Nonexpansive Mappings, Asymptotic Regularity and Successive Approximations
Nonexpansive Mappings, Asymptotic Regularity and Successive Approximations
复制标题
非扩张映射、渐近正则性和逐次逼近
DOI：
10.1112/jlms/s2-17.3.547
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发表时间：
1978
影响因子：
1.2
通讯作者：
Richard C. O'Brien
中科院分区：
文献类型：
--
作者：
M. Edelstein;Richard C. O'Brien
设C是赋范线性空间X的凸子集和C的非扩张自映射；即||/(x) -f {y)||<\\x - y\\,（x, yeC）。1990年，Krasnoselski证明了如果X是一致凸且C是紧致的，那么对于任意X ec， X在F±=\（I+ F）下的迭代序列{^"(X)}收敛于/的不动点。Schaefer[13]观察到对于任意Fx= XI+(1 - A)/，当0< A< 1时，同样成立，并且有作者证明了X的严格凸性满足[4]。严格凸性是否也能被去除这个自然的、并非不重要的问题多年来一直悬而未决，这个问题激发了本文的工作。我们得到一个肯定的答案，作为一个更深刻的发现的必然结果；即，在任何赋范线性空间X和任何有界凸集C C = X上，FA是（一致）渐近正则的（见第2节）。渐近正则性的概念是由Browder和Petryshyn[2]提出的，作为一个度量概念，它可以表述如下：度量空间（M, d）到自身的映射/:M-> M在xeM处是渐近正则的，如果d (f ' + 1 (x),/ '(*))-> 0为n-*•oo；如果它在每个xeM上都是渐近正则的，我们就说它在M上是渐近正则的。关于Fx的渐近正则性的结果是由Browder和Petryshyn在2010年首次得到的。他们证明了如果X是一致凸且/:C-> C是闭有界凸子集C上的非扩张自映射，则Fx是渐近正则的。我们的结果是更强的，一方面，它们不局限于一致凸空间，另一方面，它们涉及一个一致版本的渐近正则性。很容易看出，Fx (y)= y等价于f (y)= y，因此关于/不动点的存在性和位置的问题可以简化为Fx的类似问题，其中Fx可以根据我们的结果假设为渐近正则。（参见定理1后面的注释。）
Let C be a convex subset of a normed linear space X and/a selfmapping of C which is nonexpansive; that is,||/(x)—f {y)||<\\x—y\\,(x, yeC). In [8] Krasnoselski proved that if X is uniformly convex and C is compact then, for any xeC, the sequence {^"(x)} of iterates of x under F±=\(I+ f) converges to a fixed point of/. Schaefer [13] observed that the same holds for any Fx= XI+(1—A)/with 0< A< 1, and one of the authors proved that strict convexity of X suffices [4]. The natural, and not unimportant, question of whether strict convexity can also be removed remained open for many years and this question motivated the work presented here. We obtain an affirmative answer as a corollary to a more incisive finding; namely that, on any normed linear space X and on any bounded convex set C c= X, FA is (uniformly) asymptotically regular (see Section 2). The concept of asymptotic regularity is due to Browder and Petryshyn [2] and, as a metric notion, it can be stated as follows: a mapping/: M-> M of a metric space (M, d) into itself is said to be asymptotically regular at xeM if d (f"+ 1 (x),/"(*))-> 0 as n-*• oo; it is said to be asymptotically regular on M if it is so at each xeM. Results on the asymptotic regularity of Fx were first obtained by Browder and Petryshyn in [2]. They showed that if X is uniformly convex and/: C-> C is a nonexpansive selfmapping on a closed, bounded, convex subset C, then Fx is asymptotically regular. Our results are stronger in that, on the one hand, they are not restricted to uniformly convex spaces and, on the other, they involve a uniform version of asymptotic regularity. As is easily seen Fx (y)= y is equivalent to f (y)= y, so that problems pertaining to the existence and location of fixed points for/reduce to similar problems for Fx where Fx can be assumed to be asymptotically regular by our results.(See the remark following Theorem 1.)