Banach空间非扩张映像不动点强收敛定理

设E是具弱序列连续对偶映像自反Banach空间, C是E中闭凸集, T:C→ C是具非空不动点集F(T)的非扩张映像.给定u∈ C,对任意初值x0∈ C,实数列{αn}n∞=0,{βn}∞n=0∈ (0,1),满足如下条件:(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|<∞设{x_n}_(n_1)~∞是由下式定义的迭代序列:{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)~∞则{x_n}_(n=1)~∞强收敛于T的某不动点.

Strong Convergence Theorems for Nonexpansive Mapping in Banach Spaces

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) ≠(0\). 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) ∞∑n=0αn = ∞,αn → 0;(ii) βn ∈ 0,a) for some a ∈ (0,1);(iii) ∞∑n=0|αn+1-αn| ＜ ∞,∞∑n=0|βn+1 - βn| ＜∞.Let {xn}n=1∞ be composite process defined by{yn = βnxn + (1-βn)Txnxn+1 = αnu + (1-αn)ynThen {xn}n=1∞ converges strongly to a fixed point of T.