Random Fixed Point Theorems for Measurable Multifunctions in Banach Spaces

In this paper we prove several random fixed point theorems for measurable closed and nonclosed valued multifunctions satisfying general continuity conditions. Our work extends and sharpens earlier results by Engl, Itoh and Reich. 1. Preliminaries and definitions. The study of random fixed points was initiated by the Prague school of probabilists in the fifties. Recently the interest on this subject was revived especially after the survey article of Bharucha-Reid [3]. The theory of random fixed points has found important applications in random operator equations [2], random differential equations in Banach spaces [15] and differential inclusions [4]. Throughout this paper let (Q, E, /¿) be a complete rj-finite measure space and X a separable Banach space. Additional assumptions will be introduced as needed. By X* we will denote the topological dual of X. We will use the following notation: Pf(c)(X) = {A Ç X: nonempty, closed, (convex)} and P(UI)fc(c)(X) = {A Ç X: nonempty, (weakly)-compact, (convex)}. If A Ç X nonempty, we set \A[ = supxgA ||x||. Also by oa(-) we denote the support function of A, i.e. oa(x*) = suPi€a(x*'x) f°r an x* € X* and by (¿a(-) the distance function from A, i.e. d^(x) = \niz^A ||a¡ — z\\ for all x G X. k multifunction F: fi —> Pf(X) is said to be measurable if it satisfies any of the following equivalent conditions: (i) GrF = {(id,x) G fi x X:x G F(id)} G £ x B(X) where B(X) is the Borel tr-field of X, (ii) id —> d¡r(u) {x) is measurable for all x G X, and (iii) there exists a sequence {/n(-)}n>i of measurable selectors of F(-) s.t. F(i (Castaing's representation). If (fi, S) is a measurable space, then (ii) ■&■ (iii) => (i). For details see [4, 13, 22]. By SF we denote the set of all LX(Q) selectors of F(-), i.e. SF = {/(•) G LX(U): f(id) G F(td) /x-a.e.}. Clearly this is a closed subset LX(U) (maybe empty) and is nonempty if and only if inixeF(w) ||x|| G L\(Q). Let G: fi —> Pf(X). A multifunction F: Gr G —» Pf(X) is called "measurable multifunction with stochastic domain G(-)" if, for all x G X and all U Ç X open, {id G fi: x G G(id) and F(id,i)ni/^0}eS. A measurable function x: fi —» X s.t. for /i-almost all w G fi, x(td) G G(id) and x(td) G F(id,x(id)) is said to be a random fixed point for F(-, ■). If F is a topological space F:Y —► Pf(X) is called upper (lower) semicontinuous Received by the editors October 18, 1984 and, in revised form, February 19, 1985, July 11, 1985 and July 23, 1985. 1980 Mathematics Subject Classification. Primary 60H25, 47H10.