Abel equation

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

or

.

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

Equivalence

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The second equation can be written

 

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

 

For a known function f(x) , a problem is to solve the functional equation for the function α−1h, 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]

 

e.g., for  ,

 .     (Observe ω(x,0) = x.)

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

History

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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

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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.,

 

and so on,

 

Solutions

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The Abel equation has at least one solution on   if and only if for all   and all  ,  , where  , is the function f iterated n times.[8]

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

Let   be analytic, meaning it has a Taylor expansion. To find: real analytic solutions   of the Abel equation  .

Existence

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

  •   has no fixed points, meaning there is no   such that  .
  • The set of critical points of  , where  , is bounded above if   for all  , or bounded below if   for all  .

Uniqueness

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

  • The set of critical points of   is bounded above if   for all  , or bounded below if   for all  .
  • This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by

 

Approximate solution

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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

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References

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  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" (PDF). 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" (PDF). 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. 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.