Search

Abstract 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.

Fixed points results for rational type contractions in metric spaces have been widely studied in the literature. In the last years, many of these results are obtained in the context of partially ordered metric spaces. In this paper, we introduce a fixed point principle for a class of mappings between partially ordered metric spaces that we call orbitally order continuous. We show that the hypotheses in the statement of such a principle are not redundant and, in addition, that they cannot be weakened in order to guarantee the existence of a fixed point. Moreover, the relationship between this kind of mappings and those that are continuous and orbitally continuous is discussed. As an application, we extend many fixed point theorems for continuous contractions of rational type to the framework of those that are only orbitally order continuous. Furthermore, we get extensions of the aforementioned metric fixed point results to the framework of partial metrics. This is achieved thanks to the fact that each partial metric induces in a natural way a metric in such a way that our new principle is applicable. In both approaches, the metric and the partial metric, we show that there are orbitally order continuous mappings that satisfy all assumptions in our new fixed point principle but that they are not contractions of rational type. The explored theory is illustrated by means of appropriate examples.

In this article, we introduce the study of corona metric space. After discussing some basic properties of these metric spaces, such as completeness, boundedness, compactness and separability, we provide a necessary and sufficient condition for the existence of universal lines, and we obtain a formula for the metric dimension of corona metric spaces.

The classical notion of statistical convergence has recently been transported to the scope of real normed spaces by means of the $ f $-statistical convergence for $ f $ a modulus function. Here, we go several steps further and extend the $ f $-statistical convergence to the scope of uniform spaces, obtaining particular cases of $ f $-statistical convergence on pseudometric spaces and topological modules.

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.

We consider Fréchet sufficient dimension reduction with responses being complex random objects in a metric space and high-dimensional Euclidean predictors. We propose a novel approach, called the weighted inverse regression ensemble method, for linear Fréchet sufficient dimension reduction. The method is further generalized as a new operator defined on reproducing kernel Hilbert spaces for nonlinear Fréchet sufficient dimension reduction. We provide theoretical guarantees for the new method via asymptotic analysis. Intensive simulation studies verify the performance of our proposals, and we apply our methods to analyse handwritten digit data and real-world affective face data to demonstrate its use in real applications. [ABSTRACT FROM AUTHOR]

In this paper, we define θ-expansions on Branciari metric spaces by complementing the concept of θ-contractions introduced by Jleli and Samet (J. Inequal. Appl. 2014:38, 2014). Also, we present some new fixed point results for θ-expansion mappings on a Branciari metric space. [ABSTRACT FROM AUTHOR]

Fixed point theory is a crucial branch in mathematics with a colossal number of applications in countless subjects. It furnishes preeminent tools and resources for elucidating varied problems which at first glance do not look like a fixed point problem. Since and even before 1912 till now several authors launched the existence and uniqueness of common fixed points for pairs of single and set-valued maps satisfying certain compatibilities on different spaces. This paper proves existence and uniqueness of a common fixed point for pairs of occasionally weakly biased maps. This unique common fixed point is guaranteed under the concept of modified contractive modulus function. Our theorems improve some results existing in the fixed point literature. [ABSTRACT FROM AUTHOR]

Samples of dynamic or time-varying networks and other random object data such as time-varying probability distributions are increasingly encountered in modern data analysis. Common methods for time-varying data such as functional data analysis are infeasible when observations are time courses of networks or other complex non-Euclidean random objects that are elements of general metric spaces. In such spaces, only pairwise distances between the data objects are available and a strong limitation is that one cannot carry out arithmetic operations due to the lack of an algebraic structure. We combat this complexity by a generalized notion of mean trajectory taking values in the object space. For this, we adopt pointwise Fréchet means and then construct pointwise distance trajectories between the individual time courses and the estimated Fréchet mean trajectory, thus representing the time-varying objects and networks by functional data. Functional principal component analysis of these distance trajectories can reveal interesting features of dynamic networks and object time courses and is useful for downstream analysis. Our approach also makes it possible to study the empirical dynamics of time-varying objects, including dynamic regression to the mean or explosive behavior over time. We demonstrate desirable asymptotic properties of sample based estimators for suitable population targets under mild assumptions. The utility of the proposed methodology is illustrated with dynamic networks, time-varying distribution data and longitudinal growth data. [ABSTRACT FROM AUTHOR]

The classical notion of statistical convergence has recently been transported to the scope of real normed spaces by means of the -statistical convergence for a modulus function. Here, we go several steps further and extend the -statistical convergence to the scope of uniform spaces, obtaining particular cases of -statistical convergence on pseudometric spaces and topological modules. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove some existence and uniqueness theorems of the fixed points for Hardy-Rogers type contraction with respect to a wt-distance in b-metric spaces endowed with a graph. These results prepare a more general statement, since we apply the condition of orbitally G-continuity of mappings instead of the condition of continuity, consider b-metric spaces endowed with a graph instead general b-metric spaces and use of control functions instead of constant numbers. [ABSTRACT FROM AUTHOR]

The article examines to present sufficient conditions for the existence of solutions for nonlinear matrix equations and the second-order two-point boundary value problems for nonlinear ordinary differential equations via fixed point theorem. Topics include considered two classes of mappings called the Suzuki–Geraghty-(α, β)-F-contraction mappings and generalize Suzuki-type (α, β)-F-contraction mappings and establish some fixed point results for these classes of mappings.

Here, we prove some general results that allow us to ensure that specific representations (as well as extensions) of certain Lipschitz operators exist, provided we have some additional information about the underlying space, in the context of what we call enriched metric spaces. In this conceptual framework, we introduce some new classes of Lipschitz operators whose definition depends on the notion of metric coordinate system, which are defined by specific dominance inequalities involving summations of distances between certain points in the space. We analyze "Pietsch Theorem inspired factorizations" through subspaces of ℓ ∞ and L 1 , which are proved to characterize when a given metric space is Lipschitz isomorphic to a metric subspace of these spaces. As an application, extension results for Lipschitz maps that are obtained by a coordinate-wise adaptation of the McShane–Whitney formulas, are also given. [ABSTRACT FROM AUTHOR]

In this article, we propose a general nonlinear sufficient dimension reduction (SDR) framework when both the predictor and the response lie in some general metric spaces. We construct reproducing kernel Hilbert spaces with kernels that are fully determined by the distance functions of the metric spaces, and then leverage the inherent structures of these spaces to define a nonlinear SDR framework. We adapt the classical sliced inverse regression within this framework for the metric space data. Next we build an estimator based on the corresponding linear operators, and show that it recovers the regression information in an unbiased manner. We derive the estimator at both the operator level and under a coordinate system, and establish its convergence rate. Lastly, we illustrate the proposed method using synthetic and real data sets that exhibit non-Euclidean geometry. [ABSTRACT FROM AUTHOR]

Given uniformly homeomorphic metric spaces and , it is proven that hyperspaces and are uniformly homeomorphic, where denotes the collection of all nonempty closed subsets of , and is endowed with Hausdorff distance. Gerald Beer has proved that hyperspace is an Atsuji space when is either compact or uniformly discrete. An Atsuji space is a generalization of compact metric spaces as well as of uniformly discrete spaces. In this paper, we investigate space when is an Atsuji space, and a class of Atsuji subspaces of is obtained. Using the results, some fixed point results for continuous maps on Atsuji spaces are obtained. [ABSTRACT FROM AUTHOR]

A semiflow is eventually expansive if there is a prefixed radius up to which two orbits eventually coincide. We prove the following result: Every injective eventually expansive semiflow of a compact metric space consists of finitely many closed orbits and, in the noncompact case, the semiflow cannot have global attractors. The suspension of a continuous map is eventually expansive only if the map is positively expansive. The suspension of every positively expansive map is an eventually expansive semiflow. The topological entropy of an eventually expansive semiflow is bounded from below by the growth rate of the periodic orbits. We present some related examples. [ABSTRACT FROM AUTHOR]

In this paper, in the first step, we will introduce the concept of occasionally weakly biased mappings of type (A) which is a convenient generalization of the concept of weakly biased mappings of type (A). In the second step, we will show that this new definition coincides with our concept of occasionally weakly biased mappings given in [8]. In the third and last step we will give an example which verifies the validity of our result.

In this paper we consider a best proximity point problem whose purpose is to determine the minimum distance between two sets. It is a global optimization problem by its very nature which is solved by converting it into a problem of finding an optimal approximate solution of a fixed point inclusion for a coupled setvalued mapping. Two solutions are obtained simultaneously through an iteration. We introduce certain definitions which are used in our theorems. We investigate the data dependence property of the proximity point sets and establish a weak stability result for the proximity point sets. There are some illustrative examples. The broad area of the present study is setvalued optimization.

In 2007, in our joint work with D. Butnariu and S. Reich, we proved that if for a self-mapping of a complete metric that is uniformly continuous on bounded sets all its iterates converge uniformly on bounded sets, then this convergence is stable under the presence of small errors. In the present paper, we obtain an extension of this result for self-mappings of a metric space with a graph.

One of the tools for building new fixed-point results is the use of symmetry in the distance functions. The symmetric property of metrics is particularly useful in constructing contractive inequalities for analyzing different models of practical consequences. A lot of important invariant point results of crisp mappings have been improved by using the symmetry of metrics. However, more than a handful of fixed-point theorems in symmetric spaces are yet to be investigated in fuzzy versions. In accordance with the aforementioned orientation, the idea of Presic-type intuitionistic fuzzy stationary point results is introduced in this study within a space endowed with a symmetrical structure. The stability of intuitionistic fuzzy fixed-point problems and the associated new concepts are proposed herein to complement their corresponding concepts related to multi-valued and single-valued mappings. In the instance where the intuitionistic fuzzy-set-valued map is reduced to its crisp counterparts, our results complement and generalize a few well-known fixed-point theorems with symmetric structure, including the main results of Banach, Ciric, Presic, Rhoades, and some others in the comparable literature. A significant number of consequences of our results in the set-up of fuzzy-set- and crisp-set-valued as well as point-to-point-valued mappings are emphasized and discussed. One of our findings is utilized to assess situations from the perspective of an application for the existence of solutions to non-convex fractional differential inclusions involving Caputo fractional derivatives with nonlocal boundary conditions. Some nontrivial examples are constructed to support the assertions and usability of our main ideas.

In this paper, our primary result is on the existence of non-empty fixed point sets for a new multivalued function defined here. An admissibility condition is also postulated and used in the main theorem. In two separate sections, we present data dependence and stability results for the non-empty fixed-point sets of these mappings. Some consequences of the main theorem are discussed. The analysis is in the most general setting of metric spaces. An illustrative example is discussed, which shows that our existence theorem effectively extends some results in this line of research.

Abstract The aim of this paper is to investigate the interpolative contractions involving rational forms in the framework of b-metric spaces. We prove the existence of a fixed point of such a mapping with different combinations of the rational forms. A certain example is considered to indicate the validity of the observed result.

The so-called Higuchi’s method of fractal dimension estimation is widely used and the term Higuchi’s fractal dimension even occurs in many publications. This paper deals with this method from a mathematical point of view. Terms distance and dimension and its basic properties are explained and Higuchi’s dimension according the original source is defined. The definition of Higuchi’s dimension was compared with the mathematical definition of distance and dimension. It is shown, that the definition of Higuchi’s dimension does not satisfy axioms of distance and dimension. The so-called Higuchi’s method and Higuchi’s dimension are mathematically incorrect. Therefore, all results achieved by this method are scientifically unreliable.

A relationship between Poincaré inequalities and the topological Hausdorff dimension is exposed—a lower bound on the dimension of Ahlfors regular spaces satisfying a weak (1, p)-Poincaré inequality is given.

In this paper, the concepts of Suzuki-type ℒγ contractions and Suzuki–Berinde-type ℒγ contractions are introduced, and new fixed-point theorems for these two contractions are established. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a common fixed point theory for a pair of multivalued operators in complete metric spaces. Existence and stability results for the common fixed point problem will be discussed. The approach is based on some fixed point results for multivalued graphical contractions and for multivalued operators of Feng-Liu type. Our results extend to the multivalued case some recent results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we investigate contractions in a rational form in the context of the supermetric space, which is a very interesting generalization of the metric space. We consider an illustrative example to support this new result on supermetric space. [ABSTRACT FROM AUTHOR]

The maximum traveling salesman problem (Max TSP) consists of finding a Hamiltonian cycle with the maximum total weight of the edges in a given complete weighted graph. This problem is APX-hard in the general metric case but admits polynomial-time approximation schemes in the geometric setting, when the edge weights are induced by a vector norm in fixed-dimensional real space. We propose the first approximation scheme for Max TSP in an arbitrary metric space of fixed doubling dimension. The proposed algorithm implements an efficient PTAS which, for any fixed ε ∈ (0 , 1) , computes a (1 - ε) -approximate solution of the problem in cubic time. Additionally, we suggest a cubic-time algorithm which finds asymptotically optimal solutions of the metric Max TSP in fixed and sublogarithmic doubling dimensions. [ABSTRACT FROM AUTHOR]

Let ([n],d) be a metric space with diameter Δ and with average distance r ̄ , where [n] ≡{1,2,...,n}. We show that if Δ = o (n r ̄) , then ∑ i = 1 ⌊ n / 2 ⌋ d (π (2 i − 1) , π (2 i)) is (1 / 2 ± o (1)) n r ̄ with probability 1 − o(1) over a uniformly random permutation π: [n] → [n]. In particular, a uniformly random perfect matching in ([n],d) has size concentrating around n r ̄ / 2 when n is even and Δ = o (n r ̄) . Our result has implications for finding maximum travelling salesman tours in metric spaces. [ABSTRACT FROM AUTHOR]

We consider mappings of domains of Riemannian manifolds that admit branch points and satisfy a certain condition regarding the distortion of the modulus of families of paths. We have established logarithmic estimates of distance distortion under such mappings. A separate study relates to the situation when the mappings are defined in metric spaces, and one of them is the Loewner space. We also studied the question of equicontinuity of the families of the indicated mappings in the closure of the domain. In addition, we have established the possibility of continuous extension of these mappings to an isolated point of the boundary. [ABSTRACT FROM AUTHOR]

In this article, we introduce the study of corona metric space. After discussing some basic properties of these metric spaces, such as completeness, boundedness, compactness and separability, we provide a necessary and sufficient condition for the existence of universal lines, and we obtain a formula for the metric dimension of corona metric spaces. [ABSTRACT FROM AUTHOR]

We consider two new classes of contractions and obtain some new fixed point results in complete metric spaces. The mapping considered herein are not necessarily continuous on their domains. Many, well-known generalizations and extensions of the classical Banach contraction theorem have been extended and generalized. We present some illustrative examples to show the genuineness of our results. Finally, an application of our results to nonlinear integral equations is discussed. [ABSTRACT FROM AUTHOR]

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

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. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and prove some fixed point theorems for two maps that satisfy (φ—ψ)-contractive conditions with rational expression in partially ordered metric spaces, our results improve and unify a multitude of fixed point theorems and generalize some recent results in ordered partially metric space. [ABSTRACT FROM AUTHOR]

In this paper, we will investigate the existence and uniqueness of common fixed points of certain mappings in the frame of a metric space. The given results cover a number of unique common fixed point theorems especially a result of Phaneendra and Swatmaram [12]. We will also display two examples to illustrate our theorems. [ABSTRACT FROM AUTHOR]

A relationship between Poincaré inequalities and the topological Hausdor dimension is exposed|a lower bound on the dimension of Ahlfors regular spaces satisfying a weak (1; p)-Poincaré inequality is given. [ABSTRACT FROM AUTHOR]

In the present article, by applying our 2013 Metatheorem and the Brøndsted-Jachymski Principle, we obtain various forms generalizations of Zorn's Lemma and their applications. Such examples are our version of the Zermelo fixed point theorem, equivalent formulations of the Caristi fixed point theorem, and Jachymski's 2003 theorem on equivalent conditions when fixed point sets are same to periodic point sets. [ABSTRACT FROM AUTHOR]

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipschitz clustering property, while for some connected spaces, the lack of sufficiently short connecting curves turns out to be an obstruction. This property is shown to be invariant under quasihomogeneous maps, but not under quasisymmetric ones. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of solutions to the system of nonlinear infinite-point fractional order boundary value problems by an application of n-best proximity point theorem in a complete metric space. Further, we study Hyers-Ulam stability of the addressed system. An appropriate example is provided to check the validity of obtained results. [ABSTRACT FROM AUTHOR]