Approximating zeros of accretive operators
Approximating zeros of accretive operators
复制标题
累加算子的近似零点
DOI：
10.1090/s0002-9939-1975-0470762-1
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发表时间：
1975
期刊：
影响因子：
--
通讯作者：
S. Reich
中科院分区：
文献类型：
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作者：
S. Reich
设A是自反Banach空间E中具有Gateaux可微范数的m-增生集。对于正的r，设Jr表示A的预解式。若E的对偶映射是弱序列连续的，且0在A的值域内，则对E中的每个x，存在强limoojx，且强limoojx属于A-1（0）.这是一个扩展到Banach空间设置的结果以前只知道希尔伯特空间。设H是真实的Hilbert空间，UCHxH是极大单调算子.对于每个正的r，在H中存在唯一的yr，使得0 eYr + rU（y）。已知[4]，如果0属于U的值域，则强limroo yr存在，并且是U-1（0）中最接近0的点。本文的目的是将这一结果推广到某些Banach空间中的增生算子。根据[4]，这导致计算给定算子的零点作为迭代构造序列的极限的可能性。我们的证明方法不是希尔伯特空间证明的直接推广。然而，它仅在有限的一类Banach空间中有效。我们的定理在其他Banach空间中是否有效的问题仍然是开放的。设E* 表示真实的Banach空间E的对偶。从E到E * 的非空子集族的对偶映射J定义为J（x）= lx* c E*：（x，x*）= 11 × 12，||X*|| = ||X|}. J是单值的当且仅当E的范数是Gateaux可微的。如果A是E x E和x e E的子集，我们定义Ax = ye E：[x，yle Al和集合D（A）= IxE：Ax/0}。A的值域定义为：R（A）= U Ax：x E D（A）} 1974年10月1日由编辑接收。AMS（MOS）主题分类（1970年）。小学47 H15、47 H 05;中学47 B44、40 A05。
Let A be an m-accretive set in a reflexive Banach space E with a Gateaux differentiable norm. For positive r let Jr denote the resolvent of A. If the duality mapping of E is weakly sequentially continuous and 0 is in the range of A, then for each x in E the strong lim ooj x exists and belongs to A-1(0). This is an extension to a Banach space setting of a result previously known only for Hilbert space. Let H be a real Hilbert space and U C H x H a maximal monotone operator. For each positive r there is a unique y r in H such that 0 e Yr + rU(y). It is known [4] that if 0 belongs to the range of U, then the strong limroo yr exists and is the point of U-1(0) closest to 0. It is our purpose in this note to extend this result to accretive operators in certain Banach spaces. According to [4], this leads to the possibility of calculating a zero of the given operator as the limit of an iteratively constructed sequence. Our method of proof is not a direct generalization of the Hilbert space proof. It works, however, only in a restricted class of Banach spaces. The question of whether our theorem is valid in other Banach spaces remains open. Let E* denote the dual of a real Banach space E. The duality mapping J from E into the family of nonempty subsets of E * is defined by J(x) = lx* c E*: (x, x*) = 11 x12 and ||x*|| = ||x|}. J is single-valued if and only if the norm of E is Gateaux differentiable. If A is a subset of E x E and x e E, we define Ax = ye E: [x, yle Al and set D(A)= IxE E: Ax/0}. The range of A is defined by R(A)= U Ax: x E D( A)} Received by the editors October 1, 1974. AMS (MOS) subject classifications (1970). Primary 47H15, 47H05; Secondary 47B44, 40A05.