The random Markov-Kakutani fixed point theorem in a random locally convex module
发布时间：2024-08-20
点击次数：
发表刊物：New York J. Math.
项目来源：国家自然科学基金
关键字：Random locally convex modules, stable compactness, random Markov-Kakutani fixed point theorem, random Hahn-Banach theorem.
摘要：Based on the recently developed theory of 𝜎-stable sets and stable compactness, we first establish the random Markov-Kakutani fixed point theorem in a random locally convex module: let (𝐸, 𝒫) be a random locally convex module and 𝐺 be a nonempty stably compact 𝐿0-convex subset of 𝐸, then every commutative family of 𝒯𝑐(𝒫𝑐𝑐)-continuous 𝐿0-affine mappings from 𝐺 to 𝐺 has a common fixed point, where 𝒫𝑐𝑐 is the 𝜎-stable hull of 𝒫 and 𝒯𝑐(𝒫𝑐𝑐) is the locally 𝐿0-convex topology induced by 𝒫𝑐𝑐. Second, we prove that the random Markov-Kakutani fixed point theorem implies the algebraic form of the known random Hahn-Banach theorem. Finally, we establish a more general strict separation theorem in a random locally convex module, which provides not only a more general geometric form of the random HahnBanach theorem but also another proof for the random Markov-Kakutani fixed point theorem. Therefore, as a byproduct, the work of this paper also shows that the algebraic and geometric forms of the random Hahn-Banach theorem are equivalent. It should be pointed out that the main challenge in this paper lies in overcoming noncompactness since a stably compact set is generally noncompact.
合写作者：Xiaohuan Mu
第一作者：Qiang Tu
论文类型：期刊论文
通讯作者：Tiexin Guo
学科门类：理学
一级学科：数学
文献类型：J
卷号：30
页面范围：1196-1219
ISSN号：1076-9803
是否译文：否
发表时间：2024-08-20
收录刊物：SCI