Implicit iteration process of nonexpansive non-self-mappings

Abstract

Suppose C is a nonempty closed convex subset of real Hilbert space H. Let T:C→H be a nonexpansive non-self-mapping and P is the nearest point projection of H onto C. In this paper, we study the convergence of the sequences {xn}, {yn}, {zn} satisfying xn=(1−αn)u+αnT[(1−βn)xn+βnTxn], yn=(1−αn)u+αnPT[(1−βn)yn+βnPTyn], and zn=P[(1−αn)u+αnTP[(1−βn)zn+βnTzn]], where {αn}⊆(0,1), 0≤βn≤β<1 and αn→1 as n→∞. Our results extend and improve the recent ones announced by Xu and Yin, and many others.