View source for Schröder functional equation

The equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300702.png" /> is the unknown function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300703.png" /> is a known real-valued function of a real variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300704.png" />. I.e. one asks for the eigenvalues and eigenfunctions of the composition operator (substitution operator) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300705.png" />. Sometimes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300706.png" /> is allowed to be a function itself.

One also considers the non-autonomous Schröder functional equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300707.png" /></td> </tr></table>

The Schröder and Abel functional equations (see also [[Functional equation|Functional equation]]) have much to do with the translation functional equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300708.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s1300709.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s13007010.png" /> is a [[Semi-group|semi-group]], which asks for something like a right action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s13007011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130070/s13007012.png" />, [[#References|[a1]]], [[#References|[a4]]].

The equation was formulated by E. Schröder, [[#References|[a5]]], and there is an extensive body of literature.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Aczél,   "A short course on functional equations" , Reidel  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Kuczma,   "On the Schröder operator" , PWN  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Kuczma,   "Functional equations in a single variable" , PWN  (1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Targonski,   "Topics in iteration theory" , Vandenhoeck and Ruprecht  (1981)  pp. 82ff.</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E. Schröder,   "Uber iterierte Funktionen III"  ''Math. Ann.'' , '''3'''  (1970)  pp. 296–322</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Walorski,   "Convex solutions of the Schröder equation in Banach spaces"  ''Proc. Amer. Math. Soc.'' , '''125'''  (1997)  pp. 153–158</TD></TR></table>