非扩张映象不动点的迭代算法

设C是具有一致Gateaux可微范数的实Banach空间X中的一非空闭凸子集,T是C中不动点集F(T)≠0的一自映象．假设当t→0时,{Xt}强收敛到T的一不动点z,其中xt是C中满足对任给u∈C,xt=tu+(1-t)Txt的唯一确定元．设{αn},{βn}和{γn}是0,1]中满足下列条件的三个实数列:(i)αn+βn+γn=1;(ii) limn-∞αn=0和．对任意的x0∈C,设序列{xn}定义为xn+1=αnu+βnxn+γnTxn,则{xn}强收敛到T的不动点．

Iterative Algorithms to Fixed Point of Nonexpansive Mapping

Let C be a nonempty closed convex subset of a real Banach space X which has a uniformly Gateaux differentiable norm and T be a nonexpansive self-mapping of C with F(T)≠0: Assume that {xt} converges strongly to a fixed point z of T as t→0, where xt is the unique element of C which satisfies xt = tu + (1-t)T for arbitrary u←C. Let {αn}, {βn} and {γn} be three real sequences in 0,1] which satisfies the following conditions: (i)αn+βn+γn = 1; (ii) limn→∞αn = 0 and ; (iii) 0 < liminfn→∞βn≤limsupn→∞βn < 1. For arbitrary x0∈C, let the sequence {xn} be defined by xn+1 =αnu +βnxn +γnTxn. Then, {xn} converges strongly to a fixed point of T.