Abstract
Fixed point theorems for set valued mappings are reexamined from a unified viewpoint on local directions of the values of a mapping on a subset of a Hausdorff topological vector space to itself. Some basic fixed point theorems, such as Kakutani’s and Browder’s, are generalized so that we could apply them to game theoretic and economic equilibrium existence problems with non-ordered preferences having neither global continuity nor convexity conditions. Relations of our main results to other mathematical theorems such as Fan-Browder’s theorem, maximal element existence theorem for ℒ-majorized mappings, Eaves’ theorem, KKM and KKMS theorem, are also studied.
Research was supported by the Japanese Ministry of Education Grant 10730006. I am indebted to Professor Hukukane Nikaido, with a special bow to his works, Nikaido (1957) and (1959). The author thanks Hidetoshi Komiya (Keio University), Akira Yamazaki (Hitotsubashi University), Toru Maruyama (Keio University) and an anonymous referee for useful suggestions and comments. Preliminary versions of this paper were presented in a conference at the Research Institute for Mathematical Sciences, Kyoto University (December 2, 1998) and a seminar at the Graduate School of Economics, Osaka University (December 17, 1998). The author also thanks to the participants at these seminars.
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Urai, K. (2000). Fixed point theorems and the existence of economic equilibria based on conditions for local directions of mappings. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 2. Springer, Tokyo. https://doi.org/10.1007/978-4-431-67909-7_5
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