Extend amenability–fixed point characterizations to non-affine semihypergroup actions

Determine whether left amenability of the space AP(K) of almost periodic functions on a semihypergroup K can be characterized by the existence of common fixed points for separately continuous non-affine actions of K on compact convex subsets of separated locally convex spaces, analogous to the affine and non-expansive cases established in this work.

Background

The paper proves fixed-point characterizations of left amenability of AP(K) for two broad classes of actions: (i) separately continuous, equicontinuous affine actions on compact convex sets (Theorem 3.3), and (ii) separately continuous, non-expansive actions on compact convex sets (Theorem 3.4).

The author explicitly raises whether similar characterizations can be extended to non-affine actions, indicating that this extension remains unresolved.