On a class of nonlinear singular integral equations with shift on a closed contour
type: Item
instance of: scholarly article
title: On a class of nonlinear singular integral equations with shift on a closed contour
zbMATH Open document ID: 1063.45002
author: D. Kharzeev
published in: Applied Mathematics and Computation
publication date: 2005-01-17T00:00:00Z
review text: Assuming that \(L\) is a sufficiently well behaved closed contour in the complex plane, the authors investigate the nonlinear Cauchy singular integral equation \[ \phi(t) = \Omega \sum_{j=0}^{m-1} a_j(t) \int_L { G(\alpha_j(t), \tau, \phi(\tau)) \over \tau - \alpha_j(t)} d \tau\text{ for }t \in L. \] Here \(\alpha_0(t) = t\) and \(\alpha_j(t) = \alpha(\alpha_{j-1}(t))\) where \(\alpha\) is a homeomorphism from \(L\) to itself such that the Carleman condition \(\alpha_m(t) = t\) and \(\alpha_j(t) \neq t\) for \(1 \leq j \leq m-1\) holds. The mapping properties of the integral operator are discussed, and a Banach-type fixed point argument yields a sequence of approximate solutions that converges to the unique solution of the problem in a suitable space of functions.
reviewed by: Kai Diethelm
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45L05
Mathematics Subject Classification ID: 45E05
zbMATH DE Number: 2128818
zbMATH Keywords: nonlinear Cauchy singular integral equation
zbMATH Keywords: Carleman shift
zbMATH Keywords: approximate solutions
zbMATH Keywords: Banach-Cacciopoli theorem
MaRDI profile type: Publication
full work available at URL: j.amc.2003.08.138
OpenAlex ID: W1994816302
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DOI: 10.1016/J.AMC.2003.08.138
description: scientific article; zbMATH DE number 2128818