Fixed set theorems of Krasnoselskii type
Author:
Efe A. Ok
Journal:
Proc. Amer. Math. Soc. 137 (2009), 511518
MSC (2000):
Primary 47H04, 47H10; Secondary 47H09
Published electronically:
September 29, 2008
MathSciNet review:
2448571
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Abstract: We revisit the fixed point problem for the sum of a compact operator and a continuous function, where the domain on which these maps are defined is not necessarily convex, the former map is allowed to be multivalued, and the latter to be a semicontraction and/or a suitable nonexpansive map. In this setup, guaranteeing the existence of fixed points is impossible, but two types of invariantlike sets are found to exist.
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 C. Barroso and E. Teixeira, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Anal. 60 (2005), 625650. MR 2109150 (2005i:47087)
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 W. Dotson, Fixed point theorems for nonexpansive mappings on starshaped subsets of Banach spaces, J. London Math. Soc. (2)4 (1972), 408410. MR 0296778 (45:5837)
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 X. Wu, A new fixed point theorem and its applications, Proc. Amer. Math. Soc. 125 (1997), 17791783. MR 1397000 (97h:90014)
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Additional Information
Efe A. Ok
Affiliation:
Department of Economics, New York University, New York, New York 10012
Email:
efe.ok@nyu.edu
DOI:
http://dx.doi.org/10.1090/S0002993908093325
PII:
S 00029939(08)093325
Keywords:
Fixed sets,
Krasnoselski\u {\i } fixed point theorem,
nonexpansive maps.
Received by editor(s):
May 8, 2006
Received by editor(s) in revised form:
April 16, 2007
Published electronically:
September 29, 2008
Additional Notes:
I thank Debraj Ray for his continuous support throughout my research on fixed set theory, and Cleon Barroso for pointing me to some related references. I should also acknowledge that the comments made by an anonymous referee have improved the exposition of this paper.
Communicated by:
Marius Junge
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
