zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The Fan minimax inequality implies the Nash equilibrium theorem. (English) Zbl 1236.49014
Summary: We show that in an abstract convex space (E,D;Γ), the partial KKM principle implies the Ky Fan minimax inequality, from which we deduce a generalization of the Nash equilibrium theorem.
MSC:
49J35Minimax problems (existence)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
References:
[1]Park, S.: Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27, 187-222 (1999) · Zbl 0938.54039
[2]Park, S.: The KKM principle in abstract convex spaces: equivalent formulations and applications, Nonlinear anal. TMA 73, 1028-1042 (2010) · Zbl 1214.47042 · doi:10.1016/j.na.2010.04.029
[3]Nash, J. F.: Equilibrium points in N-person games, Proc. natl. Acad. sci. USA 36, 48-49 (1950) · Zbl 0036.01104 · doi:10.1073/pnas.36.1.48
[4]Nash, J.: Non-cooperative games, Ann. math. 54, 286-295 (1951) · Zbl 0045.08202 · doi:10.2307/1969529
[5]Fan, K.: A generalization of tychonoff’s fixed point theorem, Math. ann. 142, 305-310 (1961) · Zbl 0093.36701 · doi:10.1007/BF01353421
[6]Fan, K.: Applications of a theorem concerning sets with convex sections, Math. ann. 163, 189-203 (1966) · Zbl 0138.37401 · doi:10.1007/BF02052284
[7]S. Park, Generalizations of the Nash equilibrium theorem in the KKM theory, Takahashi Legacy, Fixed Point Theory Appl. vol. 2010, Article ID 234706, 23 pp, doi:10.1155/2010/234706. · Zbl 1185.49002 · doi:10.1155/2010/234706
[8]Park, S.: On the von Neumann–sion minimax theorem in KKM spaces, Appl. math. Lett. 23, 1269-1273 (2010) · Zbl 1195.49010 · doi:10.1016/j.aml.2010.06.011
[9]Fan, K.: O.shishaa minimax inequality and applications, inequalities III, A minimax inequality and applications, inequalities III, 103-113 (1972)
[10]Zeidler, E.: Nonlinear functional analysis and its applications, Nonlinear functional analysis and its applications 5 (1986–1990)
[11]Lin, Y. J.; Tian, G.: Minimax inequalities equivalent to the Fan–knaster–Kuratowski–mazurkiewicz theorem, Appl. math. Optim. 28, 173-179 (1993) · Zbl 0788.49015 · doi:10.1007/BF01182980
[12]Park, S.: Elements of the KKM theory on abstract convex spaces, J. korean math. Soc. 45, No. 1, 1-27 (2008) · Zbl 1149.47040 · doi:10.4134/JKMS.2008.45.1.001
[13]Park, S.: Equilibrium existence theorems in KKM spaces, Nonlinear anal. TMA 69, 4352-4364 (2008) · Zbl 1163.47044 · doi:10.1016/j.na.2007.10.058
[14]Park, S.: New foundations of the KKM theory, J. nonlinear convex anal. 9, No. 3, 331-350 (2008) · Zbl 1167.47041 · doi:http://www.ybook.co.jp/online/jncae/vol9/p331.html
[15]Luc, D. T.; Sarabi, E.; Soubeyran, A.: Existence of solutions in variational relation problems without convexity, J. math. Anal. appl. 364, 544-555 (2010) · Zbl 1183.49017 · doi:10.1016/j.jmaa.2009.10.040
[16]Park, S.: A genesis of general KKM theorems for abstract convex spaces, J. nonlinear anal. Optim. 2, No. 1, 121-132 (2011)
[17]Park, S.: New generalizations of basic theorems in the KKM theory, Nonlinear anal. TMA 74, 3000-3010 (2011) · Zbl 1225.47060 · doi:10.1016/j.na.2011.01.020
[18]S. Park, On S.-Y. Chang’s inequalities and Nash equilibria (in press).
[19]Park, S.: Generalized convex spaces, L-spaces, and FC-spaces, J. global optim. 45, 203-210 (2009) · Zbl 1225.47059 · doi:10.1007/s10898-008-9363-1
[20]Ziad, A.: A counterexample to 0-diagonal quasiconcavity in a minimax inequality, J. optim. Theory appl. 109, No. 2, 457-462 (2001) · Zbl 1013.91014 · doi:10.1023/A:1017530825781
[21]J.P. Torres-Martínez, Fixed points as Nash equilibria, Fixed Point Theory Appl. vol. 2006, Article ID 36135, 4 pp. · Zbl 1150.91303 · doi:10.1155/FPTA/2006/36135