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


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, access via your institution.


  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MATH  Google Scholar 

  4. 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)

    MathSciNet  Article  Google Scholar 

  5. 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)

    MathSciNet  Article  Google Scholar 

  6. 6.

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

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 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. 9.

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

    Google Scholar 

  10. 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. 11.

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

    Article  Google Scholar 

  12. 12.

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

  13. 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. 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. 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)

    MathSciNet  Article  Google Scholar 

  16. 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


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



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

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 (2021). https://doi.org/10.1007/s40096-021-00381-w

Download citation


  • Collage theorem
  • Fixed point
  • Hausdorff -metric space
  • Multi-valued mapping
  • Multi-valued Zamfirescu mapping
  • Weak contraction

Mathematical Subject Classifications

  • 47H10
  • 54H25