Length Spaces and Local Contractions

  • William Kirk
  • Naseer Shahzad
Chapter
First Online:

Abstract

In general, a path in a metric space \(\left (X,d\right )\) is a continuous image of the unit interval \(I = \left [0,1\right ] \subset \mathbb{R}.\) If \(S \equiv f\left (I\right )\) is a path, then its length is defined as
$$\displaystyle{ \ell\left (S\right ) =\sup _{\left (x_{i}\right )}\sum _{i=0}^{N-1}d\left (f\left (x_{ i}\right ),f\left (x_{i+1}\right )\right ) }$$
where \(\left (x_{i}\right ):= \left (0 = x_{0} < x_{1} < \cdot \cdot \cdot < x_{N} = 1\right )\) is any partition of \(\left [0,1\right ].\) If \(\ell\left (S\right ) < \infty \), then the path is said to be rectifiable .

Keywords

Banach Space Bounded Sequence Usual Sense Length Space Preserve Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Bibliography

  1. [4]
    A.G. Aksoy, M.A. Khamsi, Nonstandard Methods in Fixed Point Theory. With an introduction by W. A. Kirk. Universitext (Springer, New York, 1990)CrossRefMATHGoogle Scholar
  2. [10]
    M.A. Alghamdi, W.A. Kirk, N. Shahzad, Locally nonexpansive mappings in geodesic and length spaces. Topol. Appl. 173, 59–73 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. [27]
    L.M. Blumenthal, Remarks concerning the Euclidean four-point property. Ergebnisse Math. Kolloq. Wien 7, 7–10 (1936)Google Scholar
  4. [28]
    L.M. Blumenthal, Theory and Applications of Distance Geometry, 2nd edn. (Chelsea Publishing, New York, 1970)MATHGoogle Scholar
  5. [36]
    M. Bridson, A. Haefliger, Metric Spaces of Non-positive Curvature (Springer, Berlin/Heidelberg/New York, 1999)CrossRefMATHGoogle Scholar
  6. [54]
    P. Corazza, Introduction to metric preserving functions. Am. Math. Mon. 106(4), 309–323 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. [93]
    S.K. Hildebrand, H.W. Milnes, Minimal arcs in metric spaces. J. Austral. Math. Soc. 19(4), 426–430 (1975)MathSciNetCrossRefMATHGoogle Scholar
  8. [96]
    R.D. Holmes, Fixed points for local radial contractions (Proc. Sem., Dalhousie Univ., Halifax, NS, 1975). (Academic, New York, 1976), pp. 79–89Google Scholar
  9. [97]
    T. Hu, W.A. Kirk, Local contractions in metric spaces. Proc. Am. Math. Soc. 68(1), 121–124 (1978)MathSciNetCrossRefMATHGoogle Scholar
  10. [103]
    G. Jungck, Local radial contractions – a counter-example. Houston J. Math. 8(4), 501–506 (1982)MathSciNetMATHGoogle Scholar
  11. [121]
    W.A. Kirk, Approximate fixed points of nonexpansive maps. Fixed Point Theory 10(2), 275–288 (2009)MathSciNetMATHGoogle Scholar
  12. [128]
    W.A. Kirk, N. Shahzad, Remarks on metric transforms and fixed-point theorems. Fixed Point Theory Appl. 2013, 11 pp. (2013)Google Scholar
  13. [154]
    K. Menger, Untersuchungen über allgemeine Metrik. Math. Ann. 100(1), 75–163 (1928) (in German)MathSciNetCrossRefMATHGoogle Scholar
  14. [158]
    N. Monod, Supperrigidity for irreducible lattices and geometric splitting. J. Am. Math. Soc. 19(4), 781–814 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. [172]
    A. Petrusel, I.A. Rus, Fixed point theorems in ordered L-spaces. Proc. Am. Math. Soc. 134(2), 411–418 (2006)MathSciNetCrossRefMATHGoogle Scholar
  16. [182]
    S. Priess-Crampe, P. Ribenboim, The approximation to a fixed point. J. Fixed Point Theory Appl. 14(1), 41–53 (2013)MathSciNetCrossRefGoogle Scholar
  17. [201]
    N. Shahzad, O. Valero, On 0-complete partial metric spaces and quantitative fixed point techniques. Abstr. Appl. Anal. 2013, 11 pp. (2013). Art. ID 985095Google 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