A Note on Approximating Fixed Points of Nonexpansive Mappings by the Ishikawa Iteration Process

Abstract

LetXbe a uniformly convex Banach space that satisfies Opial's condition or whose norm is Fréchet differentiable, letCbe a bounded closed convex subset ofX, and letT:C→Cbe a nonexpansive mapping. It is shown that for any initial datax0∈C, the Ishikawa iterates {xn}, defined byxn+1=tnT(snTxn+(1-s n)xn)+(1-tn)xn,n≥0, with the restrictions thatlimnsnis less than 1, and for any subsequence {nk}∞k=0of {n}∞n=0, ∑∞k=0tnk(1-tnk) diverges, converge to a fixed point ofTweakly. Thus, such a result complements Theorem 1 of Tan and Xu (J. Math. Anal. Appl.178(1993), 301-308) and generalizes, to a certain extent, Theorem 2 of Reich (J. Math. Anal. Appl.67(1979), 274-276). © 1998 Academic Press.