Points fixes et théorèmes ergodiques dans les espaces L¹(E)

Mourad Besbes

Studia Mathematica (1992)

  • Volume: 103, Issue: 1, page 79-97
  • ISSN: 0039-3223


We prove that for each linear contraction T : X → X (∥T∥ ≤ 1), the subspace F = {x ∈ X : Tx = x} of fixed points is 1-complemented, where X is a suitable subspace of L¹(E*) and E* is a separable dual space such that the weak and weak* topologies coincide on the unit sphere. We also prove some related fixed point results.

How to cite


Besbes, Mourad. "Points fixes et théorèmes ergodiques dans les espaces L¹(E)." Studia Mathematica 103.1 (1992): 79-97. <http://eudml.org/doc/215937>.

author = {Besbes, Mourad},
journal = {Studia Mathematica},
keywords = {linear contraction; fixed points; 1-complemented},
language = {fre},
number = {1},
pages = {79-97},
title = {Points fixes et théorèmes ergodiques dans les espaces L¹(E)},
url = {http://eudml.org/doc/215937},
volume = {103},
year = {1992},

AU - Besbes, Mourad
TI - Points fixes et théorèmes ergodiques dans les espaces L¹(E)
JO - Studia Mathematica
PY - 1992
VL - 103
IS - 1
SP - 79
EP - 97
LA - fre
KW - linear contraction; fixed points; 1-complemented
UR - http://eudml.org/doc/215937
ER -


  1. [Als] D. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981), 423-424. Zbl0468.47036
  2. [A-N] E. Asplund and I. Namioka, A geometric proof of Ryll-Nardzewski's fixed point theorem, Bull. Amer. Math. Soc. 73 (1967), 443-445. Zbl0177.40404
  3. [Bal.1] E. J. Balder, Infinite-dimensional extension of a theorem of Komlós, Probab. Theory Related Fields 81 (1989), 185-188. Zbl0643.60008
  4. [Bal.2] E. J. Balder, On uniformly bounded sequences in Orlicz spaces, Bull. Austral. Math. Soc. 41 (1990), 495-502. Zbl0717.46029
  5. [Bal.3] E. J. Balder, New sequential compactness results for spaces of scalarly integrable functions, J. Math. Anal. Appl. 151 (1990), 1-16. Zbl0733.46015
  6. [Beh] E. Behrends, M-Structure and the Banach-Stone Theorem, Lecture Notes in Math. 736, Springer, Berlin 1979. 
  7. [Bes.1] M. Besbes, Points fixes des contractions définies sur un convexe L 0 -fermé de L¹, C. R. Acad. Sci. Paris Sér. I 311 (1990), 243-246. 
  8. [Bes.2] M. Besbes, Complémentation de l'ensemble des points fixes d'une contraction, preprint, 1990. 
  9. [B-D-D-L] M. Besbes, S. Dilworth, P. Dowling and C. Lennard, New convexity and fixed point properties in Hardy and Lebesgue-Bochner spaces, preprint. Zbl0804.46044
  10. [Bru] R. Bruck, Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc. 179 (1973), 251-262. 
  11. [Day] M. M. Day, Normed Linear Spaces, Springer, Berlin 1973. 
  12. [Die] J. Diestel, Sequences and Series in Banach Spaces, Springer, Berlin 1984. 
  13. [Gar] D. J. H. Garling, Subsequence principles for vector-valued random variables, Math. Proc. Cambridge Philos. Soc. 86 (1979), 301-311. Zbl0416.60009
  14. [Gin] E. Giner, Espaces intégraux de type Orlicz, thèse, Université des Sciences et Techniques du Languedoc, Perpignan 1977. 
  15. [God] G. Godefroy, Sous-espaces bien disposés de L¹. Applications, Trans. Amer. Math. Soc. 286 (1984), 227-249. Zbl0521.46012
  16. [Hof] L. D. Hoffmann, Pseudo-uniform convexity of H¹ in several variables, Proc. Amer. Math. Soc. 26 (1970), 609-614. Zbl0208.15801
  17. [Huf] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743-749. Zbl0505.46011
  18. [Kak] S. Kakutani, Two fixed point theorems concerning bicompact convex sets, Proc. Imperial Acad. Tokyo 14 (1938), 242-245. Zbl64.1101.03
  19. [K-T] M. A. Khamsi and P. Turpin, Fixed points of nonexpansive mappings in Banach lattices, Proc. Amer. Math. Soc. 105 (1) (1989), 102-110. Zbl0677.47032
  20. [Kom] J. Komlós, A generalization of a problem of Steinhaus, Acta Math. Acad. Sci. Hungar. 18 (1967), 217-229. Zbl0228.60012
  21. [Kre.1] U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin 1985. 
  22. [Kre.2] U. Krengel, On the global limit behaviour of Markov chains and of general nonsingular Markov processes, Z. Wahrsch. Verw. Gebiete 6 (1966), 302-316. Zbl0218.60060
  23. [Len] C. Lennard, A new convexity property that implies a fixed point property for L₁, Studia Math. 100 (1991), 95-108. Zbl0762.46007
  24. [Lim] T.-C. Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980), 135-143. Zbl0454.47046
  25. [Mar] A. Markov, Quelques théorèmes sur les ensembles abéliens, Dokl. Akad. Nauk SSSR (N.S.) 10 (1936), 311-313. Zbl0014.08201
  26. [Maz] P. Mazet, manuscrit non publié. 
  27. [Opi] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597. Zbl0179.19902

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.