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COMMON FIXED POINTS OF MAPS ON TOPOLOGICAL VECTOR SPACES HAVING SUFFICIENTLY MANY LINEAR FUNCTIONALS
SEIIIE PARK
Fixed point theorems for upper semicontinuous (u. s. c.) multi maps on a nonempty compact convex subset of various topological vector spaces (t. v. s.) were obtained by S. Kakutani [9J, Bohnenblust and Karlin [3J, Ky Fan [l1J, Glicksberg [6J, and others. Recently, W. K. Kim [10J and S. Park [14J generalized those results for a t. v. s. having sufficiently many linear functionals. On the other hand, Itoh and Takahashi [7J proved a common fixed point theorem for a continuous map and an u. s. c. multimap on a compact convex subset of a locally convex space (1. c. s.) under some additional conditions. In the present paper, we generalize their theorem for at. v. s. having sufficiently many linear functionals, and also obtain a generalized version of the classical Markov-Kakutani theorem. Let E be a Hausdorff t. v. s. and E* its topological dual. E is said to have sufficiently many linear functionals if for every xE E with x*,O there exists aeon tinuous linear functional lE E* such that l (x) O. By the Hahn-Banach theorem, every 1. c. s. has sufficiently many linear functionals. An example of a t. v. s. having sufficiently many linear functionals which is not locally convex is the Hardy space HP with 0
