Caristi-like condition and the existence of minima of mappings in partially ordered spaces
type: Item
instance of: scholarly article
title: Caristi-like condition and the existence of minima of mappings in partially ordered spaces
zbMATH Open document ID: 1466.49024
author: Xianqiang Yang
published in: Journal of Optimization Theory and Applications
publication date: 2019-03-06T00:00:00Z
review text: Let \((X,\preceq)\) and \((Y,\preceq)\) be partially ordered sets; \(Min (Y)\) denote the set of all minimal elements of \(Y\). It is said that a map \(U \colon X \to Y\) satisfies the Caristi-like condition if \((K)\) \(\forall x \in X,\) if \(U(x) \notin Min (Y)\), then there exists \(x^\prime \in X\) such that \(x^\prime \prec x\), \(U(x^\prime) \prec U(x)\). The authors prove the following theorem on the existence of a minimal point of a map. Let a map \(U \colon X \to Y\) satisfy conditions \((K)\) and \((S)\) for arbitrary infinite chain \(S \subset X,\) if the restriction of \(U\) to \(S\) is strictly isotone, then \(S\) has a lower bound \(w \in X\) for which \(U(w) \prec U(x)\) for all \(x \in S\), \(x \neq w\). Then \(U(X) \cap \text{Min} (Y) \neq \emptyset\) and, for each \(x_0 \in X\), there exists \(\bar{x} \in X\), \(\bar{x} \preceq x_0\) such that \(U(\bar{x}) \in \text{Min} (Y)\), \(U(\bar{x}) \preceq U(x_0)\), It is shown that the result on a minimum of a lower semicontinuous function on a metric space (see [the first author, Proc. Steklov Inst. Math. 291, 24--37 (2015; Zbl 1336.49005); translation from Tr. Mat. Inst. Steklova 291, 30--44 (2015)], Theorem 3) follows from the above theorem. It is applied also to obtain the existence of a coincidence point for a pair of maps of partially ordered spaces.
reviewed by: Valerii V. Obukhovskij
Mathematics Subject Classification ID: 49K27
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 7032509
zbMATH Keywords: Caristi theorem
zbMATH Keywords: partially ordered space
zbMATH Keywords: minimal point
zbMATH Keywords: monotone map
zbMATH Keywords: fixed point
zbMATH Keywords: coincidence point
MaRDI profile type: Publication
full work available at URL: s10957-018-1413-3
OpenAlex ID: W2896592574
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Wikidata QID: Q129087379
DOI: 10.1007/S10957-018-1413-3
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description: scientific article; zbMATH DE number 7032509