Strongly unique best approximations and centers in uniformly convex spaces
Strongly unique best approximations and centers in uniformly convex spaces
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DOI：
10.1016/0022-247x(87)90234-4
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发表时间：
1987
影响因子：
1.3
通讯作者：
B. Prus;R. Smarzewski
中科院分区：
文献类型：
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作者：
B. Prus;R. Smarzewski
设M是Banach空间X的闭凸非空子集，使得dim X>，2。元素m EM被称为M中元素x EX的最佳逼近，如果11。对于M中的所有y，ml 1 d 11-x-Al（1.1）。称最佳逼近m是元素XE X在A4中的强唯一最佳逼近，如果存在一个常数1= n（x）> 0和一个严格递增的连续函数cp：[0，K]= R++ R+;~（0）= 0，使得对M中的所有y，cp（11 ~-11），< 50（11 ~-11）-2-c~（11 ~-11）（1.2）.本文推广了最近在[13，141]中提出的一些结果。更精确地说，我们证明了如果X是一个具有幂型凸模q> 2的一致凸空间，那么M中的一个元素x EX的最佳逼近m也是M中x的强唯一最佳逼近，q（s）= s4。特别地，我们证明了当X是Lebesgue空间L或Sobolev空间Hm 3 P时，这是真的。请注意，这完全解决了Dunham提出的以下问题[S，问题411（参见。也[1，13，143）：什么是对应的强唯一性的L，空间？其次，我们分别对有界子集和有界序列的相对中心和渐近中心建立了类似的结果。最后，我们应用它们得到了一致Lipschitz映象的一个不动点定理。
Let M be a closed convex nonempty subset of a Banach space X such that dim X>, 2. An element m EM is said to be a best approximation in M to an element x EX if11.~-ml1 d 11-x-Al(1.1) for all y in M. The best approximation m is called a strongly unique best approximation in A4 to the element XE X if there exist a constant 1= n (x)> 0 and a strictly increasing continuous function cp:[0, KJ)= R++ R+;~(0)= 0, such that cp (ll~--ll),< 50 (ll~-~ lI)-2-c~ fllm--yll)(1.2) for all y in M. In this paper we extend some results which have been presented recently in [13, 141. More precisely, we prove that if X is a uniformly convex space with a modulus of convexity of power type q> 2 then a best approximation m in M to an element x EX is also a strongly unique best approximation in M to x with q (s)= s4. In particular, we show that it is true when X is the Lebesgue space L, or the Sobolev space Hm3P. Note that this solves completely the following problem posed by Dunham [S, Problem 411 (cf. also [l, 13, 143): What is the counterpart of strong uniqueness for L, spaces? Next, we establish similar results for relative centers and asymptotic centers of bounded subsets and bounded sequences, respectively. Finally, we apply them to derive a fixed point theorem for uniformly Lipschitzian mappings.