Analytic solutions of an iterative functional equation
The authors give a construction of an analytic solution \(f\) of the functional equation \[ f(p(z)+bf(z))=h(z), \] where \(b\) is a nonzero complex number and \(p\), \(h\) are given complex functions of a complex variable. Moreover, it is assumed that \(p\), \(h\) are analytic in a neighbourhood of the origin, \(p(0)=h(0)=0\), \(p'(0)=r\), \(h'(0)=s\neq 0\) and a root \(\beta\) of the equation \(z^2- rz -bs=0\) satisfies some additional conditions. Under these assumptions it is proved that for every comlex number \(\eta \neq 0\) the auxiliary functional equation \[ g(\beta^{2} z) - p(g(\beta z)) = b h(g(z)) \] has a solution which is analytic in a neighbourhood of the origin and \(g(0)=0\), \(g'(0)=\eta\). Then the function given by the formula \[ f(z)=\frac{1}{b} (g(\beta g^{-1} (z)) - p(z)) \] is a solution of the first equation which is analytic in a neighbourhood of the origin.
scientific article; zbMATH DE number 2113892