Picard iterates for positive solutions of a class of nonlinear eigenvalue problems
type: Item
instance of: scholarly article
title: Picard iterates for positive solutions of a class of nonlinear eigenvalue problems
zbMATH Open document ID: 0541.35034
DOI: 10.1007/BF00948811
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1983-01-01T00:00:00Z
review text: Positive solutions of the nonlinear eigenvalue problem \(F(u)=\lambda G(u)\) are studied with some nonlinear monotone operators F and G. In particular, the author considers the case of nonlinear elliptic differential equations of second order (nonlinear diffusion equations) \(F(u)=-div A(x,\text{grad} u)+b(x,u)=\lambda g(x,u)=\lambda G(u)\) with Dirichlet boundary conditions. Here, A(x,.) denotes a monotone vector field with respect to the second variable, while most authors restrict their considerations to the linear case with a positive definite Matrix A(x). Picard iterates are defined by \(u_ 0=0\) and \(F(u_{n+1})=\lambda G(u_ n).\) The author shows, that \(F(u)=\lambda G(u)\) has a positive solution iff \(\lambda\) belongs to some interval \((0,\lambda^*)\) or \((0,\lambda^*]\) with \(0<\lambda^*\leq +\infty\). In this case the Picard iterates converge from below to the minimal positive solution. The convergence is obtained by some comparison principles and the maximal monotonicity of the operators F and G, which replaces the usual Green's function arguments in the case of linear F. The endpoint \(\lambda^*\) is greater than the first eigenvalue of the problem linearized at the origin.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 47J10
zbMATH DE Number: 3859602
zbMATH Keywords: Positive solutions
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: monotone operators
zbMATH Keywords: nonlinear diffusion
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: Picard iterates
zbMATH Keywords: convergence
zbMATH Keywords: comparison principles
zbMATH Keywords: Green's function
zbMATH Keywords: first eigenvalue
author: Juergen Weyer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00948811
OpenAlex ID: W4238553688
description: scientific article; zbMATH DE number 3859602