The fixed point theory of multi-valued mappings in topological vector spaces

Abstract

Introduction Let K be a compact convex subset of a real topological vector space E, which we shall always assume to be separated (i.e. Hausdorff). We consider multi-valued mappings T of K into E, i.e. mappings (in the usual sense) of K into 2 e, the space of subsets of E, where for each x in K, T(x) is a non-empty closed convex subset of E. By a fixed point of such a mapping, we mean a point u of K such that u e T(u). The earliest extension of the topological theory of fixed points of continuous mappings to the case of multi-valued mappings was made by von Neumann [27] in the connection with the proof of the fundamental theorem of game theory. The extension of the Brouwer fixed point theorem to an upper semi-continuous multivalued mapping T of a n-disk into itself was carried through by Kakutani [23] and corresponding extensions of the Schauder fixed point theorem in Banach spaces were given independently by Bohnenblust-Karlin [33 and Glicksberg [193. The corresponding extension of TychonolTs theorem for locally convex topological vector spaces was proved by Ky Fan [12], who in a group of subsequent papers ([13, 14, 15, 16, 17]) refined and extended this result and considered a variety of applications. Asymptotic fixed point theorems for multi-valued mappings in Banach spaces were established in Browder [4], and parts of the Leray-Schauder theory in Banach spaces were extended to multi-valued mappings by Granas [20, 21]. It is our object in the present paper to present a new general treatment of the fixed point theory of multi-valued mappings in topological vector spaces which has the dual virtues of obtaining new and stronger results on the one hand and drastically simplifying the proofs of known results on the other. The starting point of our investigation of this theory lies in recent results of the writer in connection with the study of monotone operators and non-linear variational inequalities [6, 7, 8]. In this direction, one considers mappings S of a compact convex set K into E*, the dual space of E, rather than E itself. Instead of trying to find fixed points of a mapping T of K into E, one looks for points u in K for which S(u)= 0, or more generally, for which