I have a fixed point iteration problem in my research, as below:
$$x_{k+1}=f(x_k)$$ where $$f(x)=\frac{\lambda r}{g}\sum_{i,j}\pi_if_{ij}\left[1- \exp\left\{\displaystyle xt_{ij}\left(1-\frac{g}{r}\right)\right\}\right]$$ and $\lambda, r,g>0,\ \{t_{ij}\}>0$, $F=\{f_{ij}\}$ is the transition matrix of a finite irreducible discrete time Markov chain and $\pi=\{\pi_i\}$ is its stationary probability distribution. Also, $$\tau:=\sum_{ij}\pi_if_{ij}t_{ij}$$ and it is known that $$\lambda\tau\frac{c}{g}<1$$ where $c=r+g$.
Note that $x=0$ is one solution to the problem above.
My problem is that I need to find the solution to $f(x)=x$ other than $x=0$. Is it possible that this fixed point iteration can converge to something other than $0$ by maybe by some choice of $x_0$. Maybe this is a simple problem but right now I have many things going on and I am unable to think of some method to find the other solution. Please help.