Fixed set theorems of Krasnoselskii type
HTML articles powered by AMS MathViewer
- by Efe A. Ok
- Proc. Amer. Math. Soc. 137 (2009), 511-518
- DOI: https://doi.org/10.1090/S0002-9939-08-09332-5
- Published electronically: September 29, 2008
- PDF | Request permission
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
- Charalambos D. Aliprantis and Kim C. Border, Infinite dimensional analysis, 3rd ed., Springer, Berlin, 2006. A hitchhiker’s guide. MR 2378491
- Cleon S. Barroso and Eduardo V. Teixeira, A topological and geometric approach to fixed points results for sum of operators and applications, Nonlinear Anal. 60 (2005), no. 4, 625–650. MR 2109150, DOI 10.1016/j.na.2004.09.040
- D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464. MR 239559, DOI 10.1090/S0002-9939-1969-0239559-9
- T. A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett. 11 (1998), no. 1, 85–88. MR 1490385, DOI 10.1016/S0893-9659(97)00138-9
- T. A. Burton and Tetsuo Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynam. Systems Appl. 11 (2002), no. 4, 499–519. MR 1946140
- Eric Chandler and Gary Faulkner, Fixed points in nonconvex domains, Proc. Amer. Math. Soc. 80 (1980), no. 4, 635–638. MR 587942, DOI 10.1090/S0002-9939-1980-0587942-9
- W. G. Dotson Jr., Fixed point theorems for non-expansive mappings on star-shaped subsets of Banach spaces, J. London Math. Soc. (2) 4 (1971/72), 408–410. MR 296778, DOI 10.1112/jlms/s2-4.3.408
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- M. A. Krasnosel′skiĭ, Some problems of nonlinear analysis, American Mathematical Society Translations, Ser. 2, Vol. 10, American Mathematical Society, Providence, R.I., 1958, pp. 345–409. MR 0094731
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- Efe A. Ok, Fixed set theory for closed correspondences with applications to self-similarity and games, Nonlinear Anal. 56 (2004), no. 3, 309–330. MR 2032033, DOI 10.1016/j.na.2003.08.001
- V. M. Sehgal and S. P. Singh, A fixed point theorem for the sum of two mappings, Math. Japon. 23 (1978/79), no. 1, 71–75. MR 500289
- D. R. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, No. 66, Cambridge University Press, London-New York, 1974. MR 0467717
- Xian Wu, A new fixed point theorem and its applications, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1779–1783. MR 1397000, DOI 10.1090/S0002-9939-97-03903-8
Bibliographic Information
- Efe A. Ok
- Affiliation: Department of Economics, New York University, New York, New York 10012
- Email: efe.ok@nyu.edu
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 511-518
- MSC (2000): Primary 47H04, 47H10; Secondary 47H09
- DOI: https://doi.org/10.1090/S0002-9939-08-09332-5
- MathSciNet review: 2448571