Startseite Mathematik On the topology of partial metric spaces
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

On the topology of partial metric spaces

  • Dariusz Bugajewski und Ruidong Wang EMAIL logo
Veröffentlicht/Copyright: 13. Januar 2020
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 70 Heft 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.

References

[1] Aydi, H.—Karapinar, E.—Kumam, P.: A note on ’Modified proof of Caristi’s fixed point theorem on partial metric spaces, J. Inequal. Appl. 2013:355 (2013).10.1186/1029-242X-2013-210Suche in Google Scholar

[2] Aydi, H.—Hadj-Amor, S.—Karapinar, E.: Berinde-type generalized contractions on partial metric spaces, Abstr. Appl. Anal. 2013 (2013), Art. ID 312479.10.1155/2013/312479Suche in Google Scholar

[3] Aydi, H.—Karapinar, E.: New Meir-Keeler type tripled fixed point theorems on ordered partial metric spaces, Math. Probl. Eng. (2012), Art. ID 409872.10.1155/2012/409872Suche in Google Scholar

[4] Aydi, H.—Karapinar, E.—Rezapour, Sh.: A generalized Meir-Keeler-type contraction on partial metric spaces, Abstr. Appl. Anal. 2012 (2012), Art. ID 287127.10.1155/2012/287127Suche in Google Scholar

[5] Aydi, H.—Karapinar, E.: A Meir-Keeler common type fixed point theorem on partial metric spaces, Fixed Point Theory Appl. (2012), 2012:26.10.1186/1687-1812-2012-26Suche in Google Scholar

[6] Altun, I.—Sola, F.—Simsek, H.: Generalized contractions on partial metric spaces, Topology Appl. 157 (2010), 2778–2785.10.1016/j.topol.2010.08.017Suche in Google Scholar

[7] Arvanitakis, A. D.: A proof of the generalized Banach contraction conjecture, Proc. Amer. Math. Soc. 131(12) (2013), 3647–3656.10.1090/S0002-9939-03-06937-5Suche in Google Scholar

[8] Bugajewski, D.: Fixed point theorems in locally convex spaces, Acta Math. Hungar. 98(4) (2003), 345–355.10.1023/A:1022842429470Suche in Google Scholar

[9] Bugajewski, D.—Gan, X.-X.: Some remarks onformal power series and formal Laurent series, Math. Slovaca 67(3)(2017), 631–644.10.1515/ms-2016-0297Suche in Google Scholar

[10] Bukatin, M.—Kopperman, R.—Matthews, S.—Pajoohesh, H.: Partial metric spaces, Amer. Math. Monthly 8(116) (2009), 708–718.10.4169/193009709X460831Suche in Google Scholar

[11] Chi, K. P.—Karapinar, E.—Thanh, T. D.: A generalized contraction principle in partial metric spaces, Math. Comput. Modelling 55 (2012), 1673–1681.10.1016/j.mcm.2011.11.005Suche in Google Scholar

[12] Gan, X.-X.: Selected Topics of Formal Analysis. Lect. Notes Nonlinear Anal. 15, Juliusz Schauder Center for Nonlinear Studies, Nicolaus Copernicus University, Toruń, 2017.Suche in Google Scholar

[13] Haghi, R. H.—Rezapour, Sh.—Shahead, N.: Be careful on partial metric fixed point results, Topology Appl. 160 (2013), 450–454.10.1016/j.topol.2012.11.004Suche in Google Scholar

[14] Hitzler, P.—Seda, A.: Mathematical Aspects of Logic Programming Semantics, Chapman & Hall/CRC Studies in Informatic Series, CRC Press, 2011.Suche in Google Scholar

[15] D. Ilić, D.—Pavlović, V.—Rakočević, V.: Some new extensions of Banach’s contraction principle to partial metric space, Appl. Math. Lett. 24 (2011), 1326–1330.10.1016/j.aml.2011.02.025Suche in Google Scholar

[16] Matthews, S. G.: Partial metric topology. In: Papers on General Topology and Applications, Flushing, NY, 1992; In: Ann. New York Acad. Sci. 728, New York Acad. Sci., New York, 1994, pp. 183–197.Suche in Google Scholar

[17] Merryfield, J.—Stein Jr., J. D.: A generalization of the Banach contraction principle, J. Math. Anal. Appl. 273 (2002), 112-120.10.1016/S0022-247X(02)00215-9Suche in Google Scholar

[18] Petelas, C.—Vidalis, T.: A fixed point theorem in non-Archimedean vector spaces, Proc. Amer. Math. Soc. 118(3) (1993), 819–821.10.1090/S0002-9939-1993-1132421-2Suche in Google Scholar

[19] Schikhof, W. H.: Ultrametric Calculus. An Introduction to p-adic Analysis, Cambridge University Press, 2007.Suche in Google Scholar

[20] Smyth, M. B.—Tsaur, R.: Hyperconvex semi-metric spaces, Topology Proc. 2006 (2001/2002), 791–810.Suche in Google Scholar
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

Artikel in diesem Heft

  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
https://doi.org/10.1515/ms-2017-0338
Schlagwörter für diesen Artikel
formal power series; Generalized Banach Contraction Principle; nonexpansive mapping; partial metric space; spherical completeness; ultrametric space; weakly contractive mapping; weighted graph

Artikel in diesem Heft

  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
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 16.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0338/html
Button zum nach oben scrollen