ABSTRACT
In this chapter, we shall first prove a Tietze-Dugundji extension theorem for upper semicontinuous correspondences with non-empty compact star-shaped values. By applying this extension theorem, Eilenberg and Montgomery (1946) fixed point theorem, Ky Fan’s (1972) minimax inequality, Fan-Glicksberg (1952) fixed point theorem, or an improvement of an extension theorem of Pruszko (1997), we shall study the duals of the Gale-Mas-Colell’s (1975,1979) and Shafer-Sonnenschein's (1975) Theorems where the correspondences are upper semicontinuous instead of being lower semicontinuous. Some applications to equilibrium existence theorems for qualitative games are also given.
