Skip to main content
SpringerLink
Log in

On relaxations of contraction constants and Caristi’s theorem in b-metric spaces

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, the following facts are stated in the setting of b-metric spaces.

  1. (1)

    The contraction constant in the Banach contraction principle fully extends to [0, 1), but the contraction constants in Reich’s fixed point theorem and many other fixed point theorems do not fully extend to [0, 1), which answers the early stated question on transforming fixed point theorems in metric spaces to fixed point theorems in b-metric spaces.

  2. (2)

    Caristi’s theorem does not fully extend to b-metric spaces, which is a negative answer to a recent Kirk–Shahzad’s question (Remark 12.6) [Fixed Point Theory in Distance Spaces. Springer, 2014].

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. Aliouche A., Simpson C.: Fixed points and lines in 2-metric spaces. Adv. Math. 229, 668–690 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. T. V. An and N. V. Dung, Answers to Kirk-Shahzad’s questions on strong b-metric spaces. Preprint, arXiv:1503.08126, 2015.

  3. An T. V., Dung N.V., Hang V. T. L.: A new approach to fixed point theorems on G-metric spaces. Topology Appl. 160, 1486–1493 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. An T. V., Dung N. V., Kadelburg Z., Radenović S.: Various generalizations of metric spaces and fixed point theorems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109, 175–198 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. An T. V., Tuyen L. Q., Dung N. V.: Stone-type theorem on b-metric spaces and applications. Topology Appl. 185–186, 50–64 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boriceanu M.: Fixed point theory for multivalued contractions on a set with two b-metrics. Creat. Math. Inform. 17, 326–332 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Boriceanu M., Bota M., Petruşel A.: Multivalued fractals in b-metric spaces. Cent. Eur. J. Math. 8, 367–377 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bota M., Molnár A., Varga C.: On Ekeland’s variational principle in b-metric spaces. Fixed Point Theory 2011, 21–28 (2011)

    Article  MATH  Google Scholar 

  9. Chifu C., Petruşel G.: Fixed points for multivalued contractions in b-metric spaces with applications to fractals. Taiwanese J. Math. 18, 1365–1375 (2014)

    Article  MathSciNet  Google Scholar 

  10. Ćirić L., Abbas M., Rajović M., Ali B.: Suzuki type fixed point theorems for generalized multi-valued mappings on a set endowed with two b-metrics. Appl. Math. Comput. 219, 1712–1723 (2012)

    MathSciNet  MATH  Google Scholar 

  11. Czerwik S.: Contraction mappings in b-metric spaces. Acta Math. Univ. Ostrav. 1, 5–11 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Czerwik S.: Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Fis. Univ. Modena 46, 263–276 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Dung N. V.: Remarks on quasi-metric spaces. Miskolc Math. Notes 15, 401–422 (2014)

    MathSciNet  MATH  Google Scholar 

  14. N. V. Dung, V. T. L. Hang and S. Sedghi, Remarks on metric-type spaces and applications. Asian Bull. Math., to appear.

  15. Fagin R., Kumar R., Sivakumar D.: Comparing top k lists. SIAM J. Discrete Math. 17, 134–160 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hieu N. T., Dung N. V.: Some fixed point results for generalized rational type contraction mappings in partially ordered b-metric spaces. Facta Univ. Ser. Math. Inform. 30, 49–66 (2015)

    MathSciNet  Google Scholar 

  17. N. Hussain, Z. Kadelburg, S. Radenovic and F. A. Solamy, Comparison functions and fixed point results in partial metric spaces. Abstr. Appl. Anal. 2012 (2012), Art. ID 605781, 15 pages.

  18. M. Jovanović, Z. Kadelburg and S. Radenović, Common fixed point results in metric-type spaces. Fixed Point Theory Appl. 2010 (2010), doi:10.1155/2010/978121, 15 pages.

  19. M. A. Khamsi, Remarks on cone metric spaces and fixed point theorems of contractive mappings. Fixed Point Theory Appl. 2010 (2010), doi:10.1155/2010/315398, 7 pages.

  20. Khamsi M. A., Hussain N.: KKM mappings in metric type spaces. Nonlinear Anal. 7, 3123–3129 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Kirk and N. Shahzad, Fixed Point Theory in Distance Spaces. Springer, Cham, 2014.

  22. Kumam P., Dung N. V.: Remarks on generalized metric spaces in the Branciari’s sense. Sarajevo J. Math. 10, 209–219 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Kumam, N. V. Dung and V. T. L. Hang, Some equivalences between cone b-metric spaces and b-metric spaces Abstr. Appl. Anal. 2013 (2013), Art. ID 573740, 8 pages.

  24. P. Lo’lo’, S. M. Vaezpour and J. Esmaily, Common best proximity points theorem for four mappings in metric-type spaces. Fixed Point Theory Appl. 2015 (2015), 10.1186/s13663-015-0298-1, 7 pages.

  25. Macías R. A., Segovia C.: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Radenovic S., Kadelburg Z., Jandrlic D., Jandrlic A.: Some results on weakly contractive maps. Bull. Iran. Math. Soc. 38, 625–645 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Reich S.: Kannan’s fixed point theorem. Boll. Un. Mat. Ital. 4(4), 1–11 (1971)

    MathSciNet  MATH  Google Scholar 

  28. Reich S.: Some remarks concerning contradiction mappings. Canad. Math. Bull. 14, 121–124 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wardowski D., Dung N. V.: A note on fixed point theorems in metric spaces. Carpathian J. Math. 31, 127–134 (2015)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Research and Development, Duy Tan University, R.809, K7/25 Quang Trung, Da Nang City, Vietnam

    Nguyen Van Dung

  2. Faculty of Mathematics and Information Technology Teacher Education, Dong Thap University, 783 Pham Huu Lau Street, Ward 6, Cao Lanh City, Dong Thap Province, Vietnam

    Nguyen Van Dung

  3. Journal of Science, Dong Thap University, 783 Pham Huu Lau Street, Ward 6, Cao Lanh City, Dong Thap Province, Vietnam

    Vo Thi Le Hang

Authors
  1. Nguyen Van Dung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vo Thi Le Hang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nguyen Van Dung.

Additional information

To Professor Tran Van An, Vinh University, on the occasion of his 60th birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dung, N.V., Hang, V.T.L. On relaxations of contraction constants and Caristi’s theorem in b-metric spaces. J. Fixed Point Theory Appl. 18, 267–284 (2016). https://doi.org/10.1007/s11784-015-0273-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-015-0273-9

Mathematics Subject Classification

Keywords

Search

Navigation