A unified monotone iterative technique for semilinear elliptic boundary value problems
type: Item
instance of: scholarly article
title: A unified monotone iterative technique for semilinear elliptic boundary value problems
zbMATH Open document ID: 1140.35419
DOI: 10.1016/S0362-546X(01)00809-4
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-09-30T00:00:00Z
review text: The authors consider the elliptic problem \[ -\sum_{{i,j=1}}^n a_{i,j}(x)u_{x_{i}x_{j}} + \sum_{i=1}^{n} b_{i}(x) + c(x)u = F(x,u) \text{ in } \Omega \tag{a} \] subject to general boundary conditions in a bounded smooth domain \(\Omega\). The right-hand side is assumed to be splitted in the form \[ F(x,u) = f(x,u) + g(x,u), \] where for each \(x \in \Omega\) the function \(f(x,\cdot)\) is nonincreasing and \(g(x,\cdot)\) is nondecreasing. Starting from suitable upper and lower initial iterates of (a) it is shown with the aid of the maximum principle that the usual iterates converge monotonely to an upper and lower solution of (a), respectively. A similar result is obtained for the variational formulation of (a). Results of this type for operators that allow a monotone splitting are known since the late sixtieth. Surprisingly, the authors do not include any reference to the numerous existing publications in this field.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1810008
zbMATH Keywords: semilinear elliptic problem
zbMATH Keywords: monotone iteration
author: Semen Köksal
author: V. Lakshmikantham
reviewed by: Rolf Dieter Grigorieff
MaRDI profile type: Publication
cites work: Existence and uniqueness of positive solutions for some quasilinear elliptic problems
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cites work: Generalized quasilinearization and semilinear elliptic boundary value problems
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description: scientific article; zbMATH DE number 1810008