On sufficient conditions ensuring the norm convergence of an iterative sequence to zeros of accretive operators

Given two real sequences (rn) and (αn), we study the iterative scheme: xn+1=αnu+(1-αn)Jrnxn, for finding a zero of an accretive operator A, where u is a fixed element and Jrn denotes the resolvent of A. To ensure its convergence, the real sequence (rn) is always assumed to satisfy ∑n=0∞|rn+1-rn|<∞. In this paper we show this condition can be completely removed, which enables us to improve a result recently obtained by Saejung.