Random differential inclusions depending on a parameter

Abstract

Throughout this note (Q, 5, p) is a complete probability space; [a, b] is a compact real interval with the Lebesgue a-algebra Yt,,h,; (X, II.]I ) is a separable real Banach space, whose null element is denoted by 8,; A is a non-empty set; F is a multifunction from 52 x [a, h] x Xx A into X, with non-empty closed values. Denote by AC( [a, h], X) the space of all strongly absolutely continuous functions from [a, 61 into X which are almost everywhere strongly differen- tiable. For every .f~ AC( [a, b], X) put llfll AC’([o.h].X) = max Ilf(t)ll + (* Ilf’(~)li & ,E ra.hl (1 where j” is the strong derivative of jI Of course, AC( [a, h], X) endowed with the norm II . II AC,Ir,.bl.X, is complete; moreover, since X is separable, it is also separable. If IDl(Q, AC([u, h], X)) denotes the space of all (equivalence classes of) measurable functions from Q into AC( [a, h], X), for every