Fixed points for extrema of contractions
type: Item
instance of: scholarly article
title: Fixed points for extrema of contractions
zbMATH Open document ID: 0716.47031
DOI: 10.1016/0022-247X(90)90348-J
author: Ilya Gertsbakh
author: Baruch Cahlon
author: Irwin E. Schochetman
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: It is assumed that X is a complete metric space which is partially ordered by \(\leq\). The required compatibility between the metric and the partial order assumes that (X,\(\leq)\) is a countably Dedekind complete lattice with \(\leq\) closed in X \((R=\{(x,y)\in X\times X:\) \(x\leq y\}\) is closed in \(X\times X)\), and d is assumed to be order preserving (x\(\leq y\) and \(y\leq z\) implies d(x,y),d(y,z)\(\leq d(x,z))\). The author consider families of order preserving contraction mappings. It is shown that if an extremum (infimum or supremum) of the contractions exists, then the fixed point of the extremum contraction is less than or equal to the extremum of fixed points of the individual contractions, and sufficient conditions are given for equality to hold. It is also shown that given suitable starting points in the space, the fixed point of the extremum can be monotonically approximated from above and below, thus yielding an algorithm for approximating this fixed point with a prescribed tolerance. The authors conclude by showing how these results may be applied to Markov decision models and families of integral equations.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H07
Mathematics Subject Classification ID: 46B42
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 4180161
zbMATH Keywords: Banach lattices
zbMATH Keywords: complete metric space
zbMATH Keywords: countably Dedekind complete lattice
zbMATH Keywords: families of order preserving contraction mappings
zbMATH Keywords: fixed point
zbMATH Keywords: Markov decision models
zbMATH Keywords: families of integral equations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4180161