Effective Uniform Bounds on the Krasnoselski-Mann Iteration

This paper is a case study in proof mining applied to non-effective proofs in nonlinear functional anlysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general so-called Krasnoselski-Mann iterations. These iterations converge to fixed points of f only under special compactness conditions and even for uniformly convex spaces the rate of convergence is in general not computable in f (which is related to the non-uniqueness of fixed points). However, the iterations yield approximate fixed points of arbitrary quality for general normed spaces and bounded C (asymptotic regularity). In this paper we apply general proof theoretic results obtained in previous papers to non-effective proofs of this regularity and extract uniform explicit bounds on the rate of the asymptotic regularity. We start off with the classical case of uniformly convex spaces treated already by Krasnoselski and show how a logically motivated modification allows to obtain an improved bound. ∗Basic Research in Computer Science, Centre of the Danish National Research Foundation.