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.