Skip to main content
SpringerLink
Log in

Geometrical Background of Metric Fixed Point Theory

  • Chapter
Handbook of Metric Fixed Point Theory

Abstract

The interplay between the geometry of Banach spaces and fixed point theory has been very strong and fruitful. In particular, geometrical properties play key roles in metric fixed point problems. In this text we discuss the most basic of these geometrical properties. Since many fixed point results have a quantitative character, we place special emphasis on the scaling coefficients and functions corresponding to the properties considered. The material we cover is far from exhaustive, in particular we do not consider applications. These are treated elsewhere in the Handbook. The interested reader may also consult [5], [44] and [1].

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. A. G. Aksoy, M. A. Khamsi, Nonstandard Methods in Fixed Point Theory, Springer-Verlag, New York, 1990.

    Book  MATH  Google Scholar 

  2. J. Alonso, A. Ullán, Moduli of convexity, Functional Analysis and Approximation (Bagni di Lucca, 1988 ), 196–218, Pitagora, Bologna, 1989.

    Google Scholar 

  3. D. Amir, On Jung’s constant and related constants in normed linear spaces, Pacific J. Math. 118 (1985), 1–15.

    MathSciNet  MATH  Google Scholar 

  4. E. Asplund, Positivity of duality mappings, Bull. Amer. Math. Soc. 73 (1967), 200–203.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. M. Ayerbe, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhäuser Verlag, Basel, Boston, Berlin, 1997.

    Google Scholar 

  6. J. S. Bae, Reflexivity of a Banach space with a uniformly normal structure, Proc. Amer. Math. Soc. 90 (1984), 269–270.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Banal, On moduli of smoothness of Banach spaces, Bull. Pol. Acad. Sci. 34 (1986), 287–293.

    Google Scholar 

  8. J. Banal, Compactness conditions in the geometric theory of Banach spaces, Nonlinear Anal. 16 (1990), 669–682.

    Google Scholar 

  9. B. Beauzamy, Introduction to Banach Spaces and Their Geometry, Elsevier Science Publishers B.V., Amsterdam, 1985.

    MATH  Google Scholar 

  10. C. Benitez, K. Przeslawski, D. Yost, A universal modulus for normed spaces, Studia Math. 127 (1998), 21–46.

    MathSciNet  MATH  Google Scholar 

  11. J. Bernal, Behavior of k-dimensional Convexity Moduli, Thesis, Catholic University of America, Washington D.C., 1980.

    Google Scholar 

  12. J. Bernal, F. Sullivan, Multi-dimensional volumes, super-reflexivity and normal structure in Banach space, Illinois J. Math. 27 (1983), 501–513.

    MathSciNet  MATH  Google Scholar 

  13. M. Besbes, S. J. Dilworth, P. N. Dowling, C. Lennard, New convexity and fixed point properties in Hardy and Lebesgue-Bochner spaces, J. Funct. Anal. 119 (1993), 340–357.

    Article  MathSciNet  Google Scholar 

  14. A. Beurling, A. E. Livingston, A theorem on duality mappings in Banach spaces, Ark. Mat. 4 (1962), 405–411.

    MathSciNet  MATH  Google Scholar 

  15. H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.

    Article  MathSciNet  MATH  Google Scholar 

  16. M. S. Brodskii, D. P. Milman, On the center of a convex set, Dokl. Akad. Nauk SSSR 59 (1948), 837–840 (Russian).

    Google Scholar 

  17. M. Budzynska, W. Kaczor, M. Koter-Morgowska, T. Kuczumow, Asymptotic normal structure and the semi-Opial property, Nonlinear Anal. 30 (1997), 3505–3515.

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Budzynska, T. Kuczumow, S. Reich, Uniform asymptotic normal structure, the uniform semiOpial property, and fixed points of asymptotically regular uniformly Lipschitzian semigroups. Part I, Abstr. Appl. Anal. 3 (1998), 133–151.

    Google Scholar 

  19. W. L. Bynum, A class of Banach spaces lacking normal structure, Compositio Math. 25 (1972), 233–236.

    MathSciNet  MATH  Google Scholar 

  20. W. L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), 427–436.

    MathSciNet  MATH  Google Scholar 

  21. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990.

    Book  MATH  Google Scholar 

  22. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.

    Article  MathSciNet  Google Scholar 

  23. M. M. Day, Uniform convexity in factor and conjugate spaces, Ann. of Math. 45 (1944), 375–385.

    Article  MathSciNet  MATH  Google Scholar 

  24. M. M. Day, Normed Linear Spaces, Springer-Verlag, Berlin, Heidelberg, New York, 1973.

    Google Scholar 

  25. M. M. Day, R. C. James, S. Swaminathan, Normed linear spaces that are uniform convexity in every direction, Canad. J. Math. 23 (1971), 1051–1059.

    MathSciNet  Google Scholar 

  26. R. Deville, G. Godefroy, V. Zizler, Smoothness and Renormings in Banach Spaces, Longman Sci. and Tech., Harlow, 1993.

    MATH  Google Scholar 

  27. J. Diestel, Geometry of Banach Spaces — Selected Topics, Springer-Verlag, Berlin, New York, 1975.

    Google Scholar 

  28. J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984.

    Book  Google Scholar 

  29. T. Dominguez Benavides, Normal structure coefficients in L5-spaces, Proc. Royal Soc. Edinburgh 117A (1991), 299–303.

    MATH  Google Scholar 

  30. T. Domínguez Benavides, Modulus of nearly uniform smoothness and Lindenstrauss formulae, Glasgow J. Math. 37 (1995), 145–153.

    Google Scholar 

  31. T. Domínguez Benavides, A geometrical coefficient implying the fixed point property and stability results, Houston J. Math. 22 (1996), 835–849.

    MATH  Google Scholar 

  32. T. Dominguez Benavides, J. Garcia Falset, M. A. Japon Pineda, The r-fixed point property for nonexpansive mappings, Abstr. Appl. Anal. 3 (1998), 343–362.

    MATH  Google Scholar 

  33. D. van Du1st, Reflexive and Superreflexive Banach Spaces, Mathematische Centrum, Amsterdam, 1978.

    MATH  Google Scholar 

  34. A. Dvoretzky, Some results on convex bodies and Banach spaces, Proc. Int. Symp. Linear Spaces, Jerusalem, 1960, 123–160, Jerusalem Acad. Press, Jerusalem, 1961.

    Google Scholar 

  35. I. Ekeland, R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, Oxford, 1976.

    Google Scholar 

  36. P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math. 13 (1972), 281–288.

    Article  MathSciNet  Google Scholar 

  37. T. Figiel, On moduli of convexity and smoothness, Studia Math. 56 (1976), 121–155.

    MathSciNet  MATH  Google Scholar 

  38. J. Garcia Falset, The fixed point property in Banach spaces with NUS-property, J. Math. Anal. Appl. 215 (1997), 532–542.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. L. Garkavi, The best possible net and the best possible cross-section of a set in a normed space, Am. Math. Soc., Transl., II. Ser. 39, 111–132 (1964)

    Google Scholar 

  40. A. L. Garkavi, Translation from Izv. Akad. Nauk SSSR, Ser. Mat. 26 (1962), 87–106.

    MathSciNet  MATH  Google Scholar 

  41. R. Geremia, F. Sullivan, Multi-dimensional volumes and moduli of convexity in Banach spaces, Ann. Mat. Pura Appl. 127 (1981), 231–251.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. A. Gillespie, B. B. Williams, Fixed point theorem for nonexpansive mappings on Banach spaces with uniformly normal structure, Appl. Anal. 9 (1979), 121–124.

    MathSciNet  MATH  Google Scholar 

  43. G. Godefroy, N. J. Kelton, G. Lancien, Szlenk indices and uniform homeomorphisms, (preprint).

    Google Scholar 

  44. K. Goebel, Convexity of balls and fixed point theorem for mappings with nonexpansive square, Compositio Math. 22 (1970), 231–251.

    MathSciNet  Google Scholar 

  45. K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.

    Book  MATH  Google Scholar 

  46. K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984.

    MATH  Google Scholar 

  47. K. Goebel, T. Sgkowski, The modulus of noncompact convexity, Annales Univ. Mariae–Sklodowska 38 (1984), 41–48.

    MATH  Google Scholar 

  48. K. Goebel, T. Sgkowski, A. Stachura, Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball, Nonlinear Anal. 4 (1980), 1011–1021.

    Article  MathSciNet  MATH  Google Scholar 

  49. J. P. Cossez, E. Lami Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 565–573.

    Google Scholar 

  50. N. M. Gulevich, Fixed points of nonexpansive mappings, J. Math. Sci. 79 (1996), 755–815.

    Article  MathSciNet  MATH  Google Scholar 

  51. O. Banner, On the uniform convexity of LP and /P, Ark. Mat. 3 (1956), 239–244.

    Google Scholar 

  52. R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743–749.

    Article  MathSciNet  MATH  Google Scholar 

  53. V. I. Isträtescu, Strict Convexity And Complex Strict Convexity, Theory and Applications, Marcel Dekker, New York, Basel, 1984.

    Google Scholar 

  54. R. C. James, Uniformly non-square Banach spaces, Ann, of Math. 80 (1964), 542–550.

    Article  MathSciNet  MATH  Google Scholar 

  55. R. C. James, Super-reflexive Banach spaces, Canadian J. Math. 24 (1972), 896–904.

    Article  MATH  Google Scholar 

  56. M. A. Japon Pineda, Stability of The Fixed Point Property for Nonexpansive Mappings, Ph. D. Thesis, Seville, 1998.

    Google Scholar 

  57. M. I. Kadets, V. M. Kadets, Series in Banach Spaces, Birkhäuser Verlag, Basel, Boston, Berlin, 1997.

    Google Scholar 

  58. M. A. Khamsi, Etude de la Propriete du Point Fixe dans les Espaces Metríques et les Espaces de Banach, Ph. D. Thesis, Paris, 1987.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  60. M. A. Khamsi, On normal structure, fixed point property and contractions of type (ry), Proc. Amer. Math. Soc. 106 (1989), 995–1001.

    MathSciNet  MATH  Google Scholar 

  61. M. A. Khamsi, On uniform Opial condition and uniform Kadec-Klee property in Banach and metric spaces, Nonlinear Anal. 26 (1996), 1733–1748.

    Article  MathSciNet  MATH  Google Scholar 

  62. Y. Kijima, W. Takahashi, A fixed point theorem for nonexpansive mappings in metric spaces, Kodai Math. Sem. Rep. 21 (1969), 326–330.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  64. W. A. Kirk, The modulus of k-rotundity, Boll. U. M. I. (7) 2A (1988), 195–201.

    MathSciNet  MATH  Google Scholar 

  65. W. A. Kirk, History and methods of metric fixed point theory, Antipodal Points and Fixed Points, 21–54, Seoul, 1995.

    Google Scholar 

  66. M. G. Krein, Sur quelques questions de la géométrie des ensembles convexes situés dans un espace linéaire normé et complet, Dokl. Akad. Nauk SSSR 14 (1937), 5–7.

    Google Scholar 

  67. T. Kuczumow, Fixed points of holomorphic mappings in the Hilbert ball, Colloq. Math. 55 (1988), 101–107.

    MathSciNet  MATH  Google Scholar 

  68. T. Landes, Permanence properties of normal structure, Pacific J. Math. 110 (1984), 125–143.

    MathSciNet  MATH  Google Scholar 

  69. T. Landes, Normal structure and hereditariness properties, Functional Analysis and Approximation (Bagni di Lucca, 1988 ), 196–218, Pitagora, Bologna, 1989.

    Google Scholar 

  70. C. Lennard, A new convexity property that implies a fixed point property for L1, Studia Math. 100 (1991), 95–108.

    MathSciNet  MATH  Google Scholar 

  71. T. C. Lim, Asymptotic centers and nonexpansive mappings in some conjugate spaces, Pacific J. Math. 90 (1980), 135–143.

    Article  MathSciNet  Google Scholar 

  72. T. C. Lim, On moduli of k-convexity, (to appear in Abstr. Appl. Anal.)

    Google Scholar 

  73. P. K. Lin, k-Uniform rotundity is equivalent to k-uniform convexity, J. Math. Anal. Appl. 132 (1988), 349–355.

    Article  MathSciNet  MATH  Google Scholar 

  74. P. K. Lin, K. K. Tan, H. K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. 24 (1995), 929–946.

    Article  MathSciNet  MATH  Google Scholar 

  75. J. Lindenstrauss, On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J. 10 (1963), 241–252.

    MathSciNet  Google Scholar 

  76. J. Lindenstrauss, V. D. Milman, The local theory of normed spaces and its applications to convexity, Handbook of Convex Geometry, 1149–1220, Elsevier, 1993.

    Google Scholar 

  77. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin, New York, 1977.

    Google Scholar 

  78. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II, Function Spaces, Springer-Verlag, Berlin, New York, 1979.

    Google Scholar 

  79. V. I. Liokumovich, The existence of B-spaces with non-convex modules of convexity, Izv. Vyss. Ucebn. Zaved. Matematika 12 (1973), 43–50 (Russian).

    Google Scholar 

  80. E. Maluta, Uniformly normal structure and related coefficients, Pacific J. Math. 111 (1984), 357–369.

    Article  MathSciNet  MATH  Google Scholar 

  81. E. Maluta, S. Prus, Banach spaces which are dual to k-uniformly convex spaces, J. Math. Anal. Appl. 209 (1997), 479–491.

    Article  MathSciNet  MATH  Google Scholar 

  82. E. Maluta, S. Prus, M. Szczepanik, On Milman’s moduli for Banach spaces, (to appear).

    Google Scholar 

  83. W. Matuszewska, W. Orlicz, A note on the theory of s-normed spaces of (p-integrable functions, Studia Math. 21 (1961), 107–115.

    MathSciNet  MATH  Google Scholar 

  84. S. Mazur, Uber schwache Konvergenz in den Räumen LP, Studia Math. 4 (1933), 128–133.

    Google Scholar 

  85. V. D. Milman, Geometric theory of Banach spaces. Part II, Geometry of the unit sphere, Russian Math. Surveys 26 (1971), 79–163

    Article  MathSciNet  Google Scholar 

  86. V. D. Milman, Translation from Usp. Mat. Nauk 26 (1971), 73–149.

    MathSciNet  Google Scholar 

  87. V. D. Milman and A. Perelson, Infinite dimensional geometric moduli and type-cotype theory, Geometric aspects of Banach spaces, 11–38, London Math. Soc. Lecture Note Ser. 140. Cambridge University Press, Cambridge, 1989.

    Google Scholar 

  88. J. L. Nelson, K. L. Singh, J. H. M. Witfield, Normal structure and nonexpansive mappings in Banach spaces, Nonlinear Analysis, 433–492, World Scientific Publ. Co., Singapore, 1987.

    Google Scholar 

  89. G. Nordlander, The modulus of convexity in normed linear spaces, Ark. Mat. 4 (1960), 15–17.

    MathSciNet  MATH  Google Scholar 

  90. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  92. S. A. Pichugov, Jung’s constant for the space L,,, Math. Notes 43 (1988), 348–354

    Article  MathSciNet  MATH  Google Scholar 

  93. S. A. Pichugov, Translation from Mat. Zametki 43 (1988) 604–614.

    MathSciNet  MATH  Google Scholar 

  94. S. A. Pichugov, Jung’s relative constant of the space L P, Ukr. Math. J. 42 (1990), 111–113

    MathSciNet  MATH  Google Scholar 

  95. S. A. Pichugov, Translation from Ukr. Mat. Zh. 42 (1990), 122–125.

    Article  MathSciNet  Google Scholar 

  96. G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326–350.

    MathSciNet  MATH  Google Scholar 

  97. S. Prus, Nearly uniformly smooth Banach spaces, Boll. Un. Mat. Ital. (7) 3B (1989), 507–521.

    MathSciNet  MATH  Google Scholar 

  98. S. Prus, On Bynum’s fixed point theorem, Atti. Sem. Mat. Fis. Univ. Modena 38 (1990), 535–545.

    MathSciNet  MATH  Google Scholar 

  99. S. Prus, Some estimates for the normal structure coefficient in Banach spaces, Rend. Circ. Mat. Palermo 40 (1991), 128–135.

    Article  MathSciNet  MATH  Google Scholar 

  100. S. Prus, Banach spaces with the uniform Opial property, Nonlinear Anal. 18 (1992), 697–704.

    Article  MathSciNet  MATH  Google Scholar 

  101. S. Prus, Banach spaces and operators which are nearly uniformly convex, Dissertationes Math. (Rozprawy Mat.) 363 (1997), 1–46.

    MathSciNet  Google Scholar 

  102. B. N. Pshenichnii, Necessary Conditions for An Extremum, Marcel Dekker, New York, 1971.

    Google Scholar 

  103. N. Routledge, A result in Hilbert space, Quart. J. Math. 3 (1952), 12–18.

    Article  MathSciNet  MATH  Google Scholar 

  104. T. Shimizu, W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topological Methods in Nonlinear Anal. 8 (1996), 197–203.

    MathSciNet  MATH  Google Scholar 

  105. B. Sims, M. Smyth, On non-uniform conditions giving weak normal structure, Quaestiones Math. 18 (1995), 9–19.

    Article  MathSciNet  MATH  Google Scholar 

  106. M. A. Smith, B. Turett, Some examples concerning normal and uniform normal structure in Banach spaces, J. Austral. Math. Soc. (Series A) 48 (1990), 223–234.

    Article  MathSciNet  MATH  Google Scholar 

  107. F. Sullivan, A generalization of uniformly rotund Banach spaces, Canad. J. Math. 31 (1979), 628–636.

    MATH  Google Scholar 

  108. S. Swaminathan, Normal structure in Banach spaces and its generalizations, Fixed Points and Nonexpansive Mappings, 201–215, Contemp. Math. 18 (1983).

    Google Scholar 

  109. J. H. Wells, L. R. Williams, Embeddings and Extensions in Analysis, Springer-Verlag, New York, Heidelberg, 1975.

    Google Scholar 

  110. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), 1127–1138.

    Article  MathSciNet  MATH  Google Scholar 

  111. X. T. Yu, k-UR spaces are NUC spaces, Kexue Tongbao (Chinese) 28 (1983), 1473–1475 (Chinese).

    Google Scholar 

  112. C. Zälinescu, On uniformly convex functions, J. Math. Anal. Appl. 95 (1983), 344–374.

    Article  MathSciNet  Google Scholar 

  113. V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971), 1–33.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Maria Curie-Sktodowski University, 20-031, Lublin, Poland

    Stanisław Prus

Authors
  1. Stanisław Prus
    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

Prus, S. (2001). Geometrical Background of Metric Fixed Point Theory. 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_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1748-9_4

  • 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