A QUANTITATIVE VERSION OF A THEOREM DUE TO BORWEIN-REICH-SHAFRIR

Abstract

We give a quantitative analysis of a result due to Borwein, Reich and Shafrir on the asymptotic behaviour of the general Krasnoselski-Mann iteration for nonexpansive self-mappings of convex sets in arbitrary normed spaces. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of asymptotic regularity of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brien and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform bounds which do not depend on the starting point of the iteration and the nonexpansive function, but only depend on the error ε, an upper bound on the diameter of C and some very general information on the sequence of scalars λ k used in the iteration. Only in the special situation, where λ k := λ is constant, uniform bounds were known in that bounded case. For the unbounded case, no quantitative information was known before. Our results were obtained in a case study of analysing non-effective proofs in analysis by certain logical methods. General logical meta-theorems of the author guarantee (at least under some additional restrictions) the extractability of such bounds from proofs of a certain kind and provide an algorithm to extract them. Our results in the present paper (which we present here without any reference to that logical background) were found by applying that method to the original proof of the Borwein/Reich/Shafrir theorem. The general logical method which led to these results will be discussed (with further examples) in [22]. *Basic Research in Computer Science, funded by the Danish National Research Foundation.