Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

Chrząszcz, Katarzyna; Jachymski, Jacek; Turoboś, Filip

doi:10.1007/s00010-018-0597-9

Cited by 12 publications

(9 citation statements)

References 20 publications

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“…The observations of Chrzaszcz et al in [17] are especially noteworthy. Fundamental to that discussion is a theorem of Schroeder [86] which asserts that if (X, d) is a quasi-metric space with constant b ≤ 2 then X has the metric boundedness property in the following strong sense: There exists a metric ρ on X such that for each x, y ∈ X The study of quasi-metric spaces and their normed analogs has a long history.…”

Section: Remark 62mentioning

confidence: 80%

Hyperbolic spaces and directional contractions

Kirk

Shahzad

2019

Bull. Math. Sci.

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The axiomatic approach to metric convexity goes back to the pioneering work of Karl Menger in 1928. This is an overview of this concept and the role it plays in metric fixed point theory especially in conjunction with spaces possessing a “hyperbolic” type structures. These include the CAT(0) spaces, hyperconvex metric spaces, and [Formula: see text]-trees. Much of the discussion involves the existence of “approximate” fixed point sequences for mappings satisfying weak contractive conditions. Applications of a well-known fixed point theorem due to Caristi are also included. These involve fixed and approximate fixed points for mappings satisfying local “directional” contractive and non-expansive conditions. Convexity plays a role in this part of the discussion as well. While the paper is semi-expository in nature, some detailed proofs appear here for the first time. Also the concept of a weak [Formula: see text]-directional contraction introduced in Sec. 8 appears to be new. Several suggestions for further research are also discussed.

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Section: Remark 62mentioning

confidence: 80%

Hyperbolic spaces and directional contractions

Kirk

Shahzad

2019

Bull. Math. Sci.

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“…Remark 2.2. An equivalent definition of a quasimetric (see for example [41]) could be given by replacing condition (QM 1 ) by the following one…”

Section: Origins Of Quasimetric Spaces (B-metric Spaces)mentioning

confidence: 99%

"The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects"

Berinde¹,

Pačurar²

2022

CJM

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"A very impressive research work has been devoted in the last two decades to obtaining fixed point theorems in quasimetric spaces (also called b-metric spaces). Some incorrect and incomplete references with respect to the early developments on fixed point theory in b-metric spaces are though perpetually taking over from the existing publications to the new ones. Starting from this fact, our main aim in this note is threefold: (1) to briefly survey the early developments in the fixed point theory on quasimetric spaces (b-metric spaces); (2) to collect some relevant bibliography related to this topic; (3) to discuss some other aspects of current interest in the fixed point theory on quasimetric spaces (b-metric spaces)."

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“…3 Both β´metric and γ´polygon spaces are well-established in the literature: [1,3,6,7,10,11,12,18,20,29,37,40,43,45,46] serve just as a couple of examples. 4 As far as we know it first appeared in [12]. 5 For an example see [31].…”

Section: Framework Of Semimetric Spaces and Graph Theorymentioning

confidence: 99%

“…A brute-force solution to the travelling salesperson problem is to go over all possible Hamiltonian cycles in a given graph, calculate the total weight of each of them and choose one with the smallest value. 12 Each Hamiltonian cycle can be thought of as a permutation of nodes -for instance, the Hamiltonian cycle px 1 , x 2 , x 3 , . .…”

Section: Approximate Solutions To the Tsp On Semimetric Graphsmentioning

confidence: 99%

Comparison of approximation algorithms for the travelling salesperson problem on semimetric graphs

Krukowski,

Turoboś

2021

Preprint

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The aim of the paper is to compare different approximation algorithms for the travelling salesperson problem. We pick the most popular and widespread methods known in the literature and contrast them with a novel approach (the polygonal Christofides algorithm) described in our previous work. The paper contains a brief summary of theory behind the algorithms and culminates in a series of numerical simulations (or "experiments"), whose purpose is to determine "the best" approximation algorithm for the travelling salesperson problem on complete, weighted graphs.

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Two refinements of Frink’s metrization theorem and fixed point results for Lipschitzian mappings on quasimetric spaces

Cited by 12 publications

References 20 publications

Hyperbolic spaces and directional contractions

Hyperbolic spaces and directional contractions

"The early developments in fixed point theory on b-metric spaces: a brief survey and some important related aspects"

Comparison of approximation algorithms for the travelling salesperson problem on semimetric graphs

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