Banach空间非扩张映像不动点强收敛定理(英文)

Strong Convergence Theorems for Nonexpansive Mapping in Banach Spaces

设E是具弱序列连续对偶映像自反Banach空间，C是E中闭凸集，T：C→C是具非空不动点集F（T）的非扩张映像．给定u∈C，对任意初值x0∈C，实数列{an}n=0^∞，{βn}n=0^∞∈（0，1），满足如下条件：（i）∑n=0^ωan=∞,an→0;（ii）0≤βn＜a＜1;（iii）∑n=0^∞｜an＋1-an｜〈∞,∑n=0^∞｜βn＋1-βn｜〈∞.设{xn}n=1^∞是由下式定义的迭代序列：{yn=βnxn＋（1-βn）Txn xn＋1=anu＋（1-an）yn 则{xn}n=1^∞强收敛于T的某不动点。

Assume E is a reflexive Banach space which has a weakly continuous duality map, C is a closed convex subset of E and let T：C→C be a nonexpansive mapping such that F（T） ≠Q. Given a point u E C, the initial guess x0 ∈ C is chosen arbitrarily and given sequences {an}n=0^∞, {βn}n=0^∞ in （0, 1） ,the following conditions are satisfied （i）∑n=0^ωan=∞,an→0;（ii）βn∈[0,a）for some α∈（0,1）;（iii）∑n=0^∞｜an＋1-an｜〈∞,∑n=0^∞｜βn＋1-βn｜〈∞.Let{xn}n=1^∞ be composite process defined by {yn=βnxn＋（1-βn）Txn xn＋1=anu＋（1-an）yn Then{xn}n=1^∞ converges strongly to a fixed point of T.