Skip to main content
SpringerLink
Log in

Introduction to Hyperconvex Spaces

  • Chapter
Handbook of Metric Fixed Point Theory

Abstract

The notion of hyperconvexity is due to Aronszajn and Panitchpakdi [1] (1956) who proved that a hyperconvex space is a nonexpansive absolute retract, i.e. it is a non-expansive retract of any metric space in which it is isometrically embedded. The corresponding linear theory is well developed and associated with the names of Gleason, Goodner, Kelley and Nachbin (see for instance [19, 29, 42, 46]). The nonlinear theory is still developing. The recent interest into these spaces goes back to the results of Sine [54] and Soardi [57] who proved independently that fixed point property for nonexpansive mappings holds in bounded hyperconvex spaces. Since then many interesting results have been shown to hold in hyperconvex spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. Aronszajn, and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405–439.

    MathSciNet  MATH  Google Scholar 

  2. J. M. Ayerbe, T. Dominguez, and G. López, Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Applications, 99, Birkhäuser Verlag, Basilea 1997.

    Google Scholar 

  3. J. B. Baillon, Nonexpansive mappings and hyperconvex spaces, Contemp. Math. 72 (1988), 11–19.

    Google Scholar 

  4. D. Bugajewski, and R. Espinola, Remarks on some fixed point theorems for hyperconvex metric spaces and absolute retracts,Lecture Note in Pure and Applied Mathematic 213, New-York Marcel-Dekker 2000 (85–92).

    Google Scholar 

  5. D. Bugajewski, and E. Grzelaczyk, A fixed point theorem in hyperconvex spaces, Arch. Math. 75 (2000), 395–400.

    MathSciNet  MATH  Google Scholar 

  6. D. Cartwright, Extension of positive operators between Banach lattices, Mem. Amer. Math. Soc. V. 3 164 (1975), 1–48.

    Google Scholar 

  7. F. Cianciaruso, and E. De Pascale, Discovering the algebraic structure on the metric injective envelope of real Banach space, Topology and its Applications 78 (1997), 285–292.

    Article  MathSciNet  MATH  Google Scholar 

  8. F. Cianciaruso, and E. De Pascale, The Hausdorff measure of non hyperconvexity, Atti Sem. Mat. Fis. Univ. Modena, XLVII (1999), 261–267.

    Google Scholar 

  9. H. B. Cohen, Injective envelopes of Banach spaces, Bull. Amer. Math. Soc. 70 (1964), 723–726.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. B. Cohen, and H. E. Lacey, On injective envelopes of Banach spaces, J. Funct. Anal. 4 (1969), 11–30.

    Article  MATH  Google Scholar 

  11. G. Darbo, Punti uniti in trasformazioni a condominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92.

    MathSciNet  MATH  Google Scholar 

  12. T. Dominguez (edit.), Recent Advances on Metric Fixed Point Theory, Universidad de Sevilla, Sevilla, 1996.

    Google Scholar 

  13. J. Dugundji, and A. Granas, Fixed Point Theory, Polish Scientific Publisher, Warszawa, 1982.

    MATH  Google Scholar 

  14. R. Espinola, Darbo-Sadovski’s theorem in hyperconvex metric spaces, Supplemento ai Rendiconti del Circolo Matematico di Palermo 40 (1996), 130–139.

    Google Scholar 

  15. R. Espinola, W. A. Kirk, and G. López, Nonexpansive retractions in hyperconvex spaces, J. Math. Anal. Appl.251 (2000), 557–570.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Espinola, and G. López, Ultimately compact operators in hyperconvex metric spaces, Nonlinear Analysis and Convex Analysis, Eds: W. Takahasi and T. Tamaka, World Scientific Publishing, 1999, 142–149.

    Google Scholar 

  17. K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240.

    Article  MathSciNet  MATH  Google Scholar 

  18. K. Goebel, and W. A. Kirk, Topics on Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990.

    Google Scholar 

  19. D. B. Goodner, Projections in nonmed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89–108.

    MathSciNet  MATH  Google Scholar 

  20. B. Grünbaum, On some covering and intersection properties in Minkowski spaces, Pacific J. Math. 27 (1959), 487–494.

    Google Scholar 

  21. H. Herrlich, On hyperconvex hulls of metric spaces, Topology and its Applications 46 (1992), 13–21.

    Article  MathSciNet  Google Scholar 

  22. C. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991), 341–357.

    Article  MathSciNet  MATH  Google Scholar 

  23. O. Hustacl’ Intersection properties of balls in complex Banach spaces whose duals are L 1 spaces, Acta Math. 132 (1974), 283–213.

    Article  MathSciNet  Google Scholar 

  24. J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helvetic 39 (1964), 65–76.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. R. Isbell, Injective envelopes of Banach spaces are rigidly attached, Bull. Amer. Math. Soc. 70 (1964), 727–729.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. R. Isbell, Three remarks on injective envelopes of Banach spaces, J. Math. Anal. Appl. 27 (1969), 516–518.

    Article  MathSciNet  MATH  Google Scholar 

  27. E. Jawhari, D. Misane, and M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, Contemp. Math. 57 (1986), 175–226.

    MathSciNet  Google Scholar 

  28. J. Jaworowski, W. A. Kirk, and S. Park, Antipodal Points and Fixed Points, Notes of the Series of lectures held at the Seoul National Univesity, 1995.

    Google Scholar 

  29. J. L. Kelley, Banach spaces with the extension property, Trans. Amer. Math. Soc. 72 (1952), 323–326.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. A. Khamsi, On metric spaces with uniform normal structure, Proc. Amer. Math. Soc. 106 (1989), 723–726.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. A. Khamsi, KKM and Ky Fan Theorems in hyperconvex metric spaces, J. Math. Anal. Appl. 204 (1996), 298–306.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. A. Khamsi, One-local retract and common fixed point for commuting mappings in metric spaces, Nonlinear Anal. 27 (1996), 1307–1313.

    Article  MathSciNet  MATH  Google Scholar 

  33. M. A. Khamsi, W. A. Kirk, and C. Martinez Yânez, Fixed point and selection theorems in hyperconvex spaces, Proc. Amer. Math. Soc. 128 (2000), 3275–3283.

    Article  MathSciNet  MATH  Google Scholar 

  34. M. A. Khamsi, H. Knaust, N. T. Nguyen, and M. D. O’Neill, Lambda-hyperconvexity in metric spaces Nonlinear Anal. 43 (2000), 21–31.

    Article  MathSciNet  Google Scholar 

  35. M. A. Khamsi, M. Lin, and R. C. Sine, On the fixed points of commuting nonexpansive maps in hyperconvex spaces, J. Math. Anal. Appl. 168 (1992), 372–380.

    Article  MathSciNet  MATH  Google Scholar 

  36. M. A. Khamsi, and S. Reich, Nonexpansive mappings and semigroups in hyperconvex spaces, Math. Japonica 3 (1990), n. 11, 467–471.

    MathSciNet  Google Scholar 

  37. W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly, 72 (1965), 1004–1006.

    Article  MathSciNet  MATH  Google Scholar 

  38. W. A. Kirk, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math, Soc. 82 (1981), 640–642.

    Article  MathSciNet  MATH  Google Scholar 

  39. W. A. Kirk, Hyperconvexity of R-trees, Fundamenta Mathematicae 156 (1998), 67–72.

    MathSciNet  MATH  Google Scholar 

  40. W. A. Kirk, and S. S. Shin, Fixed point theorems in hyperconvex spaces, Houston J. Math. 23 (1997), 175–187.

    MathSciNet  MATH  Google Scholar 

  41. W. A. Kirk, B. Sims, and G. Xian-Zhi Yuan, The Knaster, Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications, Nonlinear Anal. 39 (2000), n. 5, 611–627.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. L. Lacey, The Isometric Theory of Classical Banach Spaces, Springer Verlag, Berlin, Heidelberg, New York, 1974.

    Book  Google Scholar 

  43. A. Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1–62.

    Google Scholar 

  44. M. Lin, and R. C. Sine, Retractions on the fixed point set of semigroups of nonexpanstive maps in hyperconvex spaces, Nonlinear Anal. 15 (1990), 943–954.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964), 1–112.

    MathSciNet  Google Scholar 

  46. L. Nachbin, A theorem of the Hahn-Banach type, Trans. Amer. Math. Soc. 68 (1950), 28–46.

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Park, Fixed point theorems in hyperconvex metric spaces, Nonlinear Anal. 37 (1998), n. 4, 467–472.

    Article  Google Scholar 

  48. S. Park, The Schauder type and other fixed point theorems in hyperconvex spaces, Nonlinear Analysis Forum (1998), n. 3, 1–12.

    MathSciNet  MATH  Google Scholar 

  49. J. P. Penot, Fixed point theorems without convexity, Bull. Soc. Math. France Mémoire 60 (1979), 129–152.

    Google Scholar 

  50. D. Repovs, and P. V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht, 1998.

    Book  MATH  Google Scholar 

  51. N. V. Rao, The metric injective hulls of normed spaces, Topology and its Applications 46 (1992), 13–21.

    Article  MATH  Google Scholar 

  52. B. N. Sadovskii, A fixed point principle, Functional Analysis and Applications 1 (1967), 151–153.

    Article  Google Scholar 

  53. B. N. Sadovskii, Limit-compact and condensing operators, Russ. Math. Surveys 27 (1972), 85155.

    Article  Google Scholar 

  54. R. C. Sine, On linear contraction semigroups in sup norm spaces, Nonlinear Anal. 3 (1979), 885–890.

    Article  MathSciNet  MATH  Google Scholar 

  55. R. C. Sine, Hyperconvexity and approximate fixed points, Nonlinear Anal. 13 (1989), 863–869.

    Article  MathSciNet  MATH  Google Scholar 

  56. R. C. Sine, Hyperconvexity and nonexpansive multifonctions, Trans. Amer. Math. Soc. 315 (1989), 755–767.

    Article  MathSciNet  MATH  Google Scholar 

  57. P. Soardi, Existence of fixed points for nonexpansive mappings in certain Banach lattices, Proc. Amer. Math. Soc. 73 (1979), 25–29.

    Article  MathSciNet  MATH  Google Scholar 

  58. S. Szufla, On the application of measure of noncompactness to existence theorems, Rend. Sem. Math. Univ. Padova 75 (1986), 1–14.

    MathSciNet  MATH  Google Scholar 

  59. G. Xian-Zhi Yuan, KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, Inc., New York 1999.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Análisis Matemático Facultad dí Matemáticas, Universidad de Sevilla, Aptdo 1160, Sevillo, 41080, Spain

    R. Espínola

  2. Department of Mathematical Sciences and Computer Science, The University of Texas at El Paso, El Paso, Texas, 79968, USA

    M. A. Khamsi

Authors
  1. R. Espínola
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Khamsi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, The University of Iowa, Iowa City, IA, USA

    William A. Kirk

  2. School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, Australia

    Brailey Sims

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Espínola, R., Khamsi, M.A. (2001). Introduction to Hyperconvex Spaces. In: Kirk, W.A., Sims, B. (eds) Handbook of Metric Fixed Point Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1748-9_13

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1748-9_13

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5733-4

  • Online ISBN: 978-94-017-1748-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation