On an Altman type fixed point theorem on convex cones
type: Item
instance of: scholarly article
title: On an Altman type fixed point theorem on convex cones
zbMATH Open document ID: 0868.47035
author: G. Isac
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-08-17T00:00:00Z
review text: The paper recalls the existing connection of the fixed point theory with the so-called complementarity theory. Given a continuous map \(f:K\to H\) of a closed convex cone \(K\) in a Hilbert space \(H\), the complementarity problem concerns the existence of a point \(u\in K\) such that \(f(u)\) is contained in the dual cone \(K^*\) and \(\langle u,f(u)\rangle=0\). It appears that the solvability of the complementarity problem is equivalent to the existence of zeros of \(f\) (i.e. fixed points of \(T=I-f\)). Thus, results concerning the solvability of the complementarity problem (related to the well-known Hartmann-Stampacchia theorem) yield fixed points. For instance, the author shows that given a contraction \(S\) and a compact map \(T\), the map \(f=S+T: K\to H\) has a fixed point provided \(f(K)\subset K\) and \(K,I-f\) satisfy some auxiliary natural conditions. Some other, even more general results are proved and discussed.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 47J20
zbMATH DE Number: 796896
zbMATH Keywords: convex cone
zbMATH Keywords: contraction
zbMATH Keywords: Galerkin approximation
zbMATH Keywords: fixed point theory
zbMATH Keywords: complementarity theory
zbMATH Keywords: Hartmann-Stampacchia theorem
reviewed by: Wojciech Kryszewski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072244
OpenAlex ID: W1991691297
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DOI: 10.1216/RMJM/1181072244
description: scientific article; zbMATH DE number 796896