Let G be a nonempty subset of a locally convex space E such that cl(G) is convex and quasi-complete, and f: cl(G)-+E a continuous condensing multifunction. In this paper, several fixed point theorems are established if f satisfies some conditions on the boundary of G. The results herein extend some theorems of Reich [91 and generalize some of the well-known fixed point theorems. The classical Tychonoff's fixed point theorem [lO has been extended to multifunctions by Browder [31, Fan [41, Glicksberg [51, Himmelberg [71 and several others. In a recent paper [91, Reich has proved some interesting fixed point theorems for condensing multifunctions and has given extensions of some of the results contained in [31 and [41. In this paper, we prove several fixed point theorems for condensing multifunctions in a locally convex space. The main result of this paper (Theorem 2) is motivated by Reich [91 and extends a result in [91. The results herein also generalize some wellknown results (see [21, [8]). 1. Let X and Y be topological spaces. A multifunction /: X -Y is a point to set function such that for each x e X, /(x) is a nonempty subset of Y. The multifunction ft X Y is (a) upper semicontinuous (u. s. c.) iff for each closed subset B of Y, the set f '(B) = tx: f(x) n B ? ;0 is a closed subset of X, (b) lower semicontinuous (l.s.c.)iff for each open subset U C Y, the set /l(U) is an open subset of X, (c) continuous iff it is both u.s.c. and l.s.c., (d) point-compact (closed, convex) iff for each x E X, f(x) is a compact (closed, convex) subset of Y. It follows immediately from the above that a multifunction /: X Y is u.s.c. iff for each x E X and open U C Y with /(x) C U, there is an open V C X such that x C V and f(V) = Ulf(z): z 6 VI C U. It is l.s.c. iff for each open set U C Y and each x e X with f(x) n U 7 0, there exists an Received by the editors February 16, 1974. AMS (MOS) subject classifications (1970). Primary 47H10, 54H20.