Convergence of generic infinite products of affine operators (Q1596326)

Let \((X,\|.\|)\) be a Banach space and let \(K\) be a nonempty bounded closed convex subset of \(X\) with the topology induced by the norm \(\|.\|\). An operator \(A:K\to K\) is affine if \(A(\alpha x+(1-\alpha)y)=Ax+(1-\alpha)Ay\) for each \(x,y\in K\) and all \(\alpha\in [0,1]\). The authors establish several results concerning the asymptotic behavior of random infinite products of generic sequences of affine uniformly continuous operators on bounded closed convex subsets of a Banach space. The authors show that in appropriate spaces of sequences of operators there exists a subset which is a countable intersection of open everywhere dense sets such that for each sequence belonging to this subset the corresponding random infinite product converges. In addition to a weak ergodic theorem the authors obtain convergence to a unique common fixed point and more generally, to an affine retraction.