The smoothness of solutions to nonlinear weakly singular integral equations
type: Item
instance of: scholarly article
title: The smoothness of solutions to nonlinear weakly singular integral equations
zbMATH Open document ID: 0803.45007
DOI: 10.4171/ZAA/501
author: V. Pereyra
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1994-09-01T00:00:00Z
review text: The authors consider the smoothness of nonlinear weakly singular Urysohn integral equations \(u(x) = \int_ GK(x,y,u(y))dy + f(x)\) \((x \in G)\). They establish \(u \in C^{m,\nu} (G)\) if \(u \in L^ \infty(G)\) and \(f \in C^{m,\nu}(G)\) and the Urysohn kernel \(K(x,y,u)\) satisfy the (sufficiently simple general) so-called weakly singularity conditions, where \(C^{m,\nu}(G)\) is the Banach space of all \(m\) times continuously differentiable functions \(u\) on an open bounded set \(G\) with the weight \[ w_ \lambda (x) = \begin{cases} 1 & \text{if } \lambda < 0 \\ \biggl(1 + \bigl | \log \rho (x)\bigr | \biggr)^{-1} \quad & \text{if } \lambda = 0\;(x \in G, \lambda \in \mathbb{R})\\ \rho(x)^ \lambda & \text{if } \lambda > 0 \end{cases} \] and \(\rho (x) = \text{dist} (x,\partial G)\). This theorem was formulated and partly (for \(m=2)\) proved by \textit{G. Vainikko} [Multidimensional weakly singular integral equations, Lect. Notes Math. 1549 (1993; Zbl 0789.65097)]. In this article the authors give a full proof by extending some new idea in this book. This theorem is more general even in the case of nonlinear Hammerstein equations than the results of \textit{H. Kaneko}, \textit{R. Noren} and \textit{Y. Xu} [Integral Equations Oper. Theory 13, No. 5, 660-670 (1990; Zbl 0706.45005)]. It is remarkable that all techniques for getting the smoothness of solutions in the linear case excepting the idea of G. Vainikko in his book do not work for the nonlinear case. The idea of G. Vanikko is based on the ``smallness'' of \(\int_ \Omega K(x,y,u(y))dy\), where \(\Omega \subset G\) is a small subregion; the integral over \(G \backslash \Omega\) is treated as a part of the inhomogeneity.
reviewed by: Hong Thai Nguyen
Mathematics Subject Classification ID: 45G05
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45L05
zbMATH DE Number: 627647
zbMATH Keywords: nonlinear weakly singular Urysohn integral equations
zbMATH Keywords: nonlinear Hammerstein equations
zbMATH Keywords: smoothness of solutions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/zaa/501
OpenAlex ID: W2565008213
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description: scientific article; zbMATH DE number 627647