Abel equation

The Abel equation, named after Niels Henrik Abel, is special case of functional equations which can be written in the form

:$f(h(x))\; =\; f(x\; +\; 1),!$

or

: $alpha(f(x))=alpha(x)+1!$

and shows non-trivial properties at the iteration.

Equivalence

In some sense, these equaitons are equivalent. Assuming that $~alpha~$ is invertible function,the second equation can be written as:$alpha^\{-1\}(alpha(f(x)))=alpha^\{-1\}(alpha(x)+1)!$Assuming that $~x=alpha^\{-1\}(t)$, the equation can be written as::$f(alpha^\{-1\}(t))=alpha^\{-1\}(t+1)!$Function $~f~$ can be assumed to be known, then it may be suggested to find function $~alpha^\{-1\}~$ that satisfies the equation and perhaps, some additional requiremnts; for example, $~alpha^\{-1\}(0)=1~$.

History

Initially, the equation in the more general formcite journal
url=
author= 0.Abel
coauthors=
title=Correlative of the functional equation
journal= Crelle's Journal
volume=2
pages=389
year=1827] cite journal
url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.bams/1183421988&view=body&content-type=pdf_1
author= A.R.Schweitzer
coauthors=
title=Theorems on functional equations
journal=Bulletin des Sciences Mathématiques
volume=27
issue=2
pages=31
year=1903] was reported. Then it happens that even in the case of single variable, the equaiton is not trivial, and requires special analysiscite journal
url=http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf
author= G.Belitskii
coauthors=Yu.Lubish
title=The real-analytic solutions of the Abel functional equations
journal=Studia Mathematica
volume=134
issue=2
pages=135–141
year=1999] cite journal
journal= Nonlinear Analysis: Hybrid Systems
volume= 1,
issue=1
year=2007
pages=95–102
doi=10.1016/j.nahs.2006.04.002
author=Jitka Laitochová
title =Group iteration for Abel’s functional equation
abstract=Studied is the Abel functional equation α(f(x))=α(x)+1]

In the case of linear transfer function, the solution can be expressed in compact formcite journal
author=G.Belitskii
coauthor=Yu.Lubish
title=THe Abel equation and total solvability of linear functional equaitons
journal=Studia Mathematica
volume=127
year=1998
pages=81–89]

Tetration

Equation of tetration is special case of equaiton of Abel, with $~f=exp~$.

Integer parameter

In the case of integer argument, the equation is just a recurrent procedure.

ee also

*functional equation

References

also

Crelle's Journal, volume 1 (1826), pages 11-15