巴拿赫空间中Ishikawa过程非扩张映射不动点逼近(英文)

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

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,T∶C→C是一非扩张映射,证明对于任意初始假设x_0∈C,通过x_(n+1)=t_nT(S_nTx_n+(1-S_n)x_n)+(1-t_x)x_n,n=0,1,2,…定义的Ishikawa迭代弱收敛到T的不动点,其中limsup_(n→+∞)S_n≤1,{n_k}_(k=0)^(+∞)是满足∑_(k=0)^(+∞)t_n_k(1-t_n_k)发散的{n}_(n=0)^(+∞)的子列,由此证明Zeng的定理,Tan和Xu的定理1,Reich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。

Let X be a uniformly convex Banach space X whose dual X~* has the KK property. Let C be a bounded closed convex subset of X. Let T:C^1→C be a nonexpansive mapping. It is shown that for any initial guess x_0∈C, the Ishikawa iteration {x_n}~_(n=0)^(+∞) defined by x_(n+1)=t_nT(S_nTx_n+(1-s_n)x_n)+(1+t_n)x_n, with the restriction that limsup_(n→+∞)s_n is less than 1 and sum from k=o to +∞(t_(n_k)(1-t_(n_k))) diverges for any subsequence {n_k}_(k=0)^(+∞) of {n}_(n=0)^(+∞), converges weakly to a fixed point of T. This generalizes the corresponding 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 paper [6].