A method for determining the roots of some classes of equations with analytic functions and its applications
type: Item
instance of: scholarly article
title: A method for determining the roots of some classes of equations with analytic functions and its applications
zbMATH Open document ID: 0973.30008
DOI: 10.1007/BF02674728
author: N. E. Tovmasyan
author: L. Z. Gevorkyan
published in: Siberian Mathematical Journal
publication date: 2000-01-31T00:00:00Z
review text: Let \(D\) be a simply connected plane domain with piecewise smooth boundary. Given a function \(\varphi(z)\) analytic in \(D\) and continuous in \(\overline{D}\), the problem is to find a fixed point of this function, i.e. for solving the equation \(z-\varphi(z)=0\). It is assumed that \(z-\varphi(z)\neq 0\) for \(z\in \partial D\) and that \(\varphi\) takes \(\overline{D}\) into \(\overline{D}\). It is shown that there exists a unique fixed point \(z_0\) which is representable as \[ z_0=\frac{1}{2\pi i}\int_{\partial D} z \frac{\psi'(z)-1}{\psi(z)-1} dz, \] where \(\psi(z)=\varphi\big(\overline{\varphi(\bar{z})}\big)\). This result is applied to studying the Bitsadze-Samarskij problem for the Laplace equation which can be stated as follows: Find a solution to the equation \(\Delta u=0\) in the domain \(S=\{z:|z|<1\}\) satisfying the boundary condition \[ u(z)=au(\alpha(z)) + b u(\alpha(\overline{z})) + f(z),\quad |z|=1, \] where \(a\) and \(b\) are constants, \(f\) is a Hölder function defined on \(\partial S\), and \(\alpha(z)\) is an analytic function in \(S\) continuous in \(\overline{S}\).
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1396779
zbMATH Keywords: analytic function
zbMATH Keywords: fixed point of a function
zbMATH Keywords: Bitsadze-Samarskij problem for the Laplace equation
MaRDI profile type: Publication
cites work: Q4308813
full work available at URL: https://doi.org/10.1007/bf02674728
OpenAlex ID: W2044168585
description: scientific article; zbMATH DE number 1396779