Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property

Ronald Bruck; Tadeusz Kuczumow; Simeon Reich

Colloquium Mathematicae (1993)

  • Volume: 65, Issue: 2, page 169-179
  • ISSN: 0010-1354

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Bruck, Ronald, Kuczumow, Tadeusz, and Reich, Simeon. "Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property." Colloquium Mathematicae 65.2 (1993): 169-179. <http://eudml.org/doc/210212>.

@article{Bruck1993,
author = {Bruck, Ronald, Kuczumow, Tadeusz, Reich, Simeon},
journal = {Colloquium Mathematicae},
keywords = {convergence of iterates; uniform Opial property; asymptotically nonexpansive mapping; asymptotically nonexpensive mappings in the intermediate sense; Banach space with the Opial condition; fixed points},
language = {eng},
number = {2},
pages = {169-179},
title = {Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property},
url = {http://eudml.org/doc/210212},
volume = {65},
year = {1993},
}

TY - JOUR
AU - Bruck, Ronald
AU - Kuczumow, Tadeusz
AU - Reich, Simeon
TI - Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property
JO - Colloquium Mathematicae
PY - 1993
VL - 65
IS - 2
SP - 169
EP - 179
LA - eng
KW - convergence of iterates; uniform Opial property; asymptotically nonexpansive mapping; asymptotically nonexpensive mappings in the intermediate sense; Banach space with the Opial condition; fixed points
UR - http://eudml.org/doc/210212
ER -

References

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  1. [1] J.-B. Baillon, Un théorème de type ergodique pour les contractions non linéaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Sér. A 280 (1975), 1511-1514. Zbl0307.47006
  2. [2] J.-B. Baillon and R. E. Bruck, Ergodic theorems and the asymptotic behavior of contraction semigroups, preprint. 
  3. [3] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486-490. Zbl0526.46037
  4. [4] R. E. Bruck, On the almost-convergence of iterates of a nonexpansive mapping in Hilbert space and the structure of the weak ω-limit set, Israel J. Math. 29 (1978), 1-16. Zbl0367.47037
  5. [5] R. E. Bruck, Asymptotic behavior of nonexpansive mappings, in: Contemp. Math. 18, Amer. Math. Soc., 1983, 1-47. Zbl0528.47039
  6. [6] J. M. Dye, T. Kuczumow, P.-K. Lin and S. Reich, Random products of nonexpansive mappings in spaces with the Opial property, ibid. 144, 1993, to appear. Zbl0803.47052
  7. [7] M. Edelstein and R. C. O'Brien, Nonexpansive mappings, asymptotic regularity, and successive approximations, J. London Math. Soc. (2) 1 (1978), 547-554. Zbl0421.47031
  8. [8] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174. Zbl0256.47045
  9. [9] K. Goebel and T. Kuczumow, Irregular convex sets with the fixed point property for nonexpansive mappings, Colloq. Math. 40 (1978), 259-264. Zbl0418.47031
  10. [10] J. Górnicki, Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces, Comment. Math. Univ. Carolinae 30 (1989), 249-252. Zbl0686.47045
  11. [11] J. Górnicki, Nonlinear ergodic theorems for asymptotically nonexpansive mappings in Banach spaces satisfying Opial's condition, J. Math. Anal. Appl. 161 (1991), 440-446. Zbl0756.47037
  12. [12] J. P. Gossez and E. Lami-Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 565-575. Zbl0223.47025
  13. [13] S. Ishikawa, Fixed points and iterations of nonexpansive mappings in Banach space, Proc. Amer. Math. Soc. 5 (1976), 65-71. Zbl0352.47024
  14. [14] W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17 (1974), 339-346. Zbl0286.47034
  15. [15] T. Kuczumow, Weak convergence theorems for nonexpansive mappings and semigroups in Banach spaces with Opial's property, Proc. Amer. Math. Soc. 93 (1985), 430-432. Zbl0585.47043
  16. [16] T. Kuczumow and S. Reich, Opial's property and James' quasi-reflexive spaces, preprint. Zbl0818.46019
  17. [17] C. Lennard, A new convexity property that implies a fixed point property for L 1 , Studia Math. 100 (1991), 95-108. 
  18. [18] T. C. Lim, Asymptotic center and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980), 135-143. Zbl0454.47046
  19. [19] H. Oka, Nonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive mappings, Nonlinear Anal. 18 (1992), 619-635. Zbl0753.47044
  20. [20] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. Zbl0179.19902
  21. [21] S. Prus, Banach spaces with the uniform Opial property, Nonlinear Anal. 18 (1992), 697-704. Zbl0786.46023
  22. [22] S. Reich, Nonlinear evolution equations and nonlinear ergodic theorems, ibid. 1 (1976/77), 319-330. Zbl0359.34059
  23. [23] S. Reich, Almost convergence and nonlinear ergodic theorems, J. Approx. Theory 24 (1978), 269-272. Zbl0404.47032
  24. [24] S. Reich, A note on the mean ergodic theorem for nonlinear semigroups, J. Math. Anal. Appl. 91 (1983), 547-551. Zbl0521.47034
  25. [25] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, ibid. 158 (1991), 407-413. Zbl0734.47036
  26. [26] K.-K. Tan and H.-K. Xu, A nonlinear ergodic theorem for asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 45 (1992), 25-36. 
  27. [27] D. Tingley, An asymptotically nonexpansive commutative semigroup with no fixed points, Proc. Amer. Math. Soc. 97 (1986), 107-113. Zbl0592.47049
  28. [28] H.-K. Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal. 16 (1991), 1139-1146. Zbl0747.47041

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