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.