I would like so find the function $n(z)$ that solves the following equation:
$ n(z) = \frac{.2 + .24 z \int_1^{\infty} \frac{n(z)}{z^2}dz}{.5 + .24 z \int_1^{\infty} \frac{n(z)}{z^2}dz} $
That is, $n(z)$ is defined in terms of it's own integral. Even if there's no nice analytic solution to this, Numerical approximations of some sort are good enough. In fact this is just a parametrized instance of a more general equation I am interested in. What methods should I be using to find such a solution? Does Mathematica or Matlab has nice built in routines for this kind of thing?
For your further information I am interested only in $n(z)$ on $[1,\infty]$, and it is required to be between 0 and 1, so those integrals lie between 0 and 1, so $n(z)$ should be monotonically increasing. It seems like it should be well behaved, but on the face of it Mathematica doesn't like it.
Background: it is the steady state of a dynamic system in a network; think epidemic diffusion. $n(z)$ is the infection rate among agents with z links, and in the steady state it is
$ n(z) = (g + (1-g)t v z )/(g + r + (1-g) t v z ), $ where $g$ is the new infection rate, $r$ is the cure rate,and $v$ is the new infections from your peers in a networks, and $f(z)$ is the distribution of that networks; the proportion of peers withg $z$ links. One such distribution often of interest is power-law, which is
$ f(z) = 2 z^{-3} $ and t is the infection rate among a random peer, given by $ t = \int_1^{\infty} z n(z) f(z)dz, $ which, along with a paramtetrization of the above rates, gave the problem above.
Other distributions of interest are geometric, where $f(z) = Log(4) 2^{-z}$. So we are solving for the steady state level of infection among guys with z peers.