On a variant of Schu's lemma

Abstract

In this note, we demonstrate that an incorrect statement has been propagated in multiple papers, stemming from the substitution of ``lim'' with ``limsup'' for a sequence in Lemma 1.3 of the paper [J. Schu: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull.\ Austral.\ Math.\ Soc.\ 43 (1991), 153--159]. This occurred over a span of more than 20 years, with the earliest paper we identified using this incorrect statement dating back to 2002.

The paper has now 19 pages and contains 211 references. Having in view that Prof. R.I. Bot made some important remarks on the previous version of the paper, I (C. Zalinescu) invited him to become co-author of this work and I am very satisfied that he accepted