Feigenbaum universal transition of a dynamical system (like the logistic map) to chaotic behaviour through period-doubling cascade involves the functional equation
$$g(x)=\-\frac1\lambda g(g(\lambda x)$$
with the boundary conditions
$$g(0)=1,\qquad g'(0)=0,\qquad g''(0)<0.$$
The parameter $\lambda$ for which a solution exists near $x=0$ is the inverse of the Feigenbaum constant.