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A fixed-point theorem for asymptotically contractive mappings

Author: Jean-Paul Penot
Journal: Proc. Amer. Math. Soc. 131 (2003), 2371-2377
MSC (2000): Primary 47H10, 47H09, 54H25, 55M20
Published electronically: March 11, 2003
MathSciNet review: 1974633
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Abstract | References | Similar Articles | Additional Information

Abstract: We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets.

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  • 1. A. AGADI and J.-P. PENOT, Asymptotic approximation of sets with application in mathematical programming, preprint, Univ. of Pau, February 1996.
  • 2. Y. BENYAMINI and J. LINDENSTRAUSS, Geometric Nonlinear Functional Analysis, Amer. Math. Soc. Colloquium Publications #40 Amer. Math. Soc., Providence (2000). MR 2001b:46001
  • 3. F.E. BROWDER, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. 54 (1965), 1041-1044. MR 32:4574
  • 4. F.E. BROWDER, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure Math. vol. 18, Amer. Math. Soc., Providence (1976). MR 53:8982
  • 5. J.-P. DEDIEU, Cône asymptote d'un ensemble non convexe. Application à l'optimisation, C. R. Acad. Sci. Paris 287 (1977), 501-503. MR 56:16320
  • 6. D. GÖHDE, Zum prinzip der kontraktiven abbildung, Math. Nach. 30 (1965), 251-258. MR 32:8129
  • 7. V.I. ISTRATESCU, Fixed point theory, an introduction, Reidel, Dordrecht (1981). MR 83c:54065
  • 8. W. KIRK, A fixed-point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. MR 32:6436
  • 9. W.A. KIRK, Nonexpansive mappings and asymptotic regularity, Nonlinear Anal., Theory, Methods, Appl. 40A (2000), 323-332. MR 2001m:47116
  • 10. W.A. KIRK, The fixed point property and mappings which are eventually nonexpansive, in Kartsatos, Athanassios G. (ed.), Theory and applications of nonlinear operators of accretive and monotone type, Lect. Notes Pure Appl. Math. 178, Marcel Dekker, New York (1996), 141-147. MR 97e:47099
  • 11. W.A. KIRK, C. M. YANEZ and S.S. SHIN, Asymptotically nonexpansive mappings, Nonlinear Anal., Theory, Methods, Appl. 33 (1) (1998), 1-12. MR 99g:47128
  • 12. M.A. KRASNOSELSKII, Positive solutions of operator equations, Noordhoff, Groningen (1964). MR 31:6107
  • 13. D. T. LUC, Recession maps and applications, Optimization 27 (1993), 1-15. MR 95e:49021
  • 14. D. T. LUC, Recessively compact sets: uses and properties, Set-Valued Anal. 10 (2002), 15-35.
  • 15. D. T. LUC and J.-P. PENOT, Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 (2001), 4095-4121. MR 2002e:49028
  • 16. J.-P. PENOT, Fixed point theorems without convexity, Mémoire Soc. Math. de France n ${^{\circ}}$ 60 (1977), 129-152. MR 81c:47061
  • 17. J.-P. PENOT, Compact nets, filters and relations, J. Math. Anal. Appl. 93 (1983), 400-417. MR 84h:49032
  • 18. J.-P. PENOT, What is quasiconvex analysis?, Optimization 47 (2000), 35-110. MR 2001a:49024
  • 19. J.-P. PENOT, A metric approach to asymptotic analysis, preprint, Univ. of Pau, June 2001.
  • 20. J.-P. PENOT and C. ZALINESCU, Continuity of usual operations and variational convergences, preprint, Univ. of Pau, 2000 and 2001, to appear in Set-Valued Analysis.
  • 21. B.D. ROUHANI and W.A. KIRK, Asymptotic properties of nonexpansive iterations in reflexive spaces, J. Math. Anal. Appl. 236 (2) (1999), 281-289. MR 2001a:47056
  • 22. S. SINGH, B. WATSON and P. SRIVASTAVA, Fixed point theory and best approximations: the KKM principle, Kluwer (1997). MR 99a:47087
  • 23. C. ZALINESCU, Recession cones and asymptotically compact sets, J. Optim. Theory Appl., 77 (1993), 209-220. MR 94d:90057
  • 24. E. ZEIDLER, Nonlinear Functional Analysis and its Applications, Part 1: Fixed-Point Theorems, Springer Verlag, New York (1986). MR 87f:47083

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Additional Information

Jean-Paul Penot
Affiliation: Laboratoire de Mathématiques Appliquées, CNRS F.R.E. 2570, Faculté des Sciences, av. de l’Université, 64000 Pau, France

Keywords: Asymptotic, asymptotic cone, contraction, derivative at infinity, firm asymptotic cone, fixed point, nonexpansive map
Received by editor(s): February 19, 2002
Published electronically: March 11, 2003
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society

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