Skip to main content
SpringerLink
Log in

\({S}_{H}\)-metric spaces and fixed-point theorems for multi-valued weak contraction mappings

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this note for every \(S\)-metric space \((X,S),\) we define a new \(S\)-metric \({S}_{H}\) called Hausdorff S-metric on \(CB(X)\) and show that if \((X,S)\) is complete, \((K\left(X\right), {S}_{H})\) is complete too, where K(X) is the set of all compact nonempty subsets of \(X\) and the notion of weak contraction multi-valued mappings on complete metric spaces (Kritsana Neammanee & Annop Kaewkhao, 2011) is generalized to complete \(S\)-metric spaces. This idea is used to establish some fixed-point theorems for weak contractive multi-valued mappings from \((X,S)\) into \((CB\left(X\right),{S}_{H})\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berinde, M., Berinde, V.: On a general class of multi-valued weakly picard mappings. J Math. Anal. Appl. 326, 772–782 (2007)

    Article  MathSciNet  Google Scholar 

  2. Bukatin, M., Kopperman, R., Matthews, S., Pajoohesh, H.: Partial metric spaces. Am. Math. Mon. 116(8), 708–718 (2009)

    Article  MathSciNet  Google Scholar 

  3. Dhage, B.C.: Generalized metric spaces mappings with fixed point. Bull. Cal. Math. Soc. 84, 329–336 (1992)

    MATH  Google Scholar 

  4. Dung, N.V.: On coupled common fixed points for mixed weakly monotone maps in partially ordered S-metric spaces. Fixed Point Theory Appl. 2013, 1–17 (2013)

    Article  MathSciNet  Google Scholar 

  5. Dung, N.V., Hieu, N., Radojevic, S.: Fixed point Theorem for g-monotone maps on partially ordered S-metric spaces. Filomat 28(9), 1885–1895 (2014)

    Article  MathSciNet  Google Scholar 

  6. Gahler, S.: 2-metrisch Raume und iher topoloische struktur. Math. Nachr. 26, 115–148 (1963)

    Article  MathSciNet  Google Scholar 

  7. Gupta, V., Deep, R.: Some coupled fixed point theorems in partially ordered S-metric spaces. Miskolc Math. Notes 16(1), 181–194 (2015)

    Article  MathSciNet  Google Scholar 

  8. Kumbar, M., Ingalagi, U., Patel, T., Marigoudar, P., Nadaf, H., Narasimhamurthy, S.K.: Geometric approach for Banach using Hausdorff distance. Int. J. Math. Appl. 5, 95–102 (2017)

    Google Scholar 

  9. Munkers, J.R.: Topology: a first course. Prentice-Hall, INC., Englewood Cliffs (1975)

    Google Scholar 

  10. Mustafa, Z., Sims, B.: A new approach to generalized metric spaces. J. Non-linear Convex Anal. 7(2), 289–297 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Neammanee, K., Kaewkho, A.: On multi-valued weak contraction mappings. J. Math. Res. 3(2), 151–156 (2011)

    Article  Google Scholar 

  12. Ozgur, N. Y., Tas, N., Celik, U.: New fixed-circle results on S-metric spaces. Bull. Math. Anal. Appl. 10–23 (2017).

  13. Sedghi, S., Shobe, N.: Fixed point theorem in M-fuzzy metric spaces with property (E). Adv in fuzzy Math. 1(1), 55–65 (2006)

    MATH  Google Scholar 

  14. Sedghi, S., Shobe, N., Alioche, A.: A generalization of fixed point theorems in S-metric spaces. Mat. Vesn. 64(3), 258–266 (2012)

    MathSciNet  MATH  Google Scholar 

  15. Sedghi, S., Shobe, N., Shahraki, M., Dosenovic, T.: Common fixed point of four maps in S-metric spaces. Math. Sci. 12, 137–143 (2018)

    Article  MathSciNet  Google Scholar 

  16. Shrivastava, S., Daheriya, R., Ughade, M.: S-metric space, expanding mappings & fixed point theorems. Int. J. Sci. Innov. Math. Res. 1–12, (2016)

Download references

Acknowledgements

The authors would like to express their appreciation to the referee for his/her valuable comments and suggestions that improved the original version.

Author information

Authors and Affiliations

  1. Department of Pure Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran

    A. Pourgholam & M. Sabbaghan

Authors
  1. A. Pourgholam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Sabbaghan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Sabbaghan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pourgholam, A., Sabbaghan, M. \({S}_{H}\)-metric spaces and fixed-point theorems for multi-valued weak contraction mappings. Math Sci 15, 377–385 (2021). https://doi.org/10.1007/s40096-021-00381-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-021-00381-w

Keywords

Mathematical Subject Classifications

Search

Navigation