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A variational method in fixed point results with inwardness conditions
Authors:
D. Azé and J.-N. Corvellec
Journal:
Proc. Amer. Math. Soc. 134 (2006), 3577-3583
MSC (2000):
Primary 47H10; Secondary 49J53
Posted:
June 27, 2006
MathSciNet review:
2240670
Full-text PDF Free Access
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Abstract: We generalize, in a metric space setting, the result due to Lim (2000), that a weakly inward multivalued contraction, defined on a nonempty closed subset of a Banach space, has a fixed point. The simple proof of this generalization, avoiding the use of a transfinite induction as in Lim's paper, is based on Ekeland's variational principle (1974), along the lines of Hamel (1994) and Takahashi (1991). Moreover, we give a sharp estimate for the distance from any point to the fixed point set.
References
- 1.
D. AZÉ AND J.-N. CORVELLEC, Characterizations of error bounds for lower semicontinuous functions on metric spaces, ESAIM Control, Optim. Calc. Var. 10 (2004), 409-425. MR 2084330 (2005e:49027)
- 2.
D. AZÉ AND J.-N. CORVELLEC, Variational methods in classical open mapping theorems, J. Convex Anal. 13 (2006), in press.
- 3.
J. CARISTI, Fixed point theorems for mapping satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251. MR 0394329 (52:15132)
- 4.
F. H. CLARKE, Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, Vol. 5, SIAM, Philadelphia, PA, 1990 (originally published by Wiley-Interscience, New York, 1983). MR 0709590 (85m:49002)
- 5.
I. EKELAND, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324-353. MR 0346619 (49:11344)
- 6.
A. HAMEL, Remarks to an equivalent formulation of Ekeland's variational principle, Optimization 31 (1994), 233-238. MR 1307103 (95h:49019)
- 7.
T. C. LIM, A fixed point theorem for weakly inward multivalued contractions, J. Math. Anal. Appl. 243 (2000), 323-327. MR 1766943 (2001d:47088)
- 8.
C. MART´INEZ-YAÑEZ, A remark on weakly inward contractions, Nonlinear Anal. 16 (1991), 847-848. MR 1106372 (92g:47086)
- 9.
S. B. NADLER JR., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488. MR 0254828 (40:8035)
- 10.
J.-P. PENOT, A short constructive proof of Caristi's fixed point theorem, Publ. Math. Univ. Pau (1976), 1-3.
- 11.
J.-P. PENOT, The drop theorem, the petal theorem and Ekeland's variational principle, Nonlinear Anal. 10 (1986), 813-822. MR 0856865 (87j:49031)
- 12.
W. TAKAHASHI, Existence theorems generalizing fixed point theorems for multivalued mappings, Fixed Point Theory and Applications (Marseille 1989), 397-406, Pitman Res. Notes Math. Ser., Vol. 252, Longman, Harlow, 1991. MR 1122847 (92m:54078)
- 13.
H. K. XU, Multivalued nonexpansive mappings in Banach spaces, Nonlinear Anal. 43 (2001), 693-706. MR 1808203 (2002b:47122)
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Additional Information
D. Azé
Affiliation:
UMR CNRS MIP, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 4, France
Email:
aze@mip.ups-tlse.fr
J.-N. Corvellec
Affiliation:
Laboratoire MANO, Université de Perpignan Via Domitia, 52 avenue Paul Alduy, 66860 Perpignan cedex, France
Email:
corvellec@univ-perp.fr
DOI:
http://dx.doi.org/10.1090/S0002-9939-06-08744-2
PII:
S 0002-9939(06)08744-2
Received by editor(s):
June 24, 2005
Posted:
June 27, 2006
Communicated by:
Jonathan M. Borwein
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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