I have not seen any books about solving the equation of the following form: $af^2(x)+bf(x)+cx=0$ where $a$, $b$, $c$ are constants and $f^2(x)=f(f(x))$. We are going to find an expression of the function $f$. If I substitute $f(x)=kx$ in, I can get a solution, but I am interested in how to get ALL the solutions.
Please assume continuity if necessary.
Does any one know how to solve such functional equation?