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 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) ∑n j=0(PT) xn 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 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 (2001) and many others.