|
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
Posted:
March 11, 2003
MathSciNet review:
1974633
Full-text PDF Free Access
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.
- 1.
A. AGADI and J.-P. PENOT, Asymptotic approximation of sets with application in mathematical programming, preprint, Univ. of Pau, February 1996.
- 2.
Yoav
Benyamini and Joram
Lindenstrauss, Geometric nonlinear functional analysis. Vol.
1, American Mathematical Society Colloquium Publications,
vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673
(2001b:46001)
- 3.
Felix
E. Browder, Nonexpansive nonlinear operators in a Banach
space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965),
1041–1044. MR 0187120
(32 #4574)
- 4.
Felix
E. Browder, Nonlinear operators and nonlinear equations of
evolution in Banach spaces, Nonlinear functional analysis (Proc.
Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), Amer. Math.
Soc., Providence, R. I., 1976, pp. 1–308. MR 0405188
(53 #8982)
- 5.
Jean-Pierre
Dedieu, Cône asymptote d’un ensemble non convexe.
Application à l’optimisation, C. R. Acad. Sci. Paris
Sér. A-B 285 (1977), no. 7, A501–A503
(French, with English summary). MR 0458117
(56 #16320)
- 6.
Dietrich
Göhde, Zum Prinzip der kontraktiven Abbildung, Math.
Nachr. 30 (1965), 251–258 (German). MR 0190718
(32 #8129)
- 7.
Vasile
I. Istrăţescu, Fixed point theory, Mathematics
and its Applications, vol. 7, D. Reidel Publishing Co., Dordrecht,
1981. An introduction; With a preface by Michiel Hazewinkel. MR 620639
(83c:54065)
- 8.
W.
A. Kirk, A fixed point theorem for mappings which do not increase
distances, Amer. Math. Monthly 72 (1965),
1004–1006. MR 0189009
(32 #6436)
- 9.
W.
A. Kirk, Nonexpansive mappings and asymptotic regularity,
Nonlinear Anal. 40 (2000), no. 1-8, Ser. A: Theory
Methods, 323–332. Lakshmikantham’s legacy: a tribute on his
75th birthday. MR 1768416
(2001m:47116), http://dx.doi.org/10.1016/S0362-546X(00)85019-1
- 10.
W.
A. Kirk, The fixed point property and mappings which are eventually
nonexpansive, Theory and applications of nonlinear operators of
accretive and monotone type, Lecture Notes in Pure and Appl. Math.,
vol. 178, Dekker, New York, 1996, pp. 141–147. MR 1386675
(97e:47099)
- 11.
W.
A. Kirk, Carlos
Martinez Yañez, and Sang
Sik Shin, Asymptotically nonexpansive mappings, Nonlinear
Anal. 33 (1998), no. 1, 1–12. MR 1623029
(99g:47128), http://dx.doi.org/10.1016/S0362-546X(97)00541-5
- 12.
M.
A. Krasnosel′skiĭ, Positive solutions of operator
equations, Translated from the Russian by Richard E. Flaherty; edited
by Leo F. Boron, P. Noordhoff Ltd. Groningen, 1964. MR 0181881
(31 #6107)
- 13.
D.
T. Luc, Recession maps and applications, Optimization
27 (1993), no. 1-2, 1–15. MR 1275544
(95e:49021), http://dx.doi.org/10.1080/02331939308843868
- 14.
D. T. LUC, Recessively compact sets: uses and properties, Set-Valued Anal. 10 (2002), 15-35.
- 15.
Dinh
The Luc and Jean-Paul
Penot, Convergence of asymptotic
directions, Trans. Amer. Math. Soc.
353 (2001), no. 10, 4095–4121 (electronic). MR 1837222
(2002e:49028), http://dx.doi.org/10.1090/S0002-9947-01-02664-2
- 16.
Jean-Paul
Penot, Fixed point theorems without convexity, Bull. Soc.
Math. France Mém. 60 (1979), 129–152. Analyse
non convexe (Proc. Colloq., Pau, 1977). MR 562261
(81c:47061)
- 17.
Jean-Paul
Penot, Compact nets, filters, and relations, J. Math. Anal.
Appl. 93 (1983), no. 2, 400–417. MR 700155
(84h:49032), http://dx.doi.org/10.1016/0022-247X(83)90184-1
- 18.
Jean-Paul
Penot, What is quasiconvex analysis?, Optimization
47 (2000), no. 1-2, 35–110. CODE Workshop on
Generalized Convexity and Monotonicity in Economic Modeling (Bellaterra,
1998). MR
1755580 (2001a:49024), http://dx.doi.org/10.1080/02331930008844469
- 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.
Behzad
Djafari Rouhani and W.
A. Kirk, Asymptotic properties of nonexpansive iterations in
reflexive spaces, J. Math. Anal. Appl. 236 (1999),
no. 2, 281–289. MR 1704583
(2001a:47056), http://dx.doi.org/10.1006/jmaa.1999.6423
- 22.
Sankatha
Singh, Bruce
Watson, and Pramila
Srivastava, Fixed point theory and best approximation: the KKM-map
principle, Mathematics and its Applications, vol. 424, Kluwer
Academic Publishers, Dordrecht, 1997. MR 1483076
(99a:47087)
- 23.
C.
Zălinescu, Recession cones and asymptotically compact
sets, J. Optim. Theory Appl. 77 (1993), no. 1,
209–220. MR 1222792
(94d:90057), http://dx.doi.org/10.1007/BF00940787
- 24.
Eberhard
Zeidler, Nonlinear functional analysis and its applications.
I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated
from the German by Peter R. Wadsack. MR 816732
(87f:47083)
- 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
 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
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
47H10,
47H09,
54H25,
55M20
Retrieve articles in all journals
with MSC (2000):
47H10,
47H09,
54H25,
55M20
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
Email:
jean-paul.penot@univ-pau.fr
DOI:
http://dx.doi.org/10.1090/S0002-9939-03-06999-5
PII:
S 0002-9939(03)06999-5
Keywords:
Asymptotic,
asymptotic cone,
contraction,
derivative at infinity,
firm asymptotic cone,
fixed point,
nonexpansive map
Received by editor(s):
February 19, 2002
Posted:
March 11, 2003
Communicated by:
Jonathan M. Borwein
Article copyright:
© Copyright 2003 American Mathematical Society
|
