graph for the algorithm. Our empirical study using real datasets confirms that our DOD algorithms are significantly faster than state-of-the-art ones. [ABSTRACT FROM AUTHOR]

In this paper we introduce some new types of contractive mappings by combining Caristi contraction, Ćirić-quasi contraction and weak contraction in the framework of a metric space. We prove some fixed point theorems for such type of mappings over complete metric spaces with the help of φ-diminishing property. Some examples are given in strengthening the hypothesis of our established theorems.

In this paper, we obtain a fixed point theorem ω- ψ-interpolative Hardy-Rogers contractive of Suzuki type mappings. In the following, we present an example to illustrate the new theorem is applicable. Subsequently, some results are given. These results generalize several new results present in the literature.

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.

In the commenced work, we establish some best proximity point results for multivalued generalized contractions on partially ordered complete metric spaces along with the tactic of altering distance function. Furthermore, we deliver some examples to elaborate and explain the usability of the attained results. To arouse further interest in the subject and to show its efficacy, we devote this work to recent applications of fractional calculus and also invoke our findings to the equation of motion modeling to differential equations. [ABSTRACT FROM AUTHOR]

In the context of a complete metric space, we will consider the common fixed point problem for two self operators. The operators are assumed to satisfy a general contraction type condition inspired by the Ćirić fixed point theorems. Under some appropriate conditions we establish existence, uniqueness and approximation results for the common fixed point. In the same framework, the second problem is to study various stability properties. More precisely, we will obtain sufficient conditions assuring that the common fixed point problem is well-posed and has the Ulam–Hyers stability, as well as the Ostrowski property for the considered problem. Some examples and applications are finally given in order to illustrate the abstract theorems proposed in the first part of the paper. Our results extend and complement some theorems in the recent literature. [ABSTRACT FROM AUTHOR]

We introduce a new pair of mappings (S , T) on D * -metric spaces called D S * -W.C. and D R S * -W.C. Many examples are presented to show the difference between these mappings and other types of mappings in the literature. Moreover, we obtain several common fixed point results by using these types of mappings and the (E.A) property. We then employ the fixed point results to establish the existence and uniqueness of a solution for a class of nonlinear integral equations. [ABSTRACT FROM AUTHOR]

In the present paper we study the relationships between different concepts of convex structures in metric spaces that that are related to the works of K. Menger [Menger, K. Untersuchungen über allgemeine Metrik. Math. Ann. 100 (1928), 75-163], H. Busemann [Busemann, H. The geometry of geodesics, Academic Press, 1955], I. N. Herstein; J. Milnor [Herstein, I. N.; Milnor, J. An axiomatic approach to measurable utility. Econometrica 21 (1953), 291-297], E. Michael [Michael, E. Convex structures and continuous selections. Canad. J. Math. 11 (1959), 556-575], A. Nijenhuis [Nijenhuis, A. A note on hyperconvexity in Riemannian manifolds. Canad. J. Math. 11 (1959), 576-582.], W. Takahashi; T. Shimizu [Shimizu, T.; Takahashi, W. Fixed points of multivalued mappings in certain metric spaces. Topol. Methods Nonlinear Anal. 8 (1996), no. 1, 197-203 and Takahashi, W. A convexity in metric space and nonexpansive mappings I. Kodai Math. Sem. Rep. 22 (1970), 142-149], M. Taskovi'c [Taskovi'c, M. General convex topological spaces and fixed points. Math. Moravica 1 (1997), 127-134], Yu. A. Aminov [Aminov, Yu. A. Two-Dimensional Surfaces in 3-Dimensional and 4-Dimensional Euclidean Spaces. Results and Unsolved Problems. Ukr. Math. J. 71 (2019), no. 1, 1-38.], H. V. Machado [Machado, H. V. A characterization of convex subsets of normed spaces. Kodai Math. Sem. Rep. 25 (1973), 307-320], and many other papers. Some well known examples of concrete convex structures are reexamined and the possibilities of different embeddings of metric spaces with convex structures are also studied. Corollary 6.1 states that the Bolyai-Lobachevskii plane and the Bolyai-Lobachevskii half-plane are not isometrically embedding in some strictly convex normed space. A characteristic of the invariant metric generated by a norm is presented (Proposition 4.6). [ABSTRACT FROM AUTHOR]

In this article, we introduce the concept of tripled controlled V -metric spaces and study Kannan type contractions in this new structure. Also, we extend the concept of Kannan type contractions to Kannan-Presic type contractions. Our results modify some known ones in the literature. To support our main results, an example and an application in integral equations are presented. [ABSTRACT FROM AUTHOR]

Any information is (transported as) a selection from an ordered set, which is the "domain" of the information. For example, any piece of digital information is a number sequence that represents such a selection. Its senders and receivers (with software) should know the format and domain of the number sequence in a uniform way worldwide. So far, this is not guaranteed. However, it can be guaranteed after the introduction of the new "Domain Vector" (DV) data structure: "UL plus number sequence". Thereby "UL" is a "Uniform Locator", which is an efficient global pointer to the machine-readable online definition of the number sequence. The online definition can be adapted to the application so that the DV represents the application-specific, reproducible features in a precise (one-to-one), comparable, and globally searchable manner. The systematic, nestable online definition of domains of digital information (number sequences) and the globally defined DV data structure have great technical potential and are recommended as a central focus of future computer science. [ABSTRACT FROM AUTHOR]

The trichotomy between regular, semiregular, and strongly irregular boundary points for p -harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a p -Poincaré inequality, 1 < p < ∞ . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, p -harmonic measures, removability, and semibarriers. [ABSTRACT FROM AUTHOR]

The purpose of this manuscript is to demonstrate the existence and uniqueness of triple fixed-point results for Geraghty-type contractions in ordinary metric spaces with binary relations. Moreover, the well-posedness of the tripled fixed point problem is investigated. Consequently, α -dominated mapping on such space is discussed. Ultimately, to promote our paper, some illustrative examples are derived, and the existence of the solution to a system of differential equations is obtained as an application. [ABSTRACT FROM AUTHOR]

In 2014, Jleli and Samet [6] introduced a contraction mapping called θ-contraction. After this, many authors had given their contribution on φ − θ-contraction (see [7, 8]). Motivated by [[6, 9]], we introduce the notion of generalized φ − θ contraction and establish some new fixed point theorems for this contraction in the setting of complete partial metric spaces. [ABSTRACT FROM AUTHOR]

We study mappings that satisfy the so-called inverse Poletsky inequality. Provided that the given domain has a weakly flat boundary, and the mapped domain is locally connected on it, we have established a continuous boundary extension of the indicated mappings. Under some additional conditions, the corresponding class of mappings is equicontinuous at the inner and boundary points of a given domain. In particular, we have established a result on the equicontinuity of families of mappings with the inverse Poletskii inequality and one normalization condition. As an example, we have constructed a family of corresponding mappings acting between quotient spaces along two fixed groups of Möbius mappings. [ABSTRACT FROM AUTHOR]

We study the so-called John–Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John–Nirenberg inequalities, which give weak-type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calderón–Zygmund decomposition and a good- λ inequality for medians. A John–Nirenberg inequality up to the boundary is proven by using chaining arguments. As a consequence, the integral-type and the median-type John–Nirenberg spaces coincide under a Boman-type chaining assumption. [ABSTRACT FROM AUTHOR]

Many geometric optimization problems can be reduced to finding points in space (centers) minimizing an objective function which continuously depends on the distances from the centers to given input points. Examples are k-Means, Geometric k-Median/Center, Continuous Facility Location, m-Variance, etc. We prove that, for any fixed ε > 0 , every set of n input points in fixed-dimensional space with the metric induced by any vector norm admits a set of O(n) candidate centers which can be computed in almost linear time and which contains a (1 + ε) -approximation of each point of space with respect to the distances to all the input points. It gives a universal approximation-preserving reduction of geometric center-based problems with arbitrary continuity-type objective functions to their discrete versions where the centers are selected from a fairly small set of candidates. The existence of such a linear-size set of candidates is also shown for any metric space of fixed doubling dimension. [ABSTRACT FROM AUTHOR]

In a metric space $ (X, d) $, a line induced by two distinct points $ x, x'\in X $, denoted by $ \mathcal{L}\{x, x'\} $, is the set of points given by $ \mathcal{L}\{x, x'\} = \{z\in X:\, d(x, x') = d(x, z)+d(z, x') \text{ or }d(x, x') = |d(x, z)-d(z, x')|\}. $ A line $ \mathcal{L}\{x, x'\} $ is universal whenever $ \mathcal{L}\{x, x'\} = X $. Chen and Chvátal [Discrete Appl. Math. 156 (2008), 2101-2108.] conjectured that every finite metric space on $ n\ge 2 $ points either has at least $ n $ distinct lines or has a universal line. In this paper, we prove this conjecture for some classes of metric spaces. In particular, we discuss the classes of Cartesian metric spaces, lexicographic metric spaces and corona metric spaces.

We investigate a fixed point problem for coupled Geraghty type contraction in a metric space with a binary relation. The role of the binary relation is to restrict the scope of the contraction to smaller number of ordered pairs. Such possibilities have been explored for different types of contractions in recent times which has led to the emergence of relational fixed point theory. Geraghty type contractions arose in the literatures as a part of research seeking the replacement contraction constants by appropriate functions. Also coupled fixed point problems have evoked much interest in recent times. Combining the above trends we formulate and solve the fixed point problem mentioned above. Further we show that with some additional conditions such solution is unique. Well-posedness of the problem is investigated. An illustrative example is discussed. The consequences of the results are discussed considering α-dominated mappings and graphs on the metric space. Finally we apply our result to show the existence of solution of some system of nonlinear integral equations.

In the present paper, we introduce the notion of (φ, ψ)-convex contraction mapping of order m and establish a fixed point theorem for such mappings in complete metric spaces. The present result extends and generalizes the well known result of Dutta and Choudhary (Fixed Point Theory Appl. 2008 (2008), Art. ID 406368), Rhoades (Nonlinear Anal., 47(2001), 2683-2693), Istr µescu (Ann. Mat. Pura Appl., 130(1982), 89-104) and besides many others in the existing literature. An illustrative example is also provided to exhibit the utility of our main results.

Given an n -point metric space (M , d) , metric 1 -median asks for a point p ∈ M minimizing ∑ x ∈ M d (p , x). We show that for each computable function f : Z + → Z + satisfying f (n) = ω (1) , metric 1 -median has a deterministic, o (n) -query, o (f (n) ⋅ log ⁡ n) -approximation and nonadaptive algorithm. Previously, no deterministic o (n) -query o (n) -approximation algorithms are known for metric 1 -median. On the negative side, we prove each deterministic O (n) -query algorithm for metric 1 -median to be not (δ log ⁡ n) -approximate for a sufficiently small constant δ > 0. We also refute the existence of deterministic o (n) -query O (log ⁡ n) -approximation algorithms. [ABSTRACT FROM AUTHOR]

Fixed point theory and contractive mappings are popular tools in solving a variety of problems such as control theory, economic theory, nonlinear analysis and global analysis.There are many works on different types of contractions to find a fixed point in metric spaces. Improving and extending some kind of those, in this paper, we introduce a new version of H-contradiction for four mappings in a metric space (X,d). Then, we prove the existence and uniqueness of a common best proximity point for four non-self mappings. An example is also given to support our main result. The related fixed point theorem are also proved.

We show that the standard partition of unity subordinate to an open cover of a metric space has Lipschitz constant max ⁡ (1 , M − 1) / L , where L is the Lebesgue number and M is the multiplicity of the cover. If the metric space satisfies the approximate midpoint property, as length spaces do, then the upper bound improves to (M − 1) / (2 L). These Lipschitz estimates are optimal. We also address the Lipschitz analysis of ℓ p -generalizations of the standard partition of unity, their partial sums, and their categorical products. Lastly, we characterize metric spaces with Assouad–Nagata dimension n as exactly those metric spaces for which every Lebesgue cover admits an open refinement with multiplicity n + 1 while reducing the Lebesgue number by at most a constant factor. [ABSTRACT FROM AUTHOR]

In the present study, we commenced by presenting a new class of maps, termed noncyclic (φ , R t) -enriched quasi-contractions within metric spaces equipped with a transitive relation 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. [ABSTRACT FROM AUTHOR]

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.

It is well known that of all the extensions of the Banach-Caccioppoli Contraction Principle, the most general result was established by ćirić in 1974. In this paper, we will present some results related to ćirić type operator in com-plete metric spaces. Existence and uniqueness are re-called and several stability properties (data dependence and Ostrowski stability property) are proved. Using the retraction-displacement condition, we will establish the well-posedness and the Ulam-Hyers stability property of the fixed point equation x = f(x). [ABSTRACT FROM AUTHOR]

We make a systematic study of frames for metric spaces. We prove that every separable metric space admits a metric M d -frame. Through Lipschitz-free Banach spaces we show that there is a correspondence between frames for metric spaces and frames for subsets of Banach spaces. We derive some characterizations of metric frames. We also derive stability results for metric frames. [ABSTRACT FROM AUTHOR]

In this paper, unique common fixed point theorems for pairs of subcompatible and reciprocally continuous mappings or compatible and subsequentially continuous mappings are obtained. This unique common fixed point is guaranteed under a new concept named a weak altering distance function. Our theorems improve some results especially the ones of [2]. [ABSTRACT FROM AUTHOR]

In this paper, some common fixed point theorems for pairs of subcompatible and reciprocally continuous mappings or compatible and subsequentially continuous mappings satisfying integral type are obtained. Our results improve several results especially the results of [3], Pathak et al. [13], Djoudi and Aliouche [6], Mbarki [10] and others. [ABSTRACT FROM AUTHOR]

Sufficient conditions are given for the existence of solutions of bifunction-set optimization problems, whose underlying sets are complete metric spaces. The main result of this paper is formulated in Theorem 3.1, with the help of some new data. The advantage of introducing the new data is that, by choosing suitable new data, we can obtain existence results with assumptions, that are imposed directly on the objective maps, or that are formulated via scalarization. The main result is applied successfully to Kuroiwa set optimization problems and Ky Fan vector inequality problems. We also discuss equivalences between the existence of solutions of the bifunction-set optimization problem, an Ekeland-type variational principle and a Caristi-type fixed point theorem. Examples are provided. [ABSTRACT FROM AUTHOR]

In this paper we establish a coupled fixed point theorem of a coupled multivalued mapping defined on a complete metric space. In our result we use a new contractive inequality. There are rational terms in the expression of the inequality. The contractive condition on the nonlinear map is assumed only to hold on the elements that are related by a binary relation. The main theorem has several corollaries and is illustrated with example. The main result is deduced in metric spaces. Its consequences are discussed for α-admissible mapping as well as in metric spaces with partial order and graph respectively. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this study, using a lacunary sequence we introduce the concepts of lacunary d-statistically convergent sequences of order α and lacunary d-statistically bounded sequences of order α in general metric spaces. [ABSTRACT FROM AUTHOR]

The aim of this manuscript is to propose a contraction to pursue the existence of g-best proximity point results. The finding of this manuscript generalize and unify the results of Rohen and Mlaiki by using the new contraction with P-property and prove the existence and uniqueness of quadruple best proximity point alongwith an example. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of positive radial solutions to the iterative system of nonlinear elliptic equations of the form ▵ z j + (N - 2) 2 r 0 2 N - 2 | x | 2 N - 2 z j + φ (| x |) g j (z j + 1) = 0 , R 1 < | x | < R 2 , where j ∈ { 1 , 2 , 3 , · · · , ℓ } , z 1 = z ℓ + 1 , ▵ z = div (▿ z) , N > 2 , 0 < r 0 < π / 2 , φ = ∏ i = 1 n φ i , each φ i : (r 0 , + ∞) → (0 , + ∞) is continuous, r N - 1 φ is integrable, and g j : [ 0 , + ∞) → R is continuous, by an application of various fixed point theorems in a Banach space. Further, we also establish uniqueness of the solution for the addressed system by using Rus's theorem in a complete metric space. [ABSTRACT FROM AUTHOR]

We explore the model-theoretic concept of polymorphism-homogeneity, a natural generalisation of the classical homogeneity, for certain classes of metric spaces, viewed as relational structures in an obvious way. At first, we prove that ultrametric spaces are polymorphism-homogeneous. Subsequently, we proceed to a thorough investigation of the so-called skeletal metric spaces, with respect to the diameter of their skeleton, a specific unique graph corresponding to them. Consequently, we obtain a characterization of metrically polymorphism-homogeneous connected graphs of diameter at most 3. [ABSTRACT FROM AUTHOR]

Similarity search in metric spaces is used widely in areas such as multimedia retrieval, data mining, data integration, to name but a few. To accelerate metric similarity search, pivot-based indexing is often employed. Pivot-based indexing first computes the distances between data objects and pivots and then exploits filtering techniques that use the triangle inequality on pre-computed distances to prune search space during search. The performance of pivot-based indexing depends on the quality of the pivots used, and many algorithms have been proposed for selecting high-quality pivots. We present a comprehensive empirical study of pivot selection algorithms. Specifically, we classify all existing algorithms into three categories according to the types of distances they use for selecting pivots. We also propose a new pivot selection algorithm that exploits the power law probabilistic distribution. Next, we report on a comprehensive empirical study of the search performance enabled by different pivot selection approaches, using different datasets and indexes, thus contributing new insight into the strengths and weaknesses of existing selection techniques. Finally, we offer advice on how to select appropriate pivot selection algorithms for different settings. [ABSTRACT FROM AUTHOR]

In this paper, the researcher proved the best proximity point theorem for Suzuki type mappings in the setting of metric spaces endowed a graph. In particular, some earlier results in the literature on both best proximity theory and metric fixed point theory were enriched, extended, and at last generalized. [ABSTRACT FROM AUTHOR]

Introduction/purpose: This paper establishes some new results of Piri– Kumam–Dung-type mappings in a complete metric space.Тhe goal was to improve the already published results. Methods: Using the property of a strictly increasing function as well as the known Lemma formulated in (Radenović et al, 2017), the authors have proved that a Picard sequence is a Cauchy sequence. Results: New results were obtained concerning the F – contraction mappings of S in a complete metric space. To prove it, the authors used only property (W1). Conclusion:The authors believe that the obtained results represent a significant improvement of many known results in the existing literature.

We continue the work of [4, 2, 3], in which we discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality; or that we improve upon.

Abstract The aim of this paper is to present several fixed-point results for L-contractive multivalued mappings involving θ-functions in the class of metric spaces. We also give some examples in support of the related concepts and presented results. A homotopy result is also provided.