Existence and uniqueness of solutions in \(H_{1}(\Delta)\) of a general class of non-linear functional equations
type: Item
instance of: scholarly article
title: Existence and uniqueness of solutions in \(H_{1}(\Delta)\) of a general class of non-linear functional equations
zbMATH Open document ID: 1021.39011
DOI: 10.1016/S0022-247X(03)00019-2
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: The authors study the functional equation \[ f(z)+ \sum^m_{i=1} \alpha_i (z) f(p_iz)= g(z)+ \sum^k_{j=1} \sum^\infty_{s=3} c_s^j(z)\bigl[ f(q_jz) \bigr]^{s-1}, \;z\in\mathbb{C}, \tag{1} \] where \(m,k\) are positive integers, \(p_i,q_j\) are known complex constants and \(g(z)\), \(\alpha_i(z)\), \(c_s^j(z)\) are known functions belonging to the Banach space \(H_1(\Delta)= \{f(z)= \sum^\infty_{u=1} f_nz^{n-1}\), analytic \(\Delta\) in and \(\sum^\infty_{n=1} |f_n|<\infty\}\), where \(\Delta= \{z\in\mathbb{C}, |z|<1\}\). Using a fixed point theorem concerning holomorphic functions in abstract Banach spaces, the existence and uniqueness of a solution in \(H_1(\Delta)\) for the equation (1) is proved.
reviewed by: Ion Onciulescu
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 1915762
zbMATH Keywords: functional equations for complex functions
zbMATH Keywords: analytic solutions
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: bounded solutions
zbMATH Keywords: Banach space
zbMATH Keywords: fixed point theorem
zbMATH Keywords: holomorphic functions
author: Eugenia N. Petropoulou
author: Panayiotis D. Siafarikas
MaRDI profile type: Publication
cites work: Q5604772
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full work available at URL: s0022-247x(03)00019-2
OpenAlex ID: W1500349604
description: scientific article; zbMATH DE number 1915762