On solvability of a boundary value problem for singular integral equations
type: Item
instance of: scholarly article
title: On solvability of a boundary value problem for singular integral equations
zbMATH Open document ID: 1034.45003
author: Nguyen Van Mau
author: Nguyen Tan Hoa
published in: Southeast Asian Bulletin of Mathematics
publication date: 2003-11-16T00:00:00Z
review text: Let \(\Gamma\) be a simple regular closed arc on the complex plane and let \(X= H^\mu(\Gamma)\), \(0<\mu< 1\). Consider the following boundary value problem: \[ [(K^n+ K^{n-1} K_c)\varphi](t)= f(t), \] \((F_j K^j\varphi)(t)= \varphi_j(t)\), \(j= 0,1,\dots, n-1\), \(\varphi_j(t)\in\ker K\) if \(\kappa> 0\), \((G_0 f)(t)=0\), \((G_j R_{j-1}\cdots R_0)f(t)= 0\), \(j= 1,\dots, n-1\), if \(\kappa< 0\), \(K_c\) is the operator of multiplication by a function \(c(t)\), \(c,f\in X\), \[ (K\varphi)(t)= a(t)\varphi(t)+ {b(t)\over\pi i} \int_\Gamma {\varphi(\tau)\over \tau-t}\,d\tau,\quad \kappa= \text{ind\,}K, \] \(F_j\), \(G_j\) are initial and \(\omega\)-initial operators for \(K\) induced by right (left) inverses \(R_j\) of \(K\), respectively. The problem under question is reduced to a system of algebraic linear equations and then all its solutions are given in a closed form.
Mathematics Subject Classification ID: 45E05
Mathematics Subject Classification ID: 30E25
zbMATH DE Number: 2004107
zbMATH Keywords: singular integral equation
zbMATH Keywords: Cauchy kernel
zbMATH Keywords: index
zbMATH Keywords: boundary value problem
zbMATH Keywords: right invertible operator
zbMATH Keywords: initial operator
zbMATH Keywords: co-initial operator
reviewed by: Danuta Przeworska-Rolewicz
MaRDI profile type: Publication
full work available at URL: s100120200045
OpenAlex ID: W2014749519
DOI: 10.1007/S100120200045
description: scientific article; zbMATH DE number 2004107