An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems
type: Item
instance of: scholarly article
title: An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems
zbMATH Open document ID: 1368.46070
publication date: 2017-04-21T00:00:00Z
review text: Let \(X\) be a topological space, \(Y\) a Hausdorff topological vector space ordered by a closed cone \(K\) and \(f:X\to 2^Y\setminus\{\emptyset\}.\) The author considers various semicontinuity and boundedness properties of \(f\), expressed in terms of the topologies of the spaces \(X,Y\) and of the order induced by the cone \(K\), and proves several Ekeland-type variational principles for these kinds of maps. He also considers set-valued bimaps \(f:X\times X\to 2^Y\setminus\{\emptyset\}\) satisfying a so-called \(K\)-sequential lower monotony (\(K\)-slm) condition: if \(x_n\to \bar x\) in \(X\) is such that \(f(x_n,x_{n+1})\subset -K\) for all \(n\in\mathbb{N}\), then \(f(x_n,\bar x)\subset -K\) for all \(n\in\mathbb{N}\). An equilibrium version of the Ekeland variational principle, extending one proved by \textit{X.-H. Gong} [Acta Math. Sci., Ser. B, Engl. Ed. 34, No. 4, 1179--1192 (2014; Zbl 1324.49014)], is proved and applied to equilibrium problems, i.e., the existence of \(\bar x\in X\) s.t. \(f(\bar x,y)\nsubseteq -\mathrm{int}(K)\) for all \(y\in X\). Stronger results are obtained in the case when \(X\) is a Fréchet space (i.e., a complete metrizable locally convex space). The obtained results extend those obtained for vector mappings \(f:X\to Y\) and, at the same time, improve some set-valued variational and equilibrium results.
Mathematics Subject Classification ID: 46N10
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 46A03
Mathematics Subject Classification ID: 91B50
zbMATH DE Number: 6707181
zbMATH Keywords: Ekeland variational principle
zbMATH Keywords: set-valued vector equilibrium problem
zbMATH Keywords: ordered locally convex space
zbMATH Keywords: semi-continuous set-valued map
zbMATH Keywords: lower boundedness
author: Jinghui Qiu
reviewed by: Stefan Cobzaş
MaRDI profile type: Publication
full work available at URL: s10114-016-6184-x
OpenAlex ID: W2555736318
cites work: Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory
cites work: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory
cites work: The Ekeland variational principle for equilibrium problems revisited and applications
cites work: On generalized vector equilibrium problems.
cites work: An existence result for the generalized vector equilibrium problem
cites work: Existence of equilibria via Ekeland's principle
cites work: Q4366516
cites work: Q5325780
cites work: Relative Pareto minimizers for multiobjective problems: Existence and optimality conditions
cites work: Vector variational principle
cites work: Ekeland's principle for vector equilibrium problems
cites work: Vector equilibrium problems with generalized monotone bifunctions
cites work: Vector optimization. Set-valued and variational analysis.
cites work: A unified approach to the existing three types of variational principles for vector valued functions
cites work: Ekeland's \(\varepsilon\)-variational principle for set-valued mappings
cites work: NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY
cites work: On variational principles, level sets, well-posedness, and \(\epsilon\)-solutions in vector optimization
cites work: On some nonlinear problems induced by an abstract maximal element principle
cites work: On the variational principle
cites work: Nonconvex minimization problems
cites work: Vector-valued variational principles
cites work: A Brézis-Browder principle on partially ordered spaces and related ordering theorems
cites work: A generalized variational principle and its application to equilibrium problems
cites work: Variational methods in partially ordered spaces
cites work: Ekeland's principle for set-valued vector equilibrium problems
cites work: On the vectorial Ekeland's variational principle and minimal points in product spaces
cites work: A Set-Valued Ekeland's Variational Principle in Vector Optimization
cites work: Some variants of the Ekeland variational principle for a set-valued map
cites work: Equivalents to Ekeland's variational principle in uniform spaces
cites work: Sequentially lower complete spaces and Ekeland's variational principle
cites work: Q5518756
cites work: On generalized Ekeland's variational principle and equivalent formulations for set-valued mappings
cites work: Versions of Ekeland's variational principle involving set perturbations
cites work: On Equilibrium Problems
cites work: Q5723419
cites work: Q5569793
cites work: Existence of solutions for generalized vector equilibrium problems
cites work: Ekeland's Variational Principle for Set-Valued Functions
cites work: Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces
cites work: A nonconvex vector minimization problem
cites work: Equivalents of Ekeland's principle
cites work: A generalized Ekeland vector variational principle and its applications in optimization
cites work: On Ha's version of set-valued Ekeland's variational principle
cites work: Set-valued quasi-metrics and a general Ekeland's variational principle in vector optimization
cites work: A pre-order principle and set-valued Ekeland variational principle
cites work: An equilibrium version of vectorial Ekeland variational principle and its applications to equilibrium problems
cites work: A vectorial Ekeland's variational principle with a \(w\)-distance and its equivalent theorems
cites work: A generalization of ekellandz’s variational principle
cites work: Vector variational principles for set-valued functions
cites work: Q4178057
cites work: An Ekeland's variational principle for set-valued mappings with applications
cites work: Caristi type coincidence point theorem in topological spaces
cites work: Generalized variational principle and vector optimization
DOI: 10.1007/S10114-016-6184-X
Recommended article: Q5346662
Recommended article: An Ekeland's variational principle for set-valued mappings with applications
Recommended article: Some variants of the Ekeland variational principle for a set-valued map
Recommended article: A generalized Ekeland's variational principle for vector equilibria
Recommended article: On Ha's version of set-valued Ekeland's variational principle

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems (Recommended article)
Recommended article: New stability results for Ekeland's \(\varepsilon\) variational principles for vector-valued and set-valued maps
Similarity Score: 0.74879086
Recommender Run: Recommender Run 2

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems (Recommended article)
Recommended article: Ekeland variational principles for set-valued functions with set perturbations
Similarity Score: 0.74752206
Recommender Run: Recommender Run 2

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems (Recommended article)
Recommended article: The system of generalized set-valued equilibrium problems
Similarity Score: 0.7456223
Recommender Run: Recommender Run 2

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems (Recommended article)
Recommended article: On the \(\epsilon\)-variational principle for set-valued mappings
Similarity Score: 0.74322337
Recommender Run: Recommender Run 2

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems (Recommended article)
Recommended article: Ekeland Variational Principles in Vector Equilibrium Problems
Similarity Score: 0.74207854
Recommender Run: Recommender Run 2

An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems (published in)
published in: Acta Mathematica Sinica, English Series