迭代逼近m-增生映象的零点

设E是具有一致正规结构的实Banach空间,其范数是一致Gateaux可微的.设A是m-增生映象,使得C=■是E的凸子集,数列{α_n)■[0,1],{r_n}■ (0,∞),在适当的条件下,则由(1.2)式定义的迭代序列{x_n}强收敛于A~(-1)(0)中的点.其次证明了:设E是一致凸Banach空间,其范数是Frechet可微的.设数列{α_n},{β_n)■(0,1),{r_n}■(0,∞),满足适当的条件.如果A~(-1)(0)∩B~(-1)(0)≠φ,则由(3.20)式定义的序列{x_n}弱收敛于A~(-1)(0)∩B~(-1)(0)中的点.其结果推广和改进了Kamimura,Takahashi(2000)的定理2及Xu H.K.(2006)的定理4.1,定理4.2和定理4.3:(i)Kamimura,Takahashi(2000)定理2中的假设"自反Banach空间E的每个有界闭凸子集对非扩张自映象有不动点性质"被去掉;(ii)Xu H.K.(2006)的假设"E是具有弱连续对偶映象J_φ的自反Banach空间",被本文的假设"E是具有一致正规结构且其范数是一致Gateaux可微的Banach空间"所取代.从而补充了Xu H.K.(2006)未包含的另外一些Banach空间.同时还证明了逼近两个m-增生映象的公共零点,其结果也推广和改进了Mainge的相应结果.

Approximation of the Iterative Method for a Zero of m-Accretive Mappings

Suppose E is a Banach space with uniformly normal structure and suppose E also has uniformly Gateaux differentiable norm.Let A be an m-accretive operator such that C = D(A) is a convex subset of E.Let {α_n} be a sequence in the interval (0,1) and let {r_n} be a sequence in the interval (0,∞).Then,under some suitable conditions,the sequence {x_n} defined by (1.2) converges strongly to an element of A~(-1)(0).Secondly,we also prove that:Let E be a uniformly convex Banach space whose norm is Frechet differentiable.Let {α_n},{β_n} be two squences in the interval (0,1) and let {r_n} be a sequence in the interval (0,∞).If A~(-1) (0)∩B~(-1)(0)≠0,then,under some suitable conditions,the sequence {x_n} defined by (3.20) converges weakly to an element in A~(-1)(0)∩B~(-1)(0).Our results extend and improve Theorem 2 of Kamimura, Takahashi (2000) and theorem 4.1,Theorem 4.2,Theorem 4.3 of Xu (2006):(i) In Theorem 2 of Kamimura,Takahashi (2000),the condition"Every bounded,closed and convex subset of a reflexive Banach space has the fixed point property for nonexpansive mappings"is removed;(ii) In Xu H.K.(2006),the condition"E is reflexive and has a weakly continuous duality map J_■"is replaced by"E is a Banach space with uniformly normal structure and E also has uniformly Gateaux differentiable norm."so that it contains some Banach spaces besides the Banach space which is in Xu H.K.(2006). At the same time,we state how to approximate a common zero of two m-accretive operators in E.Hence,the results also improve and unify some corresponding results.