巴拿赫空间中Ishikawa过程非扩张映射不动点逼近

X是一致凸巴拿赫空间,其对偶空间X*有KK性质.C是X的有界闭的凸子集.TC→C是一非扩张映射.证明对于任意初始假设x0∈C,通过xn+1=tnT(snTxn+(1-sn)xn)+(1-tn)xn,n=0,1,2,…定义的Ishikawa迭代弱收敛到T的不动点,其中limsupn→+∞ sn≤1,{nk}+∞ k=0是满足∑+∞ k=0 tnk(1-tnk)发散的{n}+∞ n=0的子列.由此证明Zeng6]的定理,Tan和Xu3]的定理1,Reich5]的定理.条件"X*有KK性质"比文献6]中的"有Frechet导数模"严格地弱也被强调.

Approximating fixed points of nonexpansive mapping by the Ishikawa process in Banach space

Let X be a uniformly convex Banach space X whose dual X* has the KK property. Let Cbe a bounded closed convex subset of X. Let T: C → C be a nonexpansive mapping. It is shown thatfor any initial guess x0∈C, the Ishikawa iteration {xn}n+=∞0 defined by xn+1=tnT(snTxn+ (1-sn) xn)+(1+tn)xn, with the restriction that limsupn→+∞sn is less than 1 and Σ+∞k=0 tnk(1-tnk) diverges for anysubsequence {nk}+∞ k=0 of {n}+∞ n=0, converges weakly to a fixed point of T. This generalizes the correspon-ding results in 3, 5-6]. It should be emphasized that the condition "X* has the KK property"is strictly weaker than the condition "X has a Frechet differentiable norm" which is required in paper6].