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

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) not equal circle divide. In this paper, we study the strong convergence of two sequences generated by x(n+1) = alpha(n)x+ (1- alpha(n))(1/n+ 1) Sigma(n)(j= 0)(PT)(j)(xn) and y(n+1) = (1/n+1)Sigma P-n(j= 0)(alpha(n)y + (1- alpha(n))(TP)(j)(yn)) for all n >= 0, where x, x(0), y, y(0) is an element of C, {alpha(n)} is a real sequence in an interval [ 0, 1], and P is a sunny non-expansive retraction of E onto C. We prove that {x(n)} and {y(n)} converge strongly to Qx and Qy, respectively, as n ->infinity, 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 (c) 2007.