Contraction map of geometric mean

[rubinj]

I have the following mapping (generalized geometric mean):

$$y(i)=exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N$$

where p(j|i) is a normalized conditional probability.

my question is - is this a contraction mapping?

in other words, does the following equation have a unique solution:

$$x(i)=exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N$$

thanks in advance,
rubin

 

[Eynstone]

Would you specify what x(i) stands for & what p(i|j) is the probability of?
( Lokks familiar, though. if it's something like a Markov chain or transition of probability vectors, use the Brower's fixed point theorem to get a solution. The mapping may not be contractive.)

 

[rubinj]

the p(i|j) does specify the transition in a Markov chain, and the x(i) stands for some 'value' function over the states. i know that a solution exists, but i'd like to show it's also unique. I think that for this aim, i must introduce some discount parameter $$0<\gamma<1$$:

$$x(i)=\gamma\exp\left[{\sum_j p(j|i)\log x(j)}\right]\\ ,\ i,j=1..N$$

can that be shown to have a unique solution ?