一致凸Banach空间中α-非扩张映象的强收敛定理

Strong Convergence Theorems forα-nonexpansive Mapping in Uniformly Convex Banach Spaces

设C是一致凸Banach空间中的非空闭凸子集,T:C→C是具有不动点的半紧α-非扩张映象,其中α<1。任取一点x0∈C,{xn}是由xn+1=(1-αn-βn)xn+αnTyn+βnun,yn=(1-γn-δn)xn+γnTxn+δnvn,n=0,1,2,…定义的带误差的Ishikawa迭代序列,其中0<A≤αn≤B<1/2,0≤γn≤γ<1,n=0,1,2,…,∞∑n=0βn<∞,∞∑n=0δn<∞,{un}和{vn}是C中的有界点列。本文证明了{xn}强收敛于T的某一不动点。

Let C be a nonempty closed convex subset of a uniformly convex Banach space, and let T：C→C be a semi-compact α-non- expansive mapping with fixed points, where α〈1. For given x0≥C, suppose that the sequence {x,, } is the Ishikawa iterative se- quence with errors defined by xn＋1=（1-αn-βn）xn＋αnTyn＋βnun,yn=（1-γn-δn）xn＋γnTxn＋δnvn,n=0,1,2,…, where 0〈A≤αn≤B〈1/2,0≤γn≤γ〈1,n=0,1,2,…,∞∑n=0βn〈∞,∞∑n=0δn〈∞,{un}and{vn} are two bounded sequences in C. It is provedthat the sequence {x. } strongly converges to a fixed point of T.