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
Fulltext PDF Free Access
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 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.
 1.
Charalambos
D. Aliprantis and Kim
C. Border, Infinite dimensional analysis, 3rd ed., Springer,
Berlin, 2006. A hitchhiker’s guide. MR 2378491
(2008m:46001)
 2.
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
(2005i:47087), http://dx.doi.org/10.1016/j.na.2004.09.040
 3.
D.
W. Boyd and J.
S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464. MR 0239559
(39 #916), http://dx.doi.org/10.1090/S00029939196902395599
 4.
T.
A. Burton, A fixedpoint theorem of Krasnoselskii, Appl. Math.
Lett. 11 (1998), no. 1, 85–88. MR 1490385
(98i:47053), http://dx.doi.org/10.1016/S08939659(97)001389
 5.
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
(2004c:34219)
 6.
Eric
Chandler and Gary
Faulkner, Fixed points in nonconvex
domains, Proc. Amer. Math. Soc.
80 (1980), no. 4,
635–638. MR
587942 (81k:47078), http://dx.doi.org/10.1090/S00029939198005879429
 7.
W.
G. Dotson Jr., Fixed point theorems for nonexpansive mappings on
starshaped subsets of Banach spaces, J. London Math. Soc. (2)
4 (1971/72), 408–410. MR 0296778
(45 #5837)
 8.
John
E. Hutchinson, Fractals and selfsimilarity, Indiana Univ.
Math. J. 30 (1981), no. 5, 713–747. MR 625600
(82h:49026), http://dx.doi.org/10.1512/iumj.1981.30.30055
 9.
M.
A. Krasnosel′ski&ibreve;, 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
(20 #1243)
 10.
Ernest
Michael, Topologies on spaces of
subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 0042109
(13,54f), http://dx.doi.org/10.1090/S00029947195100421094
 11.
Efe
A. Ok, Fixed set theory for closed correspondences with
applications to selfsimilarity and games, Nonlinear Anal.
56 (2004), no. 3, 309–330. MR 2032033
(2004k:47112), http://dx.doi.org/10.1016/j.na.2003.08.001
 12.
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
(80a:47089)
 13.
D.
R. Smart, Fixed point theorems, Cambridge University Press,
LondonNew York, 1974. Cambridge Tracts in Mathematics, No. 66. MR 0467717
(57 #7570)
 14.
Xian
Wu, A new fixed point theorem and its
applications, Proc. Amer. Math. Soc.
125 (1997), no. 6,
1779–1783. MR 1397000
(97h:90014), http://dx.doi.org/10.1090/S0002993997039038
 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), 625650. MR 2109150 (2005i:47087)
 3.
 D. Boyd and J. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458464. MR 0239559 (39:916)
 4.
 T. Burton, A fixedpoint theorem of Krasnoselskiı, Appl. Math. Lett. 11 (1998), 8588. 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), 499519. MR 1946140 (2004c:34219)
 6.
 E. Chandler and G. Faulkner, Fixed points in nonconvex domains, Proc. Amer. Math. Soc. 80 (1980), 635638. MR 587942 (81k:47078)
 7.
 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)
 8.
 J. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J. 30 (1981), 713747. MR 625600 (82h:49026)
 9.
 M. Krasnoselskiı, Some problems of nonlinear analysis, Amer. Math. Soc. Transl. 10 (1958), 345409. MR 0094731 (20:1243)
 10.
 E. Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152182. MR 0042109 (13:54f)
 11.
 E. A. Ok, Fixed set theory for closed correspondences with applications to selfsimilarity and games, Nonlinear Anal. 56 (2004), 309330. MR 2032033 (2004k:47112)
 12.
 V. Sehgal and S. Singh, A fixed point theorem for the sum of two mappings, Math. Japonica 23 (1978), 7175. 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), 17791783. 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:
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.
