 

Qiang Tu,
Xiaohuan Mu, and
Tiexin Guo
The random MarkovKakutani fixed point theorem in a random locally convex module
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Published: 
August 20, 2024. 
Keywords: 
Random locally convex modules, stable compactness, random MarkovKakutani fixed point theorem, random HahnBanach theorem. 
Subject [2010]: 
46H25, 41A50, 47H10, 46A22. 


Abstract
Based on the recently developed theory of σstable sets and stable
compactness, we first establish the random MarkovKakutani 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 L^{0}convex subset of E, then
every commutative family of T_{c}(P_{cc})continuous
L^{0}affine mappings from G to G has a common fixed point, where
P_{cc} is the σstable hull of P and
T_{c}(P_{cc}) is the locally L^{0}convex topology induced
by P_{cc}. Second, we prove that the random MarkovKakutani fixed point
theorem implies the algebraic form of the known random HahnBanach 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 MarkovKakutani fixed point
theorem. Therefore, as a byproduct, the work of this paper also shows that the
algebraic and geometric forms of the random HahnBanach 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.


Acknowledgements
This work was supported by the National Natural Science Foundation of
China (Grant No.12371141) and the Natural Science Foundation of Hunan Province of
China (Grant No.2023JJ30642).


Author information
Qiang Tu
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
qiangtu126@126.com
Xiaohuan Mu
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
xiaohuanmu@163.com
Tiexin Guo
School of Mathematics and Statistics
Central South University
Changsha, Hunan 410083, China
tiexinguo@csu.edu.cn

