×

zbMATH — the first resource for mathematics

Fixed point theorems for discontinuous maps on a non-convex domain. (English) Zbl 1283.91112
Summary: This paper introduces economists to some fixed point theorems for discontinuous mappings with non-convex images on a non-convex domain. These theorems have recently been developed based on a new approach by mathematical economists and mathematicians. The new method of proof is first transformed into a sort of metatheorem, which is then used to obtain a set of necessary and sufficient conditions for a map to have a fixed point. Some fixed point theorems for discontinuous maps are then explained in more concrete cases. The formulations are intended for easier applications towards economic models involving discontinuity as well as non-convexity.
MSC:
91B50 General equilibrium theory
54H25 Fixed-point and coincidence theorems (topological aspects)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexandroff, Topologie I (1935) · doi:10.1007/978-3-662-02021-0
[2] Begle, A fixed point theorem, Annals of Mathematics 51 pp 544– (1950) · Zbl 0036.38901 · doi:10.2307/1969367
[3] Berge, Topological Spaces (1963)
[4] Bich , P. 2006 Some fixed point theorems for discontinuous mapping
[5] Bich , P. 2007 Nash equilibrium existence for some discontinuous games Documents de Travail du Centre d’Economie de la Sorbonne
[6] Bich, An answer to a question by Herings et al., Operations Research Letters 36 pp 525– (2008) · Zbl 1151.91340 · doi:10.1016/j.orl.2008.01.014
[7] Bohnenblust, Contributions to the Theory of Games (1950)
[8] Brouwer, Über Abbildung von Mannigfaltigkeiten, Mathematische Annalen 71 pp 97– (1912) · JFM 42.0417.01 · doi:10.1007/BF01456931
[9] Browder, A new generalization of Schauder fixed point theorem, Mathematische Annalen 174 pp 285– (1967) · Zbl 0176.45203 · doi:10.1007/BF01364275
[10] Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Mathematische Annalen 177 pp 283– (1968) · Zbl 0176.45204 · doi:10.1007/BF01350721
[11] Bula, On the stability of the Bohl-Brouwer-Schauder theorem, Nonlinear Analysis 26 pp 1859– (1996) · Zbl 0858.47031 · doi:10.1016/0362-546X(94)00343-G
[12] Cauty, Solution du problème de point fixe de Schauder, Fundamenta Mathematicae 170 pp 231– (2001) · Zbl 0983.54045 · doi:10.4064/fm170-3-2
[13] Cauty , R. 2012 Une généralization de la conjecture de point fixe de Schauder
[14] Chichilnisky, Rothe’s fixed point theorem and the existence of equilibria in monetary economies, Journal of Mathematical Analysis and Applications 65 pp 56– (1978) · Zbl 0394.90016 · doi:10.1016/0022-247X(78)90201-9
[15] Cromme, Fixed point theorems for discontinuous functions and applications, Nonlinear Analysis, Theory, Methods and Applications 30 pp 1527– (1997) · Zbl 0896.47043 · doi:10.1016/S0362-546X(97)00058-8
[16] Cromme, Fixed point theorems for discontinuous mapping, Mathematical Programming 51 pp 257– (1991) · Zbl 0748.54015 · doi:10.1007/BF01586937
[17] Debreu, A social equilibrium existence theorem, Proceedings of the National Academy of Sciences of the United States of America 38 pp 886– (1952) · Zbl 0047.38804 · doi:10.1073/pnas.38.10.886
[18] Debreu, Theory of Value an Axiomatic Analysis of Economic Equilibrium (1959) · Zbl 0193.20205
[19] Eilenberg, Fixed point theorems for multi-valued transformations, American Journal of Mathematics 68 pp 214– (1946) · Zbl 0060.40203 · doi:10.2307/2371832
[20] Fan, Fixed-point and minimax theorems in locally convex topological linear spaces, Proceedings of the National Academy of Sciences 38 pp 121– (1952) · Zbl 0047.35103 · doi:10.1073/pnas.38.2.121
[21] Fan, Extensions of two fixed point theorems of F. E. Browder, Mathematische Zeitschrift 112 pp 234– (1969) · Zbl 0185.39503 · doi:10.1007/BF01110225
[22] Fujimoto, An extension of Tarski’s fixed point theorem and its application to isotone complementarity problems, Mathematical Programming 28 pp 116– (1984) · Zbl 0526.90084 · doi:10.1007/BF02612716
[23] Glicksberg, A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points, Proceedings of the American Mathematical Society 3 pp 170– (1952) · Zbl 0163.38301
[24] Hadamard, Introduction a la Théorie des Fonctions d’une Variable 2 (1910)
[25] Halpern , B. R. 1965 Fixed-point theorems for outward maps
[26] Halpern, A fixed-point theorem for inward and outward maps, Transaction of the American Mathematical Society 130 pp 353– (1968) · Zbl 0153.45602 · doi:10.1090/S0002-9947-1968-0221345-0
[27] Herings , J. J. van der Laan , G. Talman , D. Yang , Z. 2005 A fixed point theorem for discontinuous functions · Zbl 1135.91317
[28] Herings, A fixed point theorem for discontinuous functions, Operations Research Letters 36 pp 89– (2008) · Zbl 1135.91317 · doi:10.1016/j.orl.2007.03.008
[29] Himmelberg, Fixed points of compact multifunctions, Journal of Mathematical Analysis and Applications 38 pp 205– (1972) · Zbl 0204.23104 · doi:10.1016/0022-247X(72)90128-X
[30] Hukuhara, Sur l’existence des points invariants d’une transformation dans l’espace fonctionnel, Japanese Journal Mathematics 20 pp 1– (1950) · Zbl 0041.23801
[31] Idzik, Almost fixed point theorems, Proceedings of the American Mathematical Society 104 pp 779– (1988) · Zbl 0691.47046 · doi:10.1090/S0002-9939-1988-0964857-2
[32] Jafari, An extension to a theorem of Himmelberg, Journal of Mathematical Analysis and Applications 327 pp 298– (2007) · Zbl 1109.54025 · doi:10.1016/j.jmaa.2006.04.048
[33] Kakutani, A generalization of Brouwer’s fixed point theorem, Duke Mathematical Journal 8 pp 457– (1941) · JFM 67.0742.03 · doi:10.1215/S0012-7094-41-00838-4
[34] Klee, Stability of the fixed point property, Colloquium Mathematicum VIII pp 43– (1961)
[35] Komiya, Remarks on extensions of the Himmelberg fixed point theorem, Fixed Point Theory and Applications 2007 (2007) · Zbl 1146.54312 · doi:10.1155/2007/16028
[36] Mazur, Une théorème sur les points invariants, Annales Societatis Mathematicae Polonae 17 pp 110– (1938)
[37] Miranda, Un’osservazione su un teorema di Brouwer, Bollettino dell’Unione Matematica Italiana 3 pp 5– (1940)
[38] Morishima, Equilibrium, Stability, and Growth: A Multi-sectoral Analysis (1964)
[39] Nash, Equilibrium points in N-person games, Proceedings of the National Academy of Sciences of the United States of America 36 pp 48– (1950) · Zbl 0036.01104 · doi:10.1073/pnas.36.1.48
[40] Nash, Non-cooperative games, Annals of Mathematics 54 pp 286– (1951) · Zbl 0045.08202 · doi:10.2307/1969529
[41] Nikaido, On the classical multilateral exchange problem, Metroeconomica 8 pp 135– (1956) · doi:10.1111/j.1467-999X.1956.tb00099.x
[42] Park, New versions of the Fan-Browder fixed point theorem and existence of economic equilibria, Fixed Point Theory and Applications 2004 (2004) · Zbl 1078.54026 · doi:10.1155/S1687182004308089
[43] Rothe, Zur Theorie der topologischen Ordnung und der Vektorfelder in Banachschen Räumen, Compositio Mathematica 1 pp 177– (1938) · Zbl 0018.13304
[44] Schäfer, A fixed point theorem based on Miranda, Fixed Point Theory and Applications 2004 (2007)
[45] Schauder, Der Fixpunktsatz in Funktionalräumen, Studia Mathematica 2 pp 171– (1930) · JFM 56.0355.01
[46] Smithson, Fixed points of order preserving multifunctions, Proceedings of the American Mathematical Society 28 pp 304– (1971) · Zbl 0238.06003 · doi:10.1090/S0002-9939-1971-0274349-1
[47] Smithson, Fixed points in partially ordered sets, Pacific Journal of Mathematics 45 pp 363– (1973) · Zbl 0248.06002 · doi:10.2140/pjm.1973.45.363
[48] Tarski, A fixpoint theorem for lattices and its applications (preliminary report), Bulletin of the American Mathematical Society 55 pp 1051– (1949)
[49] Tarski, A lattice-theoretical fixpoint theorem and its applications, Pacific Journal of Mathematics 5 pp 285– (1955) · Zbl 0064.26004 · doi:10.2140/pjm.1955.5.285
[50] Termwuttipong, Fixed point theorem of half-continuous mappings on topological vector spaces, Fixed Point Theory and Applications 2010 (2010) · Zbl 1187.54044 · doi:10.1155/2010/814970
[51] Tychonoff, Ein Fixpunktsatz, Mathematische Annalen 111 pp 767– (1935) · Zbl 0012.30803 · doi:10.1007/BF01472256
[52] Urai, Fixed point theorems and the existence of economic equilibria based on conditions for local directions of mappings, RIMS 1108 pp 12– (1999)
[53] Urai, Fixed point theorems and the existence of economic equilibria based on conditions for local directions of mappings, Advances in Mathematical Economics 2 pp 87– (2000) · doi:10.1007/978-4-431-67909-7_5
[54] Urai, Fixed Points and Economic Equilibria (2010) · Zbl 1211.91006 · doi:10.1142/7117
[55] Urai, A generalization of continuity and convexity conditions for correspondences in economic equilibrium theory, Japanese Economic Review 51 pp 583– (2000) · Zbl 0985.47050 · doi:10.1111/1468-5876.00172
[56] Urai, Generalization of dual system structure on linear topological spaces and fixed point theorems for multi-valued mappings, Osaka Economic Papers 56 pp 1– (2006)
[57] Urai, Fixed point theorems in Hausdorff topological vector spaces and economic equilibrium theory, Advances in Mathematical Economics 6 pp 149– (2004) · Zbl 1092.47050 · doi:10.1007/978-4-431-68450-3_7
[58] Vives, Nash equilibrium with strategic complementarities, Journal of Mathematical Economics 19 pp 305– (1990) · Zbl 0708.90094 · doi:10.1016/0304-4068(90)90005-T
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.