consider the following mappings, G and T,

$y(s) = Gx(s)=\exp\left[\sum_{s'}p(s'|s)\log x(s') \right]$

$z(s) = Ty(s)=\sum_{s'}q(s'|s)y(s')e^{-r(s')}$

where $0< x(s)\leq 1$ ,$r(s)<0$ , $s,s'\in \{1,2,...,N\}$, and $p(s'|s)$ and $q(s'|s)$ are normalized conditional distributions.

(the first mapping is a generalized geometric mean, and the second is an arithmetic mean with some discount)

my question is - does iterating these mappings, i.e., $x_{t+1} = TG(x_t)$, converges to a unique solution?