Establish random fixed-point and equilibrium theorems in random locally convex modules

Establish the random KKM mapping principle, the general dynamic (conditional) Nash equilibrium theorem, and the random Tychonoff fixed-point theorem in random locally convex modules using the framework of stable compactness.

Background

The paper develops stable topology and stable compactness as a framework intended to extend classical compactness-based results to random functional analysis. It proves random analogues of the Banach–Alaoglu, James, and Banach–Bourbaki–Kakutani–Šmulian theorems, but the random KKM principle, dynamic Nash equilibrium theorem, and random Tychonoff fixed-point theorem are not established in the present paper.

The authors indicate that stable compactness should provide the framework for resolving these problems in random locally convex modules, while noting that their proofs involve additional challenges and are deferred to future work.

References

By the way, we would also like to point out that, based on the theory of stable compactness, we will enable to establish the random KKM mapping principle, general dynamic (or, conditional) Nash equilibrium theorem and random Tychonoff fixed point theorem in random locally convex modules, namely, the open problems posed in can be completely solved, which will be given in our forthcoming work since many other challenges will be involved.

— Stable topology and stable compactness  (2610.22185 - Guo et al., 15 Oct 2026) in Section 1.4, paragraph beginning “As an application of the theory of stably compact spaces”; also discussed in Section 1.3