Abel equation

<h2 id="equivalence">Equivalence</h2>
The second equation can be written

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

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

f
        (
        
          α
          
            −
            1
          
        
        (
        y
        )
        )
        =
        
          α
          
            −
            1
          
        
        (
        y
        +
        1
        )
        
        .
      
    
    {\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 sα(x) = Ψ(x), for a <a href="/facts/Real_number/R02gw5Pb">real</a> parameter s, brings Abel's equation into the celebrated <a href="/facts/Schr%C3%B6der%27s_equation/jWWTbrra">Schröder's equation</a>, Ψ(f(x)) = s Ψ(x) .
The further change F(x) = exp(sα(x)) into <a href="/facts/B%C3%B6ttcher%27s_equation/XDwY9GPJ">Böttcher's equation</a>, F(f(x)) = F(x)s.
The Abel equation is a special case of (and easily generalizes to) the translation equation,<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

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

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

ω
 (
 x
 ,
 u
 )
 =
 
 α
 
 −
 1
 
 
 (
 α
 (
 x
 )
 +
 u
 )
 
 
 {\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 <a href="/facts/Shift_operator/6c6lc0ks">Lie advective flows</a> (one parameter <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>).

See also: <a href="/facts/Iterated_function/Vzsx64An">Iterated function § Abelian property and Iteration sequences</a>
<h2 id="history">History</h2>
Initially, the equation in the more general form
<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
In the case of a linear transfer function, the solution is expressible compactly.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="special-cases">Special cases</h2>
The equation of <a href="/facts/Tetration/5rKeVVYv">tetration</a> 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.,

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

and so on,

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

<h2 id="solutions">Solutions</h2>
The Abel equation has at least one solution on 
 
 
 
 E
 
 
 {\displaystyle E}
 
 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> for all 
 
 
 
 x
 ∈
 E
 
 
 {\displaystyle x\in E}
 
 and all 
 
 
 
 n
 ∈
 
 N
 
 
 
 {\displaystyle n\in \mathbb {N} }
 
, 
 
 
 
 
 f
 
 n
 
 
 (
 x
 )
 ≠
 x
 
 
 {\displaystyle f^{n}(x)\neq x}
 
, where 
 
 
 
 
 f
 
 n
 
 
 =
 f
 ∘
 f
 ∘
 .
 .
 .
 ∘
 f
 
 
 {\displaystyle f^{n}=f\circ f\circ ...\circ f}
 
, is the function f <a href="/facts/Iterated_function/Vzsx64An">iterated</a> n times.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>
We have the following existence and uniqueness theorem<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>: Theorem B 
Let 
 
 
 
 h
 :
 
 R
 
 →
 
 R
 
 
 
 {\displaystyle h:\mathbb {R} \to \mathbb {R} }
 
 be <a href="/facts/Analytic_function/JhL3z8FP">analytic</a>, meaning it has a Taylor expansion. To find: real analytic solutions 
 
 
 
 α
 :
 
 R
 
 →
 
 C
 
 
 
 {\displaystyle \alpha :\mathbb {R} \to \mathbb {C} }
 
 of the Abel equation 
 
 
 
 α
 ∘
 h
 =
 α
 +
 1
 
 
 {\textstyle \alpha \circ h=\alpha +1}
 
.

<h3>Existence</h3>
A real analytic solution 
 
 
 
 α
 
 
 {\displaystyle \alpha }
 
 exists if and only if both of the following conditions hold:

<ul><li>
 
 
 
 h
 
 
 {\displaystyle h}
 
 has no fixed points, meaning there is no 
 
 
 
 y
 ∈
 
 R
 
 
 
 {\displaystyle y\in \mathbb {R} }
 
 such that 
 
 
 
 h
 (
 y
 )
 =
 y
 
 
 {\displaystyle h(y)=y}
 
.</li>
<li>The set of critical points of 
 
 
 
 h
 
 
 {\displaystyle h}
 
, where 
 
 
 
 
 h
 ′
 
 (
 y
 )
 =
 0
 
 
 {\displaystyle h'(y)=0}
 
, is bounded above if 
 
 
 
 h
 (
 y
 )
 >
 y
 
 
 {\displaystyle h(y)>y}
 
 for all 
 
 
 
 y
 
 
 {\displaystyle y}
 
, or bounded below if 
 
 
 
 h
 (
 y
 )
 <
 y
 
 
 {\displaystyle h(y)<y}
 
 for all 
 
 
 
 y
 
 
 {\displaystyle y}
 
.</li></ul>
<h3>Uniqueness</h3>
The solution is essentially unique in the sense that there exists a canonical solution 
 
 
 
 
 α
 
 0
 
 
 
 
 {\displaystyle \alpha _{0}}
 
 with the following properties:

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

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

<h3>Approximate solution</h3>
Analytic solutions (Fatou coordinates) can be approximated by <a href="/facts/Asymptotic_expansion/4W79BNve">asymptotic expansion</a> of a function defined by <a href="/facts/Power_series/vYh6sTps">power series</a> in the sectors around a <a href="/facts/Classification_of_Fatou_components/zIe8ExCJ">parabolic fixed point</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> The analytic solution is unique up to a constant.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Functional_equation/4ovU4CAt">Functional equation</a>
<ul><li><a href="/facts/Schr%C3%B6der%27s_equation/jWWTbrra">Schröder's equation</a></li>
<li><a href="/facts/B%C3%B6ttcher%27s_equation/XDwY9GPJ">Böttcher's equation</a></li></ul></li>
<li><a href="/facts/Infinite_compositions_of_analytic_functions/N1JmTkaT">Infinite compositions of analytic functions</a></li>
<li><a href="/facts/Iterated_function/Vzsx64An">Iterated function</a></li>
<li><a href="/facts/Shift_operator/6c6lc0ks">Shift operator</a></li>
<li><a href="/facts/Superfunction/WLMaT1vv">Superfunction</a></li></ul>

<ul><li>M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw (1968).</li>
<li>M. Kuczma, Iterative Functional Equations. Vol. 1017. Cambridge University Press, 1990.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Aczél, János, (1966): Lectures on Functional Equations and Their Applications, Academic Press, reprinted by Dover Publications, ISBN 0486445232 . <a href="/wiki/J%C3%A1nos_Acz%C3%A9l_(mathematician)" target="_blank">/wiki/J%C3%A1nos_Acz%C3%A9l_(mathematician)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0001&DMDID=dmdlog6" target="_blank">http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0001&DMDID=dmdlog6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.bams/1183421988&view=body&content-type=pdf_1" target="_blank">http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.bams/1183421988&view=body&content-type=pdf_1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Korkine, A (1882). "Sur un problème d'interpolation", Bull Sci Math & Astron 6(1) 228—242. 
online <a href="http://archive.numdam.org/ARCHIVE/BSMA/BSMA_1882_2_6_1/BSMA_1882_2_6_1_228_1/BSMA_1882_2_6_1_228_1.pdf" target="_blank">http://archive.numdam.org/ARCHIVE/BSMA/BSMA_1882_2_6_1/BSMA_1882_2_6_1_228_1/BSMA_1882_2_6_1_228_1.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">G. Belitskii; Yu. Lubish (1999). "The real-analytic solutions of the Abel functional equations" (PDF). Studia Mathematica. 134 (2): 135–141. <a href="http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">G. Belitskii; Yu. Lubish (1998). "The Abel equation and total solvability of linear functional equations" (PDF). Studia Mathematica. 127: 81–89. <a href="http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">R. Tambs Lyche, Sur l'équation fonctionnelle d'Abel, University of Trondlyim, Norvege <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm5/fm5132.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/fm/fm5/fm5132.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="http://link.springer.com/10.1007/s00020-014-2175-4" target="_blank">http://link.springer.com/10.1007/s00020-014-2175-4</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis <a href="http://www.math.toronto.edu/graduate/Dudko-thesis.pdf" target="_blank">http://www.math.toronto.edu/graduate/Dudko-thesis.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Classifications of parabolic germs and fractal properties of orbits by Maja Resman, University of Zagreb, Croatia <a href="https://www.birs.ca/workshops/2015/15w5082/files/resman.pdf" target="_blank">https://www.birs.ca/workshops/2015/15w5082/files/resman.pdf</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Abel equation open-in-new

Abel equation