The random Markov-Kakutani fixed point theorem in a random locally convex module

Release time:2024-08-20

Journal:New York J. Math.

Funded by:国家自然科学基金

Key Words:Random locally convex modules, stable compactness, random Markov-Kakutani fixed point theorem, random Hahn-Banach theorem.

Abstract: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.

Co-author:Xiaohuan Mu

First Author:Qiang Tu

Indexed by:Journal paper

Correspondence Author:Tiexin Guo

Discipline:Natural Science

First-Level Discipline:Mathematics

Document Type:J

Volume:30

Page Number:1196-1219

ISSN No.:1076-9803

Translation or Not:no

Date of Publication:2024-08-20

Included Journals:SCI

Links to published journals:https://nyjm.albany.edu/j/2024/30-53.html