I'm looking for the self-consistent (e.g. input needs to be the same as output) solution of $r$ in
$$ r = \frac{1}{2}\operatorname{erfc}(z(r)) $$
where $\operatorname{erfc}$ is complimentary error function (in a particular example from the neural networks, I am trying to analyse one dimension problem of $r$ and $z(r) = \frac{ \mu - (1 - g)Jr}{J^2r(1-r)(1+g^2)}$ with the conditions that $g>1$ and $ J ,\mu > 0$). I would be interested to determined the conditions that above equation has at least several real solutions, at least for the case of my particular $z(r)$. In this case it can be shown that there is always a solution $1\geq r\geq0$.