Abel equation

This article is about certain functional equations. For ordinary differential equations which are cubic in the unknown function, see Abel equation of the first kind.

The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form

${\displaystyle f(h(x))=h(x+1)}$

or

${\displaystyle \alpha (f(x))=\alpha (x)+1}$.

The forms are equivalent when α is invertible. h or α control the iteration of f.

Equivalence

The second equation can be written

${\displaystyle \alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)\,.}$

Taking x = α⁻¹(y), the equation can be written

${\displaystyle f(\alpha ^{-1}(y))=\alpha ^{-1}(y+1)\,.}$

For a known function f(x) , a problem is to solve the functional equation for the function α⁻¹ ≡ h, possibly satisfying additional requirements, such as α⁻¹(0) = 1.

The change of variables s^α(x) = Ψ(x), for a real parameter s, brings Abel's equation into the celebrated Schröder's equation, Ψ(f(x)) = s Ψ(x) .

The further change F(x) = exp(s^α(x)) into Böttcher's equation, F(f(x)) = F(x)^s.

The Abel equation is a special case of (and easily generalizes to) the translation equation,^[1]

${\displaystyle \omega (\omega (x,u),v)=\omega (x,u+v)~,}$

e.g., for ${\displaystyle \omega (x,1)=f(x)}$,

${\displaystyle \omega (x,u)=\alpha ^{-1}(\alpha (x)+u)}$.     (Observe ω(x,0) = x.)

The Abel function α(x) further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).

See also: Iterated function § Abelian property and Iteration sequences

History

Initially, the equation in the more general form ^{[2] [3]} was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.^[4][5][6]

In the case of a linear transfer function, the solution is expressible compactly.^[7]

Special cases

The equation of tetration is a special case of Abel's equation, with f = exp.

In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,

${\displaystyle \alpha (f(f(x)))=\alpha (x)+2~,}$

and so on,

${\displaystyle \alpha (f_{n}(x))=\alpha (x)+n~.}$

Solutions

The Abel equation has at least one solution on ${\displaystyle E}$ if and only if for all ${\displaystyle x\in E}$ and all ${\displaystyle n\in \mathbb {N} }$, ${\displaystyle f^{n}(x)\neq x}$, where ${\displaystyle f^{n}=f\circ f\circ ...\circ f}$, is the function f iterated n times.^[8]

We have the following existence and uniqueness theorem^{[9]: Theorem B}

Let ${\displaystyle h:\mathbb {R} \to \mathbb {R} }$ be analytic, meaning it has a Taylor expansion. To find: real analytic solutions ${\displaystyle \alpha$ :\mathbb {R} \to \mathbb {C} } of the Abel equation ${\textstyle \alpha \circ h=\alpha +1}$.

Existence

A real analytic solution ${\displaystyle \alpha }$ exists if and only if both of the following conditions hold:

• ${\displaystyle h}$ has no fixed points, meaning there is no ${\displaystyle y\in \mathbb {R} }$ such that ${\displaystyle h(y)=y}$.
• The set of critical points of ${\displaystyle h}$, where ${\displaystyle h'(y)=0}$, is bounded above if ${\displaystyle h(y)>y}$ for all ${\displaystyle y}$, or bounded below if ${\displaystyle h(y)<y}$ for all ${\displaystyle y}$.

Uniqueness

The solution is essentially unique in the sense that there exists a canonical solution ${\displaystyle \alpha _{0}}$ with the following properties:

• The set of critical points of ${\displaystyle \alpha _{0}}$ is bounded above if ${\displaystyle h(y)>y}$ for all ${\displaystyle y}$, or bounded below if ${\displaystyle h(y)<y}$ for all ${\displaystyle y}$.
• This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by

$${\displaystyle \{\alpha _{0}+\beta \circ \alpha _{0}|\beta$$ :\mathbb {R} \to \mathbb {R} {\text{ is analytic, with period 1}}\}.}

Approximate solution

Analytic solutions (Fatou coordinates) can be approximated by asymptotic expansion of a function defined by power series in the sectors around a parabolic fixed point.^[10] The analytic solution is unique up to a constant.^[11]

See also

• Functional equation
  • Schröder's equation
  • Böttcher's equation
• Infinite compositions of analytic functions
• Iterated function
• Shift operator
• Superfunction