摘要
设E是满足Opial条件的一致凸Banach空间,C是E的一非空闭凸子集,T:C→C是渐近非扩张映象.又设对任给的x1∈C,序列{xn}由下列带误差的修正的Ishikawa迭代程序生成:其中, 是C中的序列,使得 且数列 满足下列条件(i)和(ii)之一: (i)tn∈[a,b]且sn∈[O,b];(ii)tn∈[a,b]且sn∈[a,b],这里,常数a,b满足0<a≤b<1.本文证明了,T有不动点当且仅当,{xn}弱收敛且{‖xn-Txn‖}收敛到0.而且,由此即得下列结论:(1)若T有不动点,则{xn}弱收敛到T的一个不动点;(2)若T有不动点且对某个m≥1,Tm是紧的,则{xn}强收敛到T的一个不动点.
Let E be a uniformly convex Banach space satisfying Opial's condition, C be a nonempty closed convex subset of E with C + C C C and T : C →C be an asymptotically nonexpansive mapping. Suppose that for any initial data x1 in C, {xn}is defined by the modified Ishikawa iteration process with errors
where {un} and {vn} are bounded sequences in C such that
oo, {αn} and {βn} are chosen so that βn ∈ [a, b] and βn e [0,b] or α [a, 1] and βn ∈ [a, b] for some a, 6 with 0 < a ≤b < 1. It is shown that T has a fixed point if and only if {xn} is weakly convergent and {xn - Txn} is strongly convergent to zero. Furthermore, this immediately implies the following conclusions: (1) If T has a fixed point then {xn}converges weakly to a fixed point of T; (2) If T has a fixed point and Tm is compact for some m ≥1, then {xn} converges strongly to a fixed point of T.
出处
《应用数学学报》
CSCD
北大核心
2004年第4期674-681,共8页
Acta Mathematicae Applicatae Sinica
基金
高等学校优秀青年教师教学和科研奖励基金
上海市科委重大项目基金(部分)
上海市高校科技发展基金(部分)资助项目
