|
Fixed set theorems of Krasnoselskii type
Author(s):
Efe
A.
Ok
Journal:
Proc. Amer. Math. Soc.
137
(2009),
511-518.
MSC (2000):
Primary 47H04, 47H10;
Secondary 47H09
Posted:
September 29, 2008
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
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 multi-valued, 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 invariant-like sets are found to exist.
References:
-
- 1.
- C. Aliprantis and K. Border, Infinite Dimensional Analysis, 3rd Edition, Springer, Berlin, 2006. MR 2378491
- 2.
- C. Barroso and E. Teixeira, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Anal. 60 (2005), 625-650. MR 2109150 (2005i:47087)
- 3.
- D. Boyd and J. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464. MR 0239559 (39:916)
- 4.
- T. Burton, A fixed-point theorem of Krasnoselskiı, Appl. Math. Lett. 11 (1998), 85-88. MR 1490385 (98i:47053)
- 5.
- T. Burton and T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynam. Systems Appl. 11 (2002), 499-519. MR 1946140 (2004c:34219)
- 6.
- E. Chandler and G. Faulkner, Fixed points in nonconvex domains, Proc. Amer. Math. Soc. 80 (1980), 635-638. MR 587942 (81k:47078)
- 7.
- W. Dotson, Fixed point theorems for non-expansive mappings on star-shaped subsets of Banach spaces, J. London Math. Soc. (2)4 (1972), 408-410. MR 0296778 (45:5837)
- 8.
- J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713-747. MR 625600 (82h:49026)
- 9.
- M. Krasnoselskiı, Some problems of nonlinear analysis, Amer. Math. Soc. Transl. 10 (1958), 345-409. MR 0094731 (20:1243)
- 10.
- E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182. MR 0042109 (13:54f)
- 11.
- E. A. Ok, Fixed set theory for closed correspondences with applications to self-similarity and games, Nonlinear Anal. 56 (2004), 309-330. MR 2032033 (2004k:47112)
- 12.
- V. Sehgal and S. Singh, A fixed point theorem for the sum of two mappings, Math. Japonica 23 (1978), 71-75. MR 500289 (80a:47089)
- 13.
- D. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, 1974. MR 0467717 (57:7570)
- 14.
- X. Wu, A new fixed point theorem and its applications, Proc. Amer. Math. Soc. 125 (1997), 1779-1783. MR 1397000 (97h:90014)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical Society
with MSC
(2000):
47H04, 47H10,
47H09
Retrieve articles in all Journals with MSC
(2000):
47H04, 47H10,
47H09
Additional Information:
Efe
A.
Ok
Affiliation:
Department of Economics, New York University, New York, New York 10012
Email:
efe.ok@nyu.edu
DOI:
10.1090/S0002-9939-08-09332-5
PII:
S 0002-9939(08)09332-5
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
Posted:
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
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org