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Abstract.We prove a random fixed point theorem in a Banach space for set valued mappings and then derive a corollary that yields a fixed point theorem of BharuchaReid and Mukherjea, as a special case.In a recent paper [1], Bharucha-Reid and Mukherjea proved the following stochastic analogue of the well-known Schauder's fixed point theorem.Theorem 1. Let S be a compact and convex subset of a Banach space E and T: Ü X S -> S be a continuous random operator. Then T admits a random fixed point.In this paper, we shall show that Theorem 1 can be derived from a more general result. For detailed definitions and terminologies we refer to Bharucha-Reid [1] or a recent paper of Itoh [3]. Throughout, this paper, (ß, 2) is a measurable space with 2 a sigma algebra of subsets of ß. The symbol 2E denotes the class of nonempty subsets of a Banach space E. A mapping F: ß -» 2E is called measurable iff for each open set G of E,

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ABSTRACT. We prove a theorem for set valued mappings in an approximatively compact, convex subset of a locally convex space, and then derive results due to Ky Fan and S. Reich as corollaries.Let E be a locally convex Hausdorff topological vector sapee, S a nonempty subset of E and p a continuous seminorm on E. It is a well-known result (see the proof in Sehgal [8] or Ky Fan [1]) that if S is compact and convex and f:S->E is a continuous map, then there exists an a; G S satisfying (1) p(fx-x) =dp(fx,S) =min{p(fx,-y)\ yES}.Since then a number of authors have provided either an extension of the above theorem to set valued mappings or have weakened the compactness condition therein. Some of these results are (d) SEHGAL AND SINGH (1985). Let S Ç E with int(S) ^ 0 and cl(S) convex and let /: S -► 2E be a continuous condensing multifunction with convex, compact values and with a bounded range. Then for each w E int(S), there exists a continuous seminorm p = p(w) satisfying (1) [6].Our aim in this presentation is to prove (a) for multifonctions and derive some results as easy corollaries.For definitions and terminologies we refer to Reich [5] (see also [3]). DEFINITION. A subset S of E is approximatively p-compact iff for each y E E and a net {xa} in S satisfying p(xa -y)-+ dp(y, S) there is a subnet {xp} and an x E S such that Xß -► x.Clearly a compact set in E is approximatively compact. The converse, however, may fail. For example, the closed unit ball of an infinite dimensional uniformly convex Banach space is approximatively norm compact but not compact.

To our parents and especially to Shri Mahadeo Singh on the occasion of his 95th birthday.