doi:10.1155/2007/59262 Research Article Strong Convergence of Cesàro Mean Iterations for Nonexpansive Nonself-Mappings in Banach Spaces

Let E be a real uniformly convex Banach space which admits a weakly sequentially con-tinuous 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 gen-erated by xn+1 = αnx + (1 − αn)(1/n+ 1) ∑n j=0(PT) jxn and yn+1 = (1/n+ 1) ∑n j=0P(αny + (1 − αn)(TP) j yn) for all n ≥ 0, where x,x0, y, y0 ∈ C, {αn} is a real sequence in an inter-val [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 (2001) and many others. Copyright © 2007 Rabian Wangkeeree. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1