Strong convergence of approximation fixed points for nonexpansivenonself‐mapping
Rudong Chen, Zhichuan Zhu · International Journal of Mathematics and Mathematical Sciences · 2006
Let C be a closed convex subset of a uniformly smooth Banach space E, and T : C → E a nonexpansive nonself‐mapping satisfying the weakly inwardness condition such that F(T) ≠ ∅, and f : C → C a fixed contractive mapping. For t ∈ (0, 1), the implicit iterative sequence {xt} is defined by xt = P(tf(xt) + (1 − t)Txt), the explicit iterative sequence {xn} is given by xn+1 = P(αnf(xn) + (1 − αn)Txn), where αn ∈ (0, 1) and P is a sunny nonexpansive retraction of E onto C. We prove that {xt} strongly converges to a fixed point of T as t → 0, and {xn} strongly converges to a fixed point of T as αn satisfying appropriate conditions. The results presented extend and improve the corresponding results of Hong‐Kun Xu (2004) and Yisheng Song and Rudong Chen (2006).