Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 59262, 11 pages
doi:10.1155/2007/59262

Strong convergence of Cesàro mean iterations for nonexpansive nonself-mappings in Banach spaces

Abstract

Let E be a real uniformly convex Banach space which admits a weakly sequentially continuous duality mapping from E to E*, C a nonempty closed convex subset of E which is also a sunny nonexpansive retract of E, and T:C→E a non-expansive nonself-mapping with F(T)≠∅. In this paper, we study the strong convergence of two sequences generated by xn+1=αnx+(1−αn)(1/n+1)∑j=0n(PT)jxn and yn+1=(1/n+1)∑j=0nP(αny+(1−αn)(TP)jyn) for all n≥0, where x,x0,y,y0∈C, {αn} is a real sequence in an interval [0,1], and P is a sunny non-expansive retraction of E onto C. We prove that {xn} and {yn} converge strongly to Qx and Qy, respectively, as n→∞, where Q is a sunny non-expansive retraction of C onto F(T). The results presented in this paper generalize, extend, and improve the corresponding results of Matsushita and Kuroiwa and many others.