## On the computational content of the Krasnoselski and Ishikawa fixed point theorems (2000)

Citations: | 11 - 10 self |

### BibTeX

@MISC{Kohlenbach00onthe,

author = {Ulrich Kohlenbach},

title = {On the computational content of the Krasnoselski and Ishikawa fixed point theorems},

year = {2000}

}

### OpenURL

### Abstract

This paper is a case study in proof mining applied to non-effective proofs in nonlinear functional analysis. 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 under certain compactness conditions. But, as we show, already for uniformly convex spaces in general no bound on the rate of convergence can be computed uniformly in f . This is related to the non-uniqueness of fixed points. However, the iterations yield even without any compactness assumption and for arbitrary normed spaces approximate fixed points of arbitrary quality for bounded C (asymptotic regularity, Ishikawa 1976). We apply proof theoretic techniques (developed in previous papers of us) to non-effective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...