Strong Convergence Theorems for Nonexpansive Mapping in Banach Spaces
Xiaolong Qin · Acta Analysis Functionalis Applicata · 2007
Assume E is a reflexive Banach space which has a weakly continuous duality map,C is a closed convex subset of E and let T:C→C be a nonexpansive mapping such that F(T)≠φ.Given a point u∈C,the initial guess x0∈C is chosen arbitrarily and given sequences {αn}∞n=0,{βn}∞n=0 in(0,1),the following conditions are satisfied(i)sum from n=α to ∞ α_n=∞, α_n→0;(ii)β_n∈[0,α) for some α∈(0,1);(iii)sun for n=α to ∞|α_(n-1)+α_n|∞,sum from n=α|β_(n-1)-β_n|∞ Let {x_n}_(n_1)~∞ be composite process defined by{y_n=β_nx_n+(1-β_n)Tx_n x_(n+1)=α_nu+(1-α_n)y_n Then {x_n}_(n=1)~∞ converges strongly to a fixed point of T.