SOLUTIONS TO NONLINEAR RECURRENCE EQUATIONS

Abstract

Let $F(z)$ be any function. Suppose that $w$ is a fixed point of $F(z)$, that is, $F(w)=w$. Then the recurrence equation

$$x_{n+1}=F(x_n)$$

for $n=0,1,2,\dots$ has a solution of the form

$$x_n(w)=w+\sum _{i=1}^{\infty }{a}_1^i{A}_i{F}_{•1}{(w)}^{in},$$

where $F_{•1}(z)=dF(z)∕dz$. So, for each $w$ there is a set of complex $x_0$ such that $x_0(w)=x_0$. We assume that $F(z)$ is analytic at $w$. This solution appears to be new, even for such famous examples like the logistic map and the Mandelbrot equation.