You have a point on challenging the term "nonlinear" for that functional equation, in that particular solutions of it may be superposed. However, often one considers Ψ to be invertible, yielding the equivalent nonlinear equation provided below, also introduced by Schröder, h(Φ(y))=Φ(cy). In practice, power series solutions of these equations involve both linear and nonlinear functional links among coefficients of Ψ, h , and/or Φ.
So, I suspect, the least fussy outcome of your edit might be to simply drop the term "linear" or "nonlinear" altogether, if you chose to do so. Cuzkatzimhut (talk) 17:05, 21 October 2010 (UTC)
Thank you for this explanation. By changing "nonlinear" to "linear" I tried to follow the notion of linearity implied in Kuczma's books (and especially in Kuczma, Choczewski, Ger 1990). A more general functional equation is discussed there: f(h(x))=g(f(x)), and depending on whether g is linear or not the resulting functional equations (special cases of the above) are considered linear or nonlinear functional equations. The simplest choices for g are: 1) g(x)=s x, which results in: f(h(x))=s f(x), Schroeder's equation (linear) 2) g(x)=x+a, which results in: f(h(x))=f(x)+a, Abels's equation (linear) 3) g(x)=x^p, p>1, which results in: f(h(x))=[f(x)]^p, Boettcher's equation (nonlinear) However, considering your arguments I agree that leaving just "functional equation" (that is dropping "linear" or "nonlinear" altogether) might be the best decision.
Aoryst (talk) 14:42, 26 October 2010 (UTC)