WEAK CONVERGENCE OF SEQUENCE OF SUCCESSIVE APPROXIMATIONS FOR NONEXPANSIVE MAPPINGS
WEAK CONVERGENCE OF SEQUENCE OF SUCCESSIVE APPROXIMATIONS FOR NONEXPANSIVE MAPPINGS
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DOI：
10.1090/s0002-9904-1967-11761-0
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发表时间：
1967-01-01
影响因子：
1.3
通讯作者：
OPIAL, Z
中科院分区：
文献类型：
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作者：
OPIAL, Z
在最近的一篇论文[4]中，FE Browder和WV Petryshyn证明了如果Hubert空间X到自身的非扩张映射T：X->；X是渐近正则的并且至少有一个不动点，则对于X中的任意x，逐次逼近序列的弱收敛子序列的弱极限{Tnx}是T的不动点。本文的主要目的是通过证明在相同的假设下序列{Tnx}必然弱收敛来显著地加强这一结果。在§1中，我们回顾了一些基本定义，并证明了两个简单的引理。在§2中，我们证明了序列{Tnx}的弱收敛；在§3中，我们讨论了这一结果推广到具有弱连续对偶映射的Banach空间的可能性。在§4中，给出了§3中的定理2在修正的逐次逼近序列中的应用，并且在§5中，讨论了§1中的第一个关键引理的有效性极限。设C是Banach空间X中的凸闭集，映射T：C-+X称为非扩张的，如果||7a；-！T；y||g||#-y§对于C中的任意x，y，在[4]之后，称映射T：C-*C是渐近的
In a recent paper [4] FE Browder and WV Petryshyn have shown that if a nonexpansive mapping T: X—> X of a Hubert space X into itself is asymptotically regular and has at least one fixed point then, for any x in X, a weak limit of a weakly convergent subsequence of the sequence of successive approximations {Tnx} is a fixed point of T. The main object of the present note is to strengthen considerably this result by showing that under the same assumptions the sequence {Tnx} is necessarily weakly convergent. In § 1 we recall some basic definitions and prove two simple lemmas. In § 2 we prove the weak convergence of the sequence {Tnx} and in § 3 we discuss the possibility of the extension of this result to Banach spaces having weakly continuous duality mappings. In § 4 an application of Theorem 2 stated in § 3 to a modified sequence of successive approximations is given and, in § 5, limits of validity of the first key lemma of § 1 are discussed.1. Let C be a convex closed set in a Banach space X. A mapping T: C-+ X is called nonexpansive if|| 7a;—! T; y|| g||#—y § for any x, y in C. Following [4], a mapping T: C—* C is said to be asymptotically