The existence of a fixed point for the sum of two monotone operators
type: Item
instance of: scholarly article
title: The existence of a fixed point for the sum of two monotone operators
zbMATH Open document ID: 1188.47042
author: Yong-Zhuo Chen
published in: Positivity
publication date: 2009-03-27T00:00:00Z
review text: The author gives sufficient conditions for the existence of an attractive fixed point of the sum \(T=A+H\) of two monotone operators in a cone in a Banach space. \(A\) is assumed to be \(\alpha\)-concave (\(A(tx)\geq t^\alpha Ax\) for all \(t\in (\eta,1)\), for some \(\alpha,\eta\in (0,1)\)), but \(H\) is only subhomogeneous (\(H(tx)\geq tHx\) for all \(t\in (\eta,1)\)). It is known that a subhomogeneous operator can have no fixed points. \textit{R.\,D.\thinspace Nussbaum} in [``Hilbert's projective metric and iterated nonlinear maps'' (Mem.\ Am.\ Math.\ Soc.\ 391) (1988; Zbl 0666.47028)] presented a sufficient condition for the existence of a solution of the problem in terms of eigenvectors. In the present paper, the following condition is assumed: there is an \(x_0\in P\) such that \(Ax_0\in C_{x_0}\) and \(Hx_0\in C_{x_0}\), and \(\lambda_m(t)=\sup_{x\in C_{x_0}}\lambda(t,x)<1\), where \(C_{x_0}\) is a component in \(P\) containing \(x_0\), and \(\lambda(t,x):(\eta,1)\times C_{x_0}\to (\alpha,1)\) is a function satisfying \[ T(tx)\geq t^{\lambda(t,x)}Tx \text{ for all } t\in (\eta,1). \] In the proof of the main Theorem 3.1, the author checks that the map \(\lambda(t,x)\) is well defined for every \((t,x)\). Under the above assumptions, the existence of a fixed point \(x_*=Tx_*\) is proved with \(T^nx_0\to x_*\). A more verifiable condition than the above one is given in Theorem 3.4. Two applications to difference equations and integral equations, respectively, are presented as illustrations in the last section.
reviewed by: Grzegorz Gabor
Mathematics Subject Classification ID: 47H07
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5534411
zbMATH Keywords: cone
zbMATH Keywords: convex metric space
zbMATH Keywords: fixed point
zbMATH Keywords: monotone operator
zbMATH Keywords: ordered Banach space
zbMATH Keywords: Thompson's metric
MaRDI profile type: Publication
full work available at URL: s11117-008-2154-6
OpenAlex ID: W2017935843
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DOI: 10.1007/S11117-008-2154-6
description: scientific article; zbMATH DE number 5534411