Abstract
This paper extends the well-known KKM theorem and variational inequalities by relaxing the closedness of values of a correspondence and lower semicontinuity of a function. The approach adopted is based on Michael's continuous selection theorem. As applications, we provide theorems for the existence of maximum elements of a binary relation, a price equilibrium, and the complementarity problem. Thus our theorems, which do not require the openness of lower sections of the preference correspondences and the lower semicontinuity of the excess demand functions, generalize many of the existence theorems such as those in Sonnenschein (Ref. 1), Yannelis and Prabhakar (Ref. 2), and Border (Ref. 3).
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Communicated by F. Giannessi
The author is grateful to Professor Franco Giannessi for helpful comments and suggestions.
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Tian, G.Q. Generalized KKM theorems, minimax inequalities, and their applications. J Optim Theory Appl 83, 375–389 (1994). https://doi.org/10.1007/BF02190063
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DOI: https://doi.org/10.1007/BF02190063