Infinite-dimensional Gale-Nikaido-Debreu theorem and a fixed-point theorem of Tarafdar.

*(English)*Zbl 0646.47036In the first part of their paper the authors give a list of five statements on fixed points for multivalued mappings defined in linear topological spaces and prove that they imply each other. One of them, a theorem of G. Tarafdar from [Proc. Am. Math. Soc. 67, 95-98 (1977; Zbl 0369.47029)] is used in the second part to prove an infinite dimensional version of the Gale-Nikaido-Debreu theorem that occurs in mathematical economics. The theorem proved is more general than another infinite dimensional version of G.-N.-D. theorem given by N. C. Yannelis [J. Math. Anal. Appl. 108, 595-599 (1985; Zbl 0581.90010)]. One of the tools used in the proof is the Hahn-Banach theorem.

Reviewer: M.Sablik

##### MSC:

47H10 | Fixed-point theorems |

91B50 | General equilibrium theory |

54H25 | Fixed-point and coincidence theorems (topological aspects) |

##### Keywords:

fixed points for multivalued mappings defined in linear topological spaces; infinite dimensional version of the Gale-Nikaido-Debreu theorem; mathematical economics; Hahn-Banach theorem
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\textit{G. Mehta} and \textit{E. Tarafdar}, J. Econ. Theory 41, 333--339 (1987; Zbl 0646.47036)

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##### References:

[1] | Aliprantis, C; Brown, D, Equilibrium in markets with a Riesz space of commodities, J. math. econ., 11, 189-207, (1983) · Zbl 0502.90006 |

[2] | Border, K, On equilibria of excess demand correspondences, () |

[3] | Granas, A; Ben-El-Mechaiekh; Deguire, P, A non-linear alternative in convex analysis: some consequences, C. R. acad. sci. Paris, 257-259, (September 1982) |

[4] | Tarafdar, E, On nonlinear variational inequalities, (), 95-98 · Zbl 0369.47029 |

[5] | Tarafdar, E; Mehta, G, On the existence of quasi-equilibrium in a competitive economy, Int. J. sci. engr., 1, 1-12, (1984) |

[6] | Yannelis, N, On a market equilibrium theorem with an infinite number of commodities, J. math. anal. appl., 108, 595-599, (1985) · Zbl 0581.90010 |

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