Skip to main content
SpringerLink
Log in

Classical Theory of Nonexpansive Mappings

  • Chapter
Handbook of Metric Fixed Point Theory

Abstract

Mappings which are defined on metric spaces and which do not increase distances between pairs of points and their images are called nonexpansive. Thus an abstract metric space is all that is needed to define the concept. At the same time, the more interesting results seem to require some notion of topology; more specifically a topology which assures that closed metric balls are compact. This is not a serious limitation, however, because many spaces which arise naturally in functional analysis possess such topologies; most notably the weak and weak* topologies in Banach 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. A. Aksoy and M. A. Khamsi, Nonstandard Methods in Fixed Point Theory, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1990.

    Google Scholar 

  2. D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), 423 – 424.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. J. M. Ayerbe Toledano, T. Dominguez Benavides, and G. López Acedo, Measures of noncompactness in Metric Fixed Point Theory, Birkhäuser Verlag, 1997.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. J. B. Baillon, Qeulques aspects de la théorie des points fixes dans les éspaces Banach, Serninaire d’Analysis Fonctionnelle, Ecole Polytechnique I, 1978 – 79.

    Google Scholar 

  7. J. B. Baillon, Nonexpansive mappings and hyperconvex spaces, in Fixed Point Theory and its Applications (R. F. Brown, ed.), Contemporary Math. vol. 72, Amer. Math. Soc., Providence, 1988, 11 – 19.

    Google Scholar 

  8. J. B. Baillon and R. Schöneberg, Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 81 (1981), 257 – 264.

    Article  MathSciNet  MATH  Google Scholar 

  9. H. F. Bohnenblust and S. Karlin, On a theorem of Ville: in Contributions to the Theory of Games (H. W. and A. W. Tucker, eds.), Princeton Univ. Press., Princeton, 1950, pp. 155 – 160.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  11. J. Borwein, S. Reich, and I Shafrir, Krasnoselskii-Mann iterations in normed spaces, Canad. Math. Bull. 35 (1992), 21 – 28.

    Article  MathSciNet  MATH  Google Scholar 

  12. H. Brezis and F. Browder, A general principle on ordered sets in nonlinear functional analysis, Advances in Math. 21 (1976), 355 – 364.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Brondsted, Fixed points and partial orders, Proc. Amer. Math. Soc. 60 (1976), 365 – 366.

    MathSciNet  Google Scholar 

  14. F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert space, Proc. Nat. Acad. Sci. USA 53 (1965), 1272 – 1276.

    Article  MathSciNet  MATH  Google Scholar 

  15. F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA 54 (1965), 1041 – 1044.

    Article  MathSciNet  MATH  Google Scholar 

  16. F. E. Browder, On a theorem of Caristi and Kirk, in Fixed Point Theory and its Applications ( S. Swaminathan, ed.), Academic Press, New York, 1976, pp. 23 – 27.

    Google Scholar 

  17. F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in a Banach space, Bull. Amer. Math. Soc. 74 (1968), 660 – 665.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. E. Browder, Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces, Proc. Symp. Pure Math. 18, pt. 2, Amer. Math. Soc., Providence, 1976.

    Google Scholar 

  19. F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 571 – 575.

    Article  MathSciNet  MATH  Google Scholar 

  20. R. E. Bruck, Properties of fixed point sets of nonexpansive mappings, Trans. Amer. Math. Soc. 179 (1973), 251 – 262.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. E. Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math. 47 (1973), 341 – 355.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. E. Bruck, A common fixed point theorem for a commuting family of nonexpansive mappings. Pacific J. Math. 53 (1974), 59 – 71.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math. 32 (1979), 107 – 116.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. E. Bruck, Structure of the approximate fixed point sets in general Banach spaces, in Fixed Point Theory and Applications ( J. B. Baillon and M. Théra, eds.), Longman Sci. Tech., Essex, 1991, pp. 91 – 96.

    Google Scholar 

  25. R. E. Bruck and K. Goebel, Bizarre fixed-point sets, in Fixed Point Theory and Applications ( K. K. Tan, ed.), World Sci. Publishing, River Edge, NJ, 1992, pp. 67 – 70.

    Google Scholar 

  26. J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241 – 251.

    Article  MathSciNet  MATH  Google Scholar 

  27. E. Cassini and E. Maluta, Fixed points for uniformly lipschitzian mappings in spaces with uniformly normal structure, Nonlinear Anal. 9 (1985), 103 – 108.

    Article  MathSciNet  Google Scholar 

  28. M. M. Day, R. C. James, and S. Swaminathan, Normed linear spaces that are uniformly convex in every direction, Canad. J. Math. 23 (1971), 1051 – 1059.

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  30. T. Dominguez Benavides, Fixed point thoerems for uniformly lipschitzian mappings and asymptotically regular mappings, Nonlinear Anal. 32 (1998), 15 – 27.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Downing and B. Turett, Some properties of the characteristic of convexity relating to fixed poing theory, Pacific J. Math. 104 (1983), 343 – 350.

    MathSciNet  MATH  Google Scholar 

  32. D. van Du1st, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. 25 (1982), 139 – 144.

    Article  MathSciNet  MATH  Google Scholar 

  33. M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74 – 79.

    Article  MathSciNet  MATH  Google Scholar 

  34. M. Edelstein, The construction of an asymptotic center with a fixed point property, Bull. Amer. Math. Soc. 78 (1972), 206 – 208.

    MathSciNet  MATH  Google Scholar 

  35. M. Edelstein and R. C. O’Brien, Nonexpansive mappings, asymptotic regularity, and successive approximations, J. London Math. Soc. 17 (1978), 547 – 554.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Elton, Pei-Kee Lin, E. Odell and S. Szarek, Remarks on the fixed point problem for nonexpansive maps, Fixed Points and Nonexpansive Maps (R. Sine, ed.), Contemporary Math. vol. 18, Amer. Math. Soc., Princeton, 1983, pp. 87 – 120.

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. J. Garcia Falset, E. Llorens Fuster, and B. Sims, Fixed point theory for almost convex functions, Nonlinear Anal. 32 (1998), 601 – 608.

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Genel and J. Lindenstrauss, An example concerning fixed points, Israel J. Math. 22 (1975), 81 – 86.

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. G. Godefroy and N. Kalton, The ball topology and its applications, Banach Space Theory (B. L. Lin, ed.), Contemporary Math. vol. 85, Amer. Math. Soc., Providence, RI, pp. 195–237.

    Google Scholar 

  43. K. Goebel, Convexity of balls and fixed point theorems for mappings with nonexpansive square, Compositio Math. 22 (1970), 269 – 274.

    MathSciNet  MATH  Google Scholar 

  44. K. Goebel, On the structure of minimal invariant sets for nonexpansive mappings, Ann. Univ. Marine Curie-Sklodowska 29 (1975), 73 – 77.

    MathSciNet  Google Scholar 

  45. K. Goebel, On a fixed point theorem for multivalued nonexpansive mappings, Ann. Univ. Marie Curie-Sklodowska 29 (1975), 69 – 72.

    MathSciNet  Google Scholar 

  46. K. Goebel, Properties of minimal invariant sets for nonexpansive mappings, Topological Methods in Nonlinear Analysis 12 (1998), 367 – 373.

    MathSciNet  MATH  Google Scholar 

  47. K. Goebel and W. A. Kirk, A fixed point theorem for mappings whose iterates have uniform Lipschitz constant, Studia Math. 71 (1973), 135 – 140.

    MathSciNet  Google Scholar 

  48. K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, in Topological Methods in Nonlinear Functional Analysis (S. P. Singh and S. Thomeier, eds.), Contemporary Math. 21, Amer. Math. Soc., Providence, 1983, pp. 115 – 123.

    Google Scholar 

  49. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990, 244 pp.

    Google Scholar 

  50. K. Goebel, W. A. Kirk, and T. L. Thele, Uniformly lipschitzian families of transformations in Banach spaces, Canadian J. Math. 26 (1974), 1245 – 1256.

    Article  MathSciNet  MATH  Google Scholar 

  51. K. Goebel and T. Komorowski, Retracting balls onto spheres and minimal displacement problems, Fixed Points and Applications (J. B. Baillon and M. Théra, eds), Longman Scientific & Technical, Essex, 1991, pp. 155 – 172.

    Google Scholar 

  52. K. Goebel and T. Kuczumow, A contribution to the theory of nonexpansive mappings, Bull. Calcutta Math. Soc. 70 (1978), 355 – 357.

    MathSciNet  MATH  Google Scholar 

  53. K. Goebel and T. Kuczumow, Irregular convex sets with the fixed point property for nonexpansive mappings, Colloq. Math. 40 (1978), 259 – 264.

    MathSciNet  Google Scholar 

  54. K. Goebel and B. Sims, More on minimal invariant sets for nonexpansive mappings, to appear in Proceedings of the Third Congress of Nonlinear Analysts, Catania, Italy, 2000.

    Google Scholar 

  55. D. Göhde, Zum Prinzip der Kontraktiven Abbildung, Math. Nachr. 30 (1965), 251 – 258.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  57. J. Isbell, Six theorems about injective metric spaces, Comment. Math. Hely. 39 (1964), 65 – 76.

    Article  MathSciNet  MATH  Google Scholar 

  58. J. Isbell, Injective envelopes of Banach spaces are rigidly attached, Bull. Amer. Math. Soc. 70 (1984), 727 – 729.

    Article  MathSciNet  Google Scholar 

  59. R. C. James, Uniformly nonsquare Banach spaces, Ann. Math. 80 (1964), 542 – 550.

    Article  MATH  Google Scholar 

  60. E. Jawhari, D. Misane, and M. Pouzet, “Retracts: graphs and ordered sets from the metric point of view”, Combinatories and Ordered Graphs (I. Rival, ed.), Contemp. Math. 57, Amer. Math. Soc., Providence, RI, 1986, pp. 175 – 226.

    Google Scholar 

  61. J. Jaworowski, W. A. Kirk, and S. Park, Antipodal Points and Fixed Points, Research Inst. of Math. GARC, Seoul, Korea, 1995.

    Google Scholar 

  62. A. Jiménez Melado, Una propriedad geométrica de los espacios de Banach relacionada con la teoricr del punto fijo, Ph.D. Dissertation, Univ. de Malaga (Spain ), 1988.

    Google Scholar 

  63. A. Jiménez Melado and E. Lloréns Fuster, A sufficient condition for the fixed point property, Nonlinear Anal. 20 (1993), 849 – 853.

    Article  MathSciNet  MATH  Google Scholar 

  64. A. Jiménez Melado and E. Lloréns Foster, The fixed point property for some uniformly nonsquare Banach spaces, Bollettino U. M. I. (7) 10-A(1996), 587–595.

    Google Scholar 

  65. A. Jiménez Melado and E. Lloréns Faster, Opial modulus and stability of the fixed point property. Nonlinear Anal. 39 (2000), 341 – 349.

    Article  MathSciNet  MATH  Google Scholar 

  66. S. Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Math. J. 8 (1941), 457 – 459.

    Article  MathSciNet  Google Scholar 

  67. S. Kaniel, Construction of a fixed point for contractions in a Banach space, Israel J. Math. 9(1 971), 535-540.

    Google Scholar 

  68. L. A. Karlovitz, On nonexpansive mappings, Proc. Amer. Math. Soc. 55 (1976), 321 – 325.

    Article  MathSciNet  MATH  Google Scholar 

  69. L. A. Karlovitz, Existence of a fixed point for a nonexpansive map in a space without normal structure, Pacific J. Math. 66 (1976), 153 – 159.

    Article  MathSciNet  MATH  Google Scholar 

  70. S. Kasahara, On fixed points in partially ordered sets and Kirk-Caristi Theorem, Math. Sem. Notes Kobe Univ. 3 (1975), 229 – 232.

    Google Scholar 

  71. T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967), 508 – 520.

    Google Scholar 

  72. John L. Kelley, General Topology, Von Nostrand, 1955, pg. 139.

    Google Scholar 

  73. M. A. Khamsi, Points fixe de contractions dans les espaces de Banach, Seminaire d’initiation a l’Analyse 86(1986–87), Univ. Paris VI, pp. 3 – 9.

    Google Scholar 

  74. M. A. Khamsi, Etude de la propriété du point fixe dans les éspaces métriques et les éspaces de Banach, Thèse de Doctorate de l’Université Paris V I, 1987.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  76. M. A. Khamsi, On normal structure, fixed point property and contractions of type (‘y), Proc. Amer. Math. Soc. 106(198910, 995–1001.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  79. Tae Hwa Kim and W. A. Kirk, Fixed point theorems for lipschitzian mappings in Banach spaces, Nonlinear Anal. 26 (1996), 1905 – 1911.

    Article  MathSciNet  MATH  Google Scholar 

  80. 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 

  81. W. A. Kirk, Krasnoselskii’s iteration process in hyperbolic space, Numer. Funct. Anal. and Optimiz.,4(1981/82), 371–381.

    Google Scholar 

  82. W. A. Kirk, Nonexpansive mappings in metric and Banach spaces, Rend. Sem. Mat. Fis. Milano 11 (1981), 133 – 144

    Article  MathSciNet  Google Scholar 

  83. W. A. Kirk, Nonexpansive mappings in product spaces, set valued mappings, and k-uniform convexity, in Nonlinear Functional Analysis and its Applications (F. Browder, ed.), Proc. Symp. Pure Math. vol. 45, pt 2, Amer. Math. Soc., Providence, 1986, pp. 51 – 64.

    Google Scholar 

  84. W. A. Kirk, Nonexpansive mappings and nonexpansive retracts, Fixed Point Theory and Applications (K. K. Tan, ed.), World Scientific, Singapore, 1992, pp. 137 – 155.

    Google Scholar 

  85. W. A. Kirk, Compactness and countable compactness in weak topologies, Studia Math. 112 (1995), 143 – 250.

    MathSciNet  Google Scholar 

  86. W. A. Kirk, Nonexpansive mappings in separable product spaces, Bollettino U.M.I. (7) 9A (1995), 239 – 244.

    MathSciNet  MATH  Google Scholar 

  87. W. A. Kirk, The fixed point property and mappings which are eventually nonexpansive, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (A. G. Kartsatos, ed.), Marcel Dekker, New York, Basel, Hong Kong, 1996, pp. 141 – 147.

    Google Scholar 

  88. W. A. Kirk, Nonexpansive mappings and asymptotic regularity, Nonlinear Anal. 40(2000), 323–332.

    Google Scholar 

  89. R. Maríka, Some forms of the axiom of choice, Jahrb. Kurt Gödel Ges. vol. 1, Wien, 1988, pp. 24 – 34.

    Google Scholar 

  90. W. A. Kirk and C. Martinez Yanez, Nonexpansive mappings and locally nonexpansive mappings in product spaces, Nonlinear Anal. 12 (1988), 719 – 725.

    Article  MathSciNet  MATH  Google Scholar 

  91. W. A. Kirk and R. Schöneberg, Zeros of m-accretive operators, Israel J. Math. 35 (1980), 1 – 8.

    MATH  Google Scholar 

  92. W. A. Kirk and Y. Sternfeld, The fixed point property for nonexpansive mappings in certain product spaces, Houston J. Math. 10 (1984), 207 – 214.

    MathSciNet  MATH  Google Scholar 

  93. W. A. Kirk and B. G. Kang, A fixed point theorem revisited, J. Korean Math. Soc. 34 (1997), 285 – 291.

    MathSciNet  MATH  Google Scholar 

  94. W. A. Kirk, C. Martinez Yanez, and S. S. Shin, Asymptotically nonexpansive mappings, Nonlinear Anal. 33 (1998), 1 – 12.

    Article  MathSciNet  MATH  Google Scholar 

  95. T. Kuczumow, Fixed point theorems in product spaces, Proc. Amer. Math. Soc., 108 (1990), 727 – 729.

    Article  MathSciNet  MATH  Google Scholar 

  96. T. Kuczumow and A. Stachura, Bruck’s retraction method, Fixed Point Theory and Applications (J. B. Baillon and M. Théra, eds.), Longman Scientific & Technical, Essex, 1991, pp. 285 – 292.

    Google Scholar 

  97. J. Kulesza and T. C. Lim, On weak compactness and sequential weak compactness in fixed point theory, Proc. Amer. Math. Soc.124(1996), 3345–3349..

    Google Scholar 

  98. H. Lacey, The Isometric Theory of Classical Banach Spaces, Springer-Verlag, New York, Heidelberg, Berlin, 1974.

    Google Scholar 

  99. E. Lami Dozo, Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer. Math. Soc. 38 (1973), 286 – 292.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  101. E. A. Lifschitz, Fixed point theorems for operators in strongly convex spaces, Voronez Gos. Univ. Trudy Mt. Fak. 16 (1975), 23 – 28.

    Google Scholar 

  102. T. C. Lim, Characterizations of normal structure, Proc. Amer. Math. Soc. 43 (1974), 313 – 319.

    Article  MathSciNet  MATH  Google Scholar 

  103. T. C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80 (1974), 1123 – 1126.

    Article  MathSciNet  MATH  Google Scholar 

  104. T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179 – 182.

    Article  MathSciNet  Google Scholar 

  105. T. C. Lim, Fixed point theorems for uniformly Lipschitz mappings in L 5 spaces, Nonlinear Anal. 7 (1983), 555 – 563.

    Article  MathSciNet  MATH  Google Scholar 

  106. T. C. Lim, Asymptotic centers and nonexpansive mappings in some conjugate spaces, Pacific J. Math. 116 (1985), 135 – 143.

    Google Scholar 

  107. P. K. Lin, Unconditional bases and fixed points of nonexpansive mappings, Pacific J. Math. 116 (1985), 69 – 76.

    Article  MathSciNet  MATH  Google Scholar 

  108. P. K. Lin, The Browder-Göhde property in product spaces, Houston J. Math. 13 (1987), 235 – 240.

    MathSciNet  MATH  Google Scholar 

  109. P. K. Lin, Stability of the fixed point property of Hilbert space, Proc. Amer. Math. Soc. 127 (1999), 3573 – 3581.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  111. J. T. Markin, A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc. 60 (1978), 639 – 640.

    MathSciNet  Google Scholar 

  112. B. Maurey, Points fixes des contractions sur un convexe fermé de Li, Seminaire d’Analyse Fonctionelle, vol. 80–81, Ecole Polytechnique Palaiseau, 1981.

    Google Scholar 

  113. J. Molony, Construction of a sequence strongly converging to a fixed point of an asymptotically nonexpansive mapping, J. Math. Anal. Appl. 182 (1994), 589 – 593.

    Article  MathSciNet  Google Scholar 

  114. 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 

  115. S. Park, Equivalent formulations of Ekeland’s variational principle for approximate solutions of minimization problems and their applications, Operator Equations and Fixed Point Theorems (S. P. Singh, et al, eds.), MSRI-Korea Publ. vol. 1, 1986, pp. 55 – 68.

    Google Scholar 

  116. S. Park, On extensions of the Caristi-Kirk fixed point theorem, J. Korean Math. Soc. 19 (1983), 143 – 151.

    MathSciNet  MATH  Google Scholar 

  117. J. P. Penot, Fixed point theorems without convexity, Analyse Nonconvex ( 1977 Pau), Bull. Soc. Math. France, Memoire 60 (1980), 129 – 152.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  119. 5. Prus, Banach spaces which are uniformly noncreasy, Nonlinear Anal. 30 (1997), 2317 – 1214.

    Google Scholar 

  120. W. O. Ray, Nonexpansive mappings on unbounded convex domains, Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys. 26 (1978), 241 – 245.

    MATH  Google Scholar 

  121. W. O. Ray, The fixed point property and unbounded sets in Hilbert space, Tram. Amer. Math. Soc. 258 (1980), 531 – 537.

    Article  MATH  Google Scholar 

  122. W. O. Ray and R. Sine, Nonexpansive mappings with precompact orbits, Fixed Point Theory (E. Fadell and G. Fournier, eds.), Lecture Notes no. 86, Springer-Verlag, New York, Berlin, 1981, pp. 409 – 416.

    Google Scholar 

  123. S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274 – 276.

    Article  MathSciNet  MATH  Google Scholar 

  124. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), 287 – 292.

    Article  MathSciNet  MATH  Google Scholar 

  125. S. Reich, The almost fixed point property for nonexpansive mappings, Proc. Amer. Math. Soc. 88 (1983), 44 – 46.

    Article  MathSciNet  MATH  Google Scholar 

  126. I. Shafrir, The approximate fixed point property in Banach and hyperbolic spaces, Israel J. Math. 71 (1990), 211 – 223.

    Article  MathSciNet  MATH  Google Scholar 

  127. B. Sims, “Ultra”-techniques in Banach Space Theory, Queens’s Papers in Pure and Applied Mathematics No. 160, Queen’s University, Kingston, Ontario, 1982, 117 pp.

    Google Scholar 

  128. B. Sims, Geometric conditions sufficient for the weak and weak& fixed point property, in Fixed Point Theory and Applications ( K. K. Tan, ed.) World Sci. Publ., River Edge, NH, 1992, pp. 278 – 290.

    Google Scholar 

  129. R. Sine, Hyperconvexity and approximate fixed points, Nonlinear Anal. 3 (1979), 863 – 869.

    Article  MathSciNet  Google Scholar 

  130. R. Sine, On nonlinear contractions in sup norm spaces, Nonlinear Anal. 3 (1979), 885 – 890.

    Article  MathSciNet  MATH  Google Scholar 

  131. R. Sine, Hyperconvexity and nonexpansive multifunctions, Trans. Amer. Math. Soc. 315 (1989), 755 – 767.

    Article  MathSciNet  MATH  Google Scholar 

  132. R. Smarzewski, Strongly unique minimization of functionals in Banach spaces with applications to theory of approximation and fixed points, J. Math. Anal. Appl. 115 (1986), 155 – 172.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  135. F. Sullivan, Ordering and completeness of metric spaces, Nieuw Arch. Wisk. 29 (1981), 178 – 193.

    MathSciNet  MATH  Google Scholar 

  136. S. Troyanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173 – 180.

    MathSciNet  MATH  Google Scholar 

  137. B. Turett, A dual view of a theorem of Baillon, in Nonlinear Analysis and Applications (S. P. Singh and J. H. Burry,eds.), Marcel Dekker, New York, 1982, pp. 279 – 286.

    Google Scholar 

  138. van de Vel, M. L. J., Theory of Convexity Structures, North-Holland, Amsterdam, London, New York, Tokyo, 1993.

    Google Scholar 

  139. S. Willard, General Topology, Addison Wesley, Reading, Massachusetts, 1970.

    Google Scholar 

  140. C. S. Wong, Close-to-normal structure and its applications, J. Fanct. Anal. 16 (1974), 353 – 358.

    Article  MATH  Google Scholar 

  141. H. K. Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal. 16 (1991), 1139 – 1146.

    Article  MathSciNet  MATH  Google Scholar 

  142. E. Zeidler, Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems, Springer-Verlag, Berlin, 1986.

    Book  MATH  Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Maria Curie-Skłodowska University, 20-031, Lublin, Poland

    Kazimierz Goebel

  2. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1419, USA

    W. A. Kirk

Authors
  1. Kazimierz Goebel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. A. Kirk
    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

Goebel, K., Kirk, W.A. (2001). Classical Theory of Nonexpansive Mappings. 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_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-1748-9_3

  • 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