Analytic solutions of a class of linear and nonlinear functional equations (Q5946939)

The functional equation in the unit disk of the complex plane NEWLINE\[NEWLINEz^{-r} \varphi (z) - \sum_{i=1}^m G_i(z) \varphi (s_i z)= G(z,\varphi (z))+p(z)NEWLINE\]NEWLINE is studied. Here \(r\) is a nonnegative integer, \(s_i\) are complex numbers with \(|s_i|<1\); \(G_i(z)\), \(p(z)\) and \(G(z,w)\) are suitably defined functions. Applying a functional analysis method which is based on the Fredholm alternative theorem the author studies the existence and uniqueness of the solution of equations. The proposed method can be efficient in studying functional equations with singularities in various classes of functions. NEWLINENEWLINENEWLINEA reference to the book of \textit{V. V. Mityushev} and \textit{S. V. Rogosin} [Constructive methods for linear and nonlinear boundary value problems for analytic functions: theory and applications. (2000; Zbl 0957.30002)] should be made, where a similar nonlinear equation is studied.