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On the topology of partial metric spaces

  • Dariusz Bugajewski and Ruidong Wang EMAIL logo
Published/Copyright: January 13, 2020
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 70 Issue 1

Abstract

In this paper, we give some necessary and sufficient conditions under which the topology generated by a partial metric is equivalent to the topology generated by a suitably defined metric. Next, we study some new extensions of the Generalized Banach Contraction Principle to partial metric spaces. Moreover, we draw a particular attention to the space of all sequences showing, in particular, that some well-known fixed point theorems for ultrametric spaces, can be used for operators acting in that space. We illustrate our considerations by suitable examples and counterexamples.

Keywords: formal power series; Generalized Banach Contraction Principle; nonexpansive mapping; partial metric space; spherical completeness; ultrametric space; weakly contractive mapping; weighted graph
MSC 2010: Primary 54E35; 54H25; Secondary 13F25; 16W60; 46S10

This work was supported by the Natural Science Foundation of China Grant No. 11301384.

  1. Communicated by L’ubica Holá

Acknowledgement

This research was partly supported by the Natural Science Foundation of China (Grant Nos. 11201337, 11201338, 11371201, 11301384). The authors thank the anonymous referee for multiple stylistical improvements. The second author is grateful to the Faculty of Mathematics and Computer Science, Adam Mickiewicz University in Poznań, for the excellent working conditions during the visit from February 2018 to August 2018.

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Received: 2019-04-05
Accepted: 2019-07-11
Published Online: 2020-01-13
Published in Print: 2020-02-25

© 2020 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0338
Keywords for this article
formal power series; Generalized Banach Contraction Principle; nonexpansive mapping; partial metric space; spherical completeness; ultrametric space; weakly contractive mapping; weighted graph

Articles in the same Issue

  1. Regular papers
  2. Doc. RNDr. Roman Frič, DrSc. – 3/4 C?
  3. Sugihara algebras and Sugihara monoids: Multisorted dualities
  4. Two by two squares in set partitions
  5. Identifying quantum logics by numerical events
  6. Some remarks on divisible polyhedral MV-algebras
  7. Remarks on b-metrics, ultrametrics, and metric-preserving functions
  8. Estimates of essential norm of generalized weighted composition operators from Bloch type spaces to nth weighted type spaces
  9. Maximum term of transcendental entire function and spider’s web
  10. Entire solutions of certain type of non-linear differential equations
  11. On the oscillatory behavior of third order differential equations with a sublinear neutral term
  12. Eigenvalue problems for a class of nonlinear Hadamard fractional differential equations with p-Laplacian operator
  13. The amalgam space L(p,q)π(G) on IN-groups
  14. On the topology of partial metric spaces
  15. The trace of the commutator of the Cesàro operator and a compact operator
  16. Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold
  17. On 4-th root metrics of isotropic scalar curvature
  18. On ultrametric-preserving functions
  19. Porosity of 𝓢-approximately continuous functions
  20. On a modified Burr XII distribution having flexible hazard rate shapes
  21. A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums
  22. An alternative estimate for the numerical radius of Hilbert space operators
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