Skip to main content
SpringerLink
Log in

Quasisymmetric mappings in b-metric spaces

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

Considering quasisymmetric mappings between b-metric spaces we have found a new estimation for the ratio of diameters of two subsets which are images of two bounded subsets. This result generalizes the well-known Tukia–Väisälä inequality. The condition under which the image of a b-metric space under a quasisymmetric mapping is also a b-metric space is established. Moreover, the latter question is investigated for additive metric spaces.

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. A. Beurling and L. V. Ahlfors, “The boundary correspondence under quasiconformal mappings,” Acta Math., 96, 125–142 (1956).

    Article  MathSciNet  Google Scholar 

  2. P. Tukia and J. Väisälä, “Quasisymmetric embeddings of metric spaces,” Ann. Acad. Sci. Fenn., Ser. A I, Math., 5, 97–114 (1980).

  3. V. V. Aseev, D. G. Kuzin, and A. V. Tetenov, “Angles between sets and the gluing of quasisymmetric mappings in metric spaces”, Russ. Math., 49(10), 1–10 (2005).

    MathSciNet  MATH  Google Scholar 

  4. J. Tyson, “Quasiconformality and quasisymmetry in metric measure spaces,” Ann. Acad. Sci. Fenn., Math., 23(2), 525–548 (1998).

    MathSciNet  MATH  Google Scholar 

  5. X. Huang and J. Liu, “Quasihyperbolic metric and quasisymmetric mappings in metric spaces,” Trans. Am. Math. Soc., 367(9), 6225–6246 (2015).

    Article  MathSciNet  Google Scholar 

  6. M. Bonk and S. Merenkov, “Quasisymmetric rigidity of square Sierpiński carpets,” Ann. Math. (2), 177(2), 591–643 (2013).

    Article  MathSciNet  Google Scholar 

  7. Y. Li, M. Vuorinen, and Q. Zhou, “Weakly quasisymmetric maps and uniform spaces,” Comput. Methods Funct. Theory, 18(4), 689–715 (2018).

    Article  MathSciNet  Google Scholar 

  8. E. S. Afanaseva and V. V. Bilet, “Some properties of quasisymmetries in metric spaces,” Ukr. Mat. Bull., 16(1), 2–9 (2019).

    Article  MathSciNet  Google Scholar 

  9. L. M. Blumenthal, Theory and applications of distance geometry. Clarendon Press, Oxford (1953).

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. S. Czerwik, “Nonlinear set-valued contraction mappings in b-metric spaces,” Atti Semin. Mat. Fis. Univ. Modena, 46(2), 263–276 (1998).

    MathSciNet  MATH  Google Scholar 

  12. M. A. Khamsi and N. Hussain, “KKM mappings in metric type spaces,” Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods, 73(9), 3123–3129 (2010).

    Article  MathSciNet  Google Scholar 

  13. H. Aydi, M. F. Bota, E. Karapınar, and S. Mitrović, “A fixed point theorem for set-valued quasi-contractions in b-metric spaces,” Fixed Point Theory Appl., 2012(88), 1–8 (2012).

    MathSciNet  MATH  Google Scholar 

  14. H. Aydi, M. F. Bota, E. Karapınar, and S. Moradi, “A common fixed point for weak φ-contractions on b-metric spaces,” Fixed Point Theory, 13(2), 1–10 (2012).

    MathSciNet  MATH  Google Scholar 

  15. R. P. Agrawal, N. Hussain, and R. Saadati, “On the topology and wt-distance on metric type spaces,” Fixed Point Theory Appl., 2014(88), 1–14 (2014).

    MathSciNet  MATH  Google Scholar 

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

  17. W. Kirk and N. Shahzad, Fixed point theory in distance spaces. Springer, Cham (2014).

    Book  Google Scholar 

  18. N. Hussainn and M. H. Shah, “KKM mappings in cone-metric spaces,” Comput. Math. Appl., 62, 1677–1684 (2011).

    Article  MathSciNet  Google Scholar 

  19. M. A. Alghamdi, N. Hussain, and P. Salimi, “Fixed point and coupled fixed point theorems on b-metric-like spaces,” J. Inequal. Appl, 2013, 1–25 (2013).

    Article  MathSciNet  Google Scholar 

  20. S. Shukla, “Partial b-metric spaces and fixed point theorems,” Mediterr. J. Math., 11, 703–711 (2014).

    Article  MathSciNet  Google Scholar 

  21. Z. Mustafa, J. R. Roshan, V. Parvaneh, and Z. Kadelburg, “Some common fixed point results in ordered partial b-metric spaces,” J. Inequal. Appl., 2013, 1–26 (2013).

    Article  MathSciNet  Google Scholar 

  22. T. Van An, N. Van Dung, Z. Kadelburg, and S. Radenović, “Various generalizations of metric spaces and fixed point theorems,” Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat., 109, 175–198 (2015).

  23. J. Heinonen, Lectures on analysis on metric spaces. Springer, New York (2001).

    Book  Google Scholar 

  24. J. Dobos and Z. Piotrowski, “A note on metric preserving functions,” Int. J. Math. Math. Sci., 19, 199–200 (1996).

    Article  MathSciNet  Google Scholar 

  25. O. Dovgoshey and O. Martio, “Functions transferring metrics to metrics,” Beitr. Algebra Geom., 54, 237–261 (2013).

    Article  MathSciNet  Google Scholar 

  26. T. Khemaratchatakumthorn and P. Pongsriiam, “Remarks on b-metric and metric-preserving functions,” Math. Slovaca, 68(5), 1009–1016 (2018).

    Article  MathSciNet  Google Scholar 

  27. S. Samphavat, T. Khemaratchatakumthorn, and P. Pongsriiam, “Remarks on b-metrics, ultrametrics, and metric-preserving functions,” Math. Slovaca, 70(1), 61–70 (2020).

    Article  MathSciNet  Google Scholar 

  28. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities. Cambridge University Press, Cambridge (1952).

  29. M. M. Deza and E. Deza, Encyclopedia of Distances. Fourth edition, Springer, Berlin (2016).

    Book  Google Scholar 

  30. P. Buneman, “A note on the metric properties of trees,” J. Comb. Theory, Ser. B, 17, 48–50 (1974).

  31. C. Semple and M. Steel, Phylogenetics, vol. 24. Oxford University Press, Oxford (2003).

    MATH  Google Scholar 

  32. N. Ailon and M. Charikar, Fitting tree metrics: hierarchical clustering and phylogeny, SIAM J. Comput., 40(5), 1275–1291 (2011).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, Slavyansk, Ukraine

    Evgeniy A. Petrov

  2. Institute of Mathematics of the NAS of Ukraine, Kyiv, Ukraine

    Ruslan R. Salimov

Authors
  1. Evgeniy A. Petrov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruslan R. Salimov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evgeniy A. Petrov.

Additional information

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 1, pp. 60–70, January–March, 2021.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Petrov, E.A., Salimov, R.R. Quasisymmetric mappings in b-metric spaces. J Math Sci 256, 770–778 (2021). https://doi.org/10.1007/s10958-021-05459-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-021-05459-3

Keywords

Search

Navigation