A Generalization of Continuity and Convexity Conditions for Correspondences in Economic Equilibrium Theory

Abstract

Fixed-point theorems for multi-valued mappings and economic equilibrium existence theorems are generalized from the viewpoint that the continuity and/or convexity assumptions on a mapping may be replaced by weaker local conditions on the vector ®eld de®ned by the mapping. The generalization gives us natural conditions on individual (possibly non-ordered) preferences or aggregated demand behaviours so that we may obtain several extensions of recent results in social and game-theoretic equilibrium theories. JEL Classi®cation Numbers: C60, C62, C72, D50, D51.

Fixed Point Theorems for Discontinuous Maps on a Non‐convex Domain

“…These authors dealt with almost continuous mappings, and they have proved that approximate continuous mappings admit approximate fixed points. Then came a series of remarkable generalizations to truly discontinuous correspondences by Urai and his students, in Urai (), Urai (), Urai and Hayashi (), Urai and Yoshimachi () and Urai and Yokota () . Key properties in these contributions are a locally common element and a locally fixed direction, so named in Urai and Yokota (, p. 4 and p. 5).…”