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Infinite-dimensional Gale-Nikaido-Debreu theorem and a fixed-point theorem of Tarafdar. (English) Zbl 0646.47036

In 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)
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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, (Social Science Working Paper, 460 (1983), California Institute of Technology: California Institute of Technology Pasadena)
[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) · Zbl 0521.47027
[4] Tarafdar, E., On nonlinear variational inequalities, (Proc. Amer. Math. Soc., 67 (1977)), 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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