关于Halpern公开问题的一个回答

An answer to the Halpern’s open question

设E是一致光滑的Banach空间,C是E中一非空闭凸子集,T:C→C是一非扩张映像且不动点集非空,u∈C是一给定的点,而x 0∈C是任一初始点.如果{αn}是(0.1)中之一实数列,满足条件limn→∞αn=0和∞∑n=1αn=∞而且由x n+1=αnu+(1-αn)Txn定义的序列{x n}满足条件■Tzn-xn‖-‖zn-xn■=ο(αn),这里zn=αnu+(1-αn)Tzn,则{xn}∞n=0强收敛于T在C中之一的不动点.

Let E be a uniformly smooth Banach space and C be a nonempty closed convex subset of E. Let T ： C→C be a nonexpansive mapping such that T（T）≠Ф.Given uЕC and the initial guess XoЕC is chosen arbitrarily.Let an be a real sequence in （0,1）satisfying the conditions lim an =0and Еan=∞.If he sequence xn defined by xn＋1=anu＋（1-an）Txn also satisfies the condition Tzn-xn-xn=0（an）,where zn=anu＋（1-an）Tzn,then xnn=0strongly converges to a fixed point of T.