Assuming that the absence of perturbations guarantees weak or strong convergence to a common fixed point, we study the behavior of perturbed products of an infinite family of nonexpansive operators. Our main result indicates that the convergence rate of unperturbed products is essentially preserved in the presence of perturbations. This, in particular, applies to the linear convergence rate of dynamic string-averaging projection methods, which we establish here as well. Moreover, we show how this result can be applied to the superiorization methodology.

For the spaces DLp, Lp and M1, we consider the topology of uniform convergence on absolutely convex compact subsets of their (pre‐)dual space. Following the notation of J. Horváth's book we call these topologies κ‐topologies. They are given by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre‐)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi‐norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of semi‐norms generating the κ‐topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence‐space representation of the spaces DLp equipped with the κ‐topology, which complements a result of J. Bonet and M. Maestre. As a byproduct, we give a characterisation of the compact subsets of the spaces DLp′, Lp and M1.

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Abstract. We consider the space of non-expansive mappings on a bounded, closed and convex subset of a Banach space equipped with the metric of uniform convergence.We show that the set of strict contractions is a σ-porous subset. If the underlying Banach space is separable, we exhibit a σ-porous subset of the space of non-expansive mappings outside of which all mappings attain the maximal Lipschitz constant one at typical points of their domain.