Relaxation theorem for set-valued functions with decomposable values (Q2785637)

Let \((T,{\mathfrak A},\mu)\) be a separable nonatomic probability measure space and let \(F:B\to L(T,\mathbb{R}^n)\) be a multifunction integrably bounded by some function \(m\in L(T,\mathbb{R})\) with closed, decomposable values, where \(B=\{f\in L(T,\mathbb{R}^n): \mu(\{t: |f(t)|>m(t)=0\}\). Suppose that there exists a function \(k\in L(T,\mathbb{R}^+)\) with \(|k|<1\) such that \({\mathcal H}_E(F(x),F(y))\leq|k|_E \sup|\int_E (x-y)d\mu|\) for \(x,y\in B\) where \({\mathcal H}_E\) denotes the Hausdorff distance relativized to \(E\in{\mathfrak A}\) and sup is taken over all such \(E\in{\mathfrak A}\). It is proved that \(F\) has a fixed point and that the closed convex hull of \(F\) is the weak closure of the set of fixed points of \(F\). This theorem is applied in order to get a relaxation theorem for fixed point sets of multivalued stochastic processes.