On weakly nonlinear contractions
type: Item
instance of: scholarly article
title: On weakly nonlinear contractions
zbMATH Open document ID: 0551.54026
author: Kun-Jen Chung
published in: Kodai Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: The author generalizes some known fixed point theorems to cone-valued metric spaces. The main theorem reads as follows: Theorem. Let \((X,d)\) be a complete, convex, \(K\)-metric space (where \(K\) denotes a strongly normed cone in a reflexive Banach space \(E\)) and \(S\) a non-empty closed subset of \(X\) such that for each \(x\in S\), there exists an element \(U\) of \(S\) with \(d(x,u)+d(u,Tx)=d(x,Tx)\). Suppose that \(T: S\to X\) satisfies (i) \(d(Tx,Ty)\leq \phi(d(x,y)),\) \(x\neq y\in S\), (ii) \(\phi (t)<t\) for any \(t\in cl\{d(x,y):x,y\in S\}=P_ 1(S),\) where \(\phi:P_ 1(S)\to K\) is upper semicontinuous, (iii) \(Tx\in S\) for every \(x\) of the boundary of \(S\). Then \(T\) has a unique fixed point in \(S\). Some important corollaries of this theorem are shown to be themain results of various authors.
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 3877954
zbMATH Keywords: cone in a normed space
zbMATH Keywords: reflexive Banach space
zbMATH Keywords: strongly normal cone
zbMATH Keywords: upper semi-continuous functions
zbMATH Keywords: metrically convex space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138036797
OpenAlex ID: W1966742514
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DOI: 10.2996/KMJ/1138036797
description: scientific article; zbMATH DE number 3877954