复合Halpern迭代逼近可数个增生映像的公共零点

Composite Halpern Iteration for Approximating Common Zero of Countably Infinite Family of m-accretive Mappings

设E是严格凸和自反的实Banach空间且其范数是一致Gateaux可微的,K是E中非空闭凸子集,Ai∶K→E(i∈N)是m-增生映像且公共不动点集非空,u∈K是给定点,x1∈K是任一初始点,{αn}n∞=1、{βn}n∞=1是[0,1]中的实数列且满足如下条件:(i)limn→∞αn=0,∑n∞=1αn=∞,∑n∞=1|αn+1-αn|<∞;(ii)limn→∞βn=0,∑n∞=1|βn+1-βn|<∞。设{xn}n∞=1是由下面复合Halpern格式定义的迭代序列:yn=βnxn+(1-βn)Sxn,n≥0xn+1=αnu+(1-αn)yn≥其中S=∑∞i=1ξi JAi,JAi=(I+Ai)-1(i∈N),那么{xn}n∞=1强收敛于{Ai}i∈N的公共0点。本文的结果改进和推广了Zegeye和Shahzad,Ofoedu以及其他作者的相应结果。

Let K be a closed convex nonempty subset of a strictly convex and real reflexive Banach space E which has a uniformly GCtteaux differentiable norm. Let Ai：K→E（i∈N）be a countably infinite family of m-accretive mappings such that ∩^∞ i=1N（Ai）≠Ф. For arbitrary u,x1∈K, {αn}^∞ n=1 and {βn}^∞ n=1 are real sequences in the [0,1] satisfying the conditions：（i）lim n→∞αn=0,∑^∞ n=1 αn=∞,∑^∞ n=1｜αn＋1-αn｜〈∞;（ii）lim n→∞βn=0,∑^∞ n=1｜βn＋1-βn｜〈∞.{Xn}^∞ n=1 be composite I-Ialpern iteration definited by：{yn=βnXn＋（1-βn）SXn,n≥0 Xn＋1=αnu＋（1-αn）yn Where S=∑∞ i=1ξiJAi,JAi=（1＋Ai）^-1（i∈N）,then,{Xn}^∞ n=1 converges strongly to a common zero of {Ai}i∈N .The results presented in this paper improve and extend the correspoinding ones of Hegeye and Shahzad and Ofoedu and others.