Partial quasi-metrics

“…On account of [7], a partial quasi-metric space (X, q) is said to be complete provided that the associated quasi-metric space (X, d q ) is bicomplete. Moreover, in the same reference the Matthews fixed point theorem (Theorem 2 in Section 1) has been extended to the context of partial quasi-metric spaces in the following way:…”

Section: The Baire Partial Quasi-metricmentioning

confidence: 99%

“…However, in the same section we show that the Baire partial metric can not be employed, in general, to obtain an asymptotic upper bound, and thus the complexity class, of the running time of computing of an algorithm. Inspired by the last fact, we introduce in Section 3 a new Baire partial metric framework whose basis resides in the partial quasi-metric approach introduced recently in [7]. We show that this new partial metric approach presents a relevant advantage with respect to the Schellekens one.…”

Section: Introductionmentioning

confidence: 99%

The Baire Partial Quasi-Metric Space: A Mathematical Tool for Asymptotic Complexity Analysis in Computer Science

2011

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In 1994, S.G. Matthews introduced the notion of partial metric space in order to obtain a suitable mathematical tool for program verification [Ann. New York Acad. Sci. 728 (1994), 183-197]. He gave an application of this new structure to parallel computing by means of a partial metric version of the celebrated Banach fixed point theorem [Theoret. Comput. Sci. 151 (1995), 195-205]. Later on, M.P. Schellekens introduced the theory of complexity (quasi-metric) spaces as a part of the development of a topological foundation for the asymptotic complexity analysis of programs and algorithms [Electronic Notes in Theoret. Comput. Sci. 1 (1995), 211-232]. The applicability of this theory to the asymptotic complexity analysis of Divide and Conquer algorithms was also illustrated by Schellekens. In * Corresponding author e-mail, telephone and fax numbers: o.valero@uib.es, +34 971 259817, +34 971 173003 1 particular, he gave a new proof, based on the use of the aforenamed Banach fixed point theorem, of the well-known fact that Mergesort algorithm has optimal asymptotic average running time of computing. In this paper, motivated by the utility of partial metrics in Computer Science, we discuss whether the Matthews fixed point theorem is a suitable tool to analyze the asymptotic complexity of algorithms in the spirit of Schellekens. Specifically, we show that a slight modification of the well-known Baire partial metric on the set of all words over an alphabet constitutes an appropriate tool to carry out the asymptotic complexity analysis of algorithms via fixed point methods without the need for assuming the convergence condition inherent to the definition of the complexity space in the Shellekens framework. Finally, in order to illustrate and to validate the developed theory we apply our results to analyze the asymptotic complexity of Quicksort, Mergesort and Largesort algorithms. 2010 AMS Classification: 47H10, 54E50, 54F05, 68Q25, 68W40.

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“…He introduced this notion to solve some difficulties of the domain theory and showed the Banach's contraction principle [7] can be generalized in context of partial metric spaces for applications in program verifications (see for example [13,21,24,27,31,32,33,36]. Now, we recall definition and properties of partial metric spaces.…”

Section: Introductionmentioning

confidence: 99%

A coincident point and common fixed point theorem for weakly compatible mappings in partial metric spaces

Akram¹,

Shamaila²

2016

J. Nonlinear Sci. Appl.

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In this paper, we prove the existence of a coincident point and a common fixed point for two self mappings defined on a complete partial metric space X. We will consider generalized cyclic representation of the set X with respect to the two self maps defined on X and a contractive condition involving a generalized distance altering function. Our results generalizes several corresponding results in the existing literature. c 2015 All rights reserved.

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Partial quasi-metrics

Cited by 78 publications

References 15 publications

Ćirić types nonunique fixed point theorems on partial metric spaces

Ćirić types nonunique fixed point theorems on partial metric spaces

The Baire Partial Quasi-Metric Space: A Mathematical Tool for Asymptotic Complexity Analysis in Computer Science

A coincident point and common fixed point theorem for weakly compatible mappings in partial metric spaces

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