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

f(h(x)) = h(x + 1)

or

\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

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

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

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 sα(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(sα(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]

\omega( \omega(x,u),v)=\omega(x,u+v) ~,

e.g., for \omega(x,1) = f(x),

\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.,

\alpha(f(f(x)))=\alpha(x)+2 ~,

and so on,

\alpha(f_n(x))=\alpha(x)+n ~.

Solutions

The Abel equation has at least one solution on E if and only if for all x \in E and all n \in \mathbb{N}^*, f^{n}(x) \neq x, where 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]

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

Existence

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

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

Uniqueness

The solution is essentially unique in the sense that there exists a canonical solution \alpha_0 with the following properties:

  • The set of critical points of \alpha_0 is bounded above if h(y) > y for all y, or bounded below if h(y) < y for all y.
  • This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by
\{\alpha_0 + \beta\circ \alpha_0 | \beta : \R \to \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

References

  1. ^ Aczél, János, (1966): Lectures on Functional Equations and Their Applications, Academic Press, reprinted by Dover Publications, ISBN 0486445232 .
  2. ^ Abel, N.H. (1826). "Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, ...". Journal für die reine und angewandte Mathematik. 1: 11–15.
  3. ^ A. R. Schweitzer (1912). "Theorems on functional equations". Bull. Amer. Math. Soc.. 19 (2): 51–106. doi:10.1090/S0002-9904-1912-02281-4
  4. ^ Korkine, A (1882). "Sur un problème d'interpolation", Bull Sci Math & Astron 6(1) 228—242. online
  5. ^ G. Belitskii & Yu. Lubish (1999). "The real-analytic solutions of the Abel functional equations". Studia Mathematica. 134 (2): 135–141.
  6. ^ Jitka Laitochová (2007). "Group iteration for Abel’s functional equation". Nonlinear Analysis: Hybrid Systems. 1 (1): 95–102. doi:10.1016/j.nahs.2006.04.002
  7. ^ G. Belitskii & Yu. Lubish (1998). "The Abel equation and total solvability of linear functional equations". Studia Mathematica. 127: 81–89.
  8. ^ R. Tambs Lyche, Sur l'équation fonctionnelle d'Abel, University of Trondlyim, Norvege
  9. ^ Bonet, José & Domański, Paweł (April 2015). "Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions". Integral Equations and Operator Theory. 81 (4): 455–482. doi:10.1007/s00020-014-2175-4. hdl:10251/71248. ISSN 0378-620X
  10. ^ Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis
  11. ^ Classifications of parabolic germs and fractal properties of orbits by Maja Resman, University of Zagreb, Croatia
  • M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw (1968).
  • M. Kuczma, Iterative Functional Equations. Vol. 1017. Cambridge University Press, 1990.