b-Metric Spaces

  • William Kirk
  • Naseer Shahzad
Chapter

Abstract

In 1993 another axiom for semimetric spaces, which is weaker than the triangle inequality, was put forth by Czerwik [58] with a view of generalizing the Banach contraction mapping theorem. This same relaxation of the triangle inequality is also discussed in Fagin et al. [79], who call this new distance measure nonlinear elastic matching (NEM). The authors of that paper remark that this measure has been used, for example, in [55] for trademark shapes and in [153] to measure ice floes. Later Q.

Bibliography

  1. [7]
    S.M.A. Aleomraninejad, Sh. Rezapour, N. Shahzad, Some fixed point results on a metric space with a graph. Topol. Appl. 159(3), 659–663 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. [18]
    I.A. Bakhtin, The contraction mapping principle in almost metric space. Funct. Anal. [Ulyanovsk. Gos. Ped. Inst., Ulyanovsk] 30, 26–37 (1989) (Russian)Google Scholar
  3. [20]
    G. Beer, A.L. Dontchev, The weak Ekeland variational principle and fixed points. Nonlinear Anal. 102, 91–96 (2014)MathSciNetCrossRefMATHGoogle Scholar
  4. [23]
    V. Berinde, Contractii Generalizate si Aplicatii, vol. 22 (Editura Cub Press, Baia Mare, 1997) (in Romanian)MATHGoogle Scholar
  5. [24]
    V. Berinde, Generalized contractions in quasimetric spaces, in Seminar on Fixed Point Theory, Babes-Bolyai Univ., Cluj-Napoca. Preprint 93-3 (1993), pp. 3–9MathSciNetGoogle Scholar
  6. [29]
    F. Bojor, Fixed point of ϕ-contraction in metric spaces endowed with a graph. An. Univ. Craiova Ser. Mat. Inform. 37(4), 85–92 (2010)MathSciNetMATHGoogle Scholar
  7. [31]
    M. Bota, A. Molnár, C. Varga, On Ekeland’s variational principle in b-metric spaces. Fixed Point Theory 12(2), 21–28 (2011)MathSciNetMATHGoogle Scholar
  8. [55]
    G. Cortelazzo, G. Mian, G. Vezzi, P. Zamperoni, Trademark shapes description by string matching techniques. Pattern Recognit. 27(8), 1005–1018 (1994)CrossRefGoogle Scholar
  9. [57]
    S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces. Atti Sem. Mat. Fis. Univ. Modena 46(2), 263–276 (1998)MathSciNetMATHGoogle Scholar
  10. [58]
    S. Czerwik, Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostraviensis 1, 5–11 (1993)MathSciNetMATHGoogle Scholar
  11. [62]
    A.L. Dontchev, W.W. Hager, An inverse mapping theorem for set-valued maps. Proc. Am. Math. Soc. 121(2), 481–489 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. [78]
    R. Fagin, R. Kumar, D. Sivakumar, Comparing top k lists. SIAM J. Discrete Math. 17(1), 134–160 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. [79]
    R. Fagin, L. Stockmeyer, Relaxing the triangle inequality in pattern matching. Int. J. Comput. Vis. 30(3), 219–231 (1998)CrossRefGoogle Scholar
  14. [92]
    J. Heinonen, Lectures on Analysis on Metric Spaces. Universitext (Springer, New York, 2001)CrossRefMATHGoogle Scholar
  15. [100]
    J. Jachymski, The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 136(4), 1359–1373 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. [148]
    J. Martínez-Maurica, M.T. Pellón, Non-archimedean Chebyshev centers. Nederl. Akad. Wetensch. Indag. Math. 49(4), 417–421 (1987)MathSciNetCrossRefMATHGoogle Scholar
  17. [153]
    R. McConnell, R. Kwok, J. Curlander, W. Kober, S. Pang, Ψ-S correlation and dynamic time warping: two methods for tracking ice floes. IEEE Trans. Geosci. Remote Sens. 29(6), 1004–1012 (1991)CrossRefGoogle Scholar
  18. [159]
    J. Mycielski, On the existence of a shortest arc between two points of a metric space. Houston J. Math. 20(3), 491–494 (1994)MathSciNetMATHGoogle Scholar
  19. [165]
    S. Oltra, O. Valero, Banach’s fixed point theorem for partial metric spaces. Rend. Istid. Math. Univ. Trieste 36(1–2), 17–26 (2004)MathSciNetMATHGoogle Scholar
  20. [171]
    C. Petalas, T. Vidalis, A fixed point theorem in non-archimedean vector spaces. Proc. Am. Math. Soc. 118(3), 819–821 (1993)MathSciNetCrossRefMATHGoogle Scholar
  21. [191]
    S. Romaguera, On Nadler’s fixed point theorem for partial metric spaces. Math. Sci. Appl. E-Notes 1, 7 pp. (2013)Google Scholar
  22. [192]
    I.A. Rus, Generalized Contractions and Applications (Cluj University Press, Cluj-Napoca, 2001)MATHGoogle Scholar
  23. [193]
    M. Samreen, T. Kamran, Fixed point theorems for integral G-contractions. Fixed Point Theory Appl. 2013, 11 pp. (2013)Google Scholar
  24. [218]
    C.S. Wong, On a fixed point theorem of contractive type. Proc. Am. Math. Soc. 57(2), 283–284 (1976)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • William Kirk
    • 1
  • Naseer Shahzad
    • 2
  1. 1.Department of MathematicsUniversity of IowaIowa CityUSA
  2. 2.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations