×

Fixed points for Kakutani factorizable multifunctions. (English) Zbl 0719.47043

Author’s summary: A multifunction \(\Gamma\) is called a Kakutani multifunction if there exist two nonempty convex sets X and Y, each in a Hausdorff topological vector space, such that \(\Gamma\) : \(X\to Y\) is upper semi-continuous and nonempty convex compact valued. Let \(\Gamma\) : \(X\to X\) be a multifunction from a simplex X into itself. If \(\Gamma\) can be factorized by a finite number of Kakutani multifunctions, then \(\Gamma\) has a fixed point. The proof relies on a simplicial approximation technique and the Brouwer fixed point theorem. Extensions to infinite-dimensional spaces and applications to game theory are given.

MSC:

47H10 Fixed-point theorems
47H04 Set-valued operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ben-El-Mechaiekh, H.; Deguire, P.; Granas, A., Points fixes et coïncidences pour les applications multivoques (applications de Ky Fan), C.R. Acad. Sci. Paris Sér. I Math., 295, 337-340 (1982) · Zbl 0525.47042
[2] Ben-El-Mechaiekh, H.; Deguire, P.; Granas, A., Points fixes et coïncidences pour les fonctions multivoques III (Applications de type \(M et \(M^∗)\), C.R. Acad. Sci. Paris Sér. I Math., 305, 381-384 (1987) · Zbl 0638.47054
[3] Berge, C., Espaces topologiques, Fonctions multivoques (1959), Dunod: Dunod Paris · Zbl 0088.14703
[4] Browder, F. E., The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann., 177, 283-301 (1968) · Zbl 0176.45204
[5] Debreu, G., A social equilibrium existence theorem, (Proc. Nat. Acad. Sci. U.S.A., 38 (1952)), 886-893 · Zbl 0047.38804
[6] Dugundji, J.; Granas, A., KKM maps and variational inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5, 679-682 (1978) · Zbl 0396.47037
[7] Fan, K., Fixed point and minimax theorems in locally convex topological linear spaces, (Proc. Nat. Acad. Sci. U.S.A., 38 (1952)), 121-126 · Zbl 0047.35103
[8] Fan, K., A generalization of Tychonoff’s fixed point theorem, Math. Ann., 142, 305-310 (1961) · Zbl 0093.36701
[9] Fan, K., Sur un théorème minimax, C.R. Acad. Sci. Paris Sér. I Math., 259, 3925-3928 (1964) · Zbl 0138.37304
[10] Fan, K., Applications of a theorem concerning sets with convex sections, Math. Ann., 163, 189-203 (1966) · Zbl 0138.37401
[11] Glicksberg, I. L., A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points, (Proc. Amer. Math. Soc., 3 (1952)), 170-174 · Zbl 0046.12103
[12] Górniewicz, L., A Lefschetz-type fixed point theorem, Fund. Math., 88, 103-115 (1975) · Zbl 0306.55007
[13] Górniewicz, L.; Granas, A., Some general theorems in coincidence theory I, J. Math. Pures Appl., 60, 361-373 (1981) · Zbl 0482.55002
[14] Granas, A.; Liu, F. C., Coincidences for set-valued maps and minimax inequalities, J. Math. Pures Appl., 65, 119-148 (1986) · Zbl 0659.49007
[15] Ha, C. W., Minimax and fixed point theorems, Math. Ann., 248, 73-77 (1980) · Zbl 0413.47042
[16] Himmelberg, C. J., Fixed points of compact multifunctions, J. Math. Anal. Appl., 38, 205-207 (1972) · Zbl 0225.54049
[17] Hukuhara, M., Sur l’existence des points invariants d’une transformation de l’espace fonctionnel, Japan J. Math. (N.S.), 20, 1-4 (1950) · Zbl 0041.23801
[18] Kakutani, S., A generalization of Brouwer’s fixed-point theorem, Duke Math. J., 8, 457-459 (1941) · JFM 67.0742.03
[19] Knaster, B.; Kuratowski, C.; Mazurkiewicz, S., Ein Beweis des Fixpunktsatzes für \(n\)-dimensionale Simplexe, Fund. Math., 14, 132-137 (1929) · JFM 55.0972.01
[20] Lassonde, M., On the use of KKM multifunctions in fixed point theory and related topics, J. Math. Anal. Appl., 97, 151-201 (1983) · Zbl 0527.47037
[21] Nash, J., Equilibrium points in \(N\)-person games, (Proc. Nat. Acad. Sci. U.S.A., 36 (1950)), 48-49 · Zbl 0036.01104
[22] von Neumann, J., A model of general economic equilibrium, Rev. Econom. Stud., 13, 1-9 (1945-1946), Translation · JFM 68.0029.02
[23] Powers, M., Lefschetz fixed point theorem for a new class of multivalued maps, Pacific J. Math., 42, 211-220 (1972) · Zbl 0244.55007
[24] Tychonoff, A., Ein Fixpunktsatz, Math. Ann., 111, 767-776 (1935) · Zbl 0012.30803
[25] Simons, S., Cyclical coincidences of multivalued maps, J. Math. Soc. Japan, 38, 515-525 (1986) · Zbl 0616.47044
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.