On topological groups with an approximate fixed point property
  A topological group $G$ has the Approximate Fixed Point (AFP) property on a bounded convex subset $C$ of a locally convex space if every continuous affine action of $G$ on $C$ admits a net $(x_i)$, $x_i\in C$, such that $x_{i}-gx_{i}\longrightarrow 0$ for all $g\in G$. We study the relationship of this property with amenability. 
Cleon S. Barroso, Brice R. Mbombo and Vladimir G. Pestov
math.GR