Vector Quasi-equilibrium Problems Research Articles

Generalized Vector Quasi-Equilibrium Problems

The aim of this paper is to present new existence results for solutions to a generalized quasi-equilibrium problem with set-valued mappings and moving cones. The key to this approach is a new Browder-type fixed point theorem, which permits working in a new direction with the milder condition of transfer open-valued mapping and considering weaker assumptions on the coving cone. These results are applied to some generalized vector quasi-equilibrium problems with trifunctions and to a vector quasi-equilibrium problem with fuzzy mappings in a fuzzy environment.

Existence of solutions of a generalized bifunction-set optimization problem

<p style='text-indent:20px;'>An existence result is established for a generalized bifunction-set optimization problem, under some verifiable assumptions. Its proof is based on a Browder fixed point theorem and the KKMFan-Lemma. As applications of the main result of this paper, we obtain new existence results for some generalized versions of Kuroiwa set optimization problems and strong vector quasiequilibrium problems. Examples are provided. One of them is a non-cooperative generalized game with utility multifunctions.</p>

Existence Results for Generalized Vector Quasi-Equilibrium Problems in Hadamard Manifolds

The purpose of this article was to establish manifold versions of existence theorems for generalized vector quasi-equilibrium problems in locally compact and σ-compact spaces without any continuity assumption. The fixed-point theorem in a product Hadamard manifold is the key focus of our discussion. We further applied our theorems to saddle point and minimax problems.

Extragradient method with Bregman distances for solving vector quasi-equilibrium problems

Extragradient method with Bregman distances for solving vector quasi-equilibrium problems

Technical note on the existence of solutions for generalized symmetric set-valued quasi-equilibrium problems utilizing improvement set

In this paper, we establish some existence results for the solution of the generalized symmetric set-valued quasi-equilibrium problem (GSSQEP). The new forms of GSSQEP via improvement set and scalar generalized symmetric set-valued quasi-equilibrium problem (GSSQEP for short) are also introduced. By using Kakutani–Fan–Glicksberg fixed point method, maximal element principle and nonlinear scalarization technique, we develop two classes of sufficient conditions for the existence of solutions to GSSQEP. The drawback of some existing work for this problem is overcome. Moreover, some applications to the symmetric set-valued equilibrium problem (SSEP), symmetric vector quasi-equilibrium problem (SVQEP) and the set-valued equilibrium problem (SEP) are also given in this paper. Our results improve a few existing ones in Fakhar and Zafarani [Generalized symmetric vector quasiequilibrium problems. J Optim Theory Appl. 2008;136:397–409], Farajzadeh et al. [On the existence of solutions of symmetric vector equilibrium problems via nonlinear scalarization. Bull Iran Math Soc. 2019;45:35–58], Fu [Symmetric vector quasi-equilibrium problems. J Math Anal Appl. 2003;285:708–713], Han et al. [Existence of solutions for symmetric vector set-valued quasi-equilibrium problems with applications. Pacific J Optim. 2018;14:31–49].

Solving Equilibrium Problems with Bifunctions Having the Property of Negative Transitivity

In this article, we introduce the notion of bifunction with the property of negative transitivity and, using the Berge-Klee intersection theorem, we establish existence theorems of the solutions for the equilibrium problems, when the involved bifunction has this property. Then, using standard scalarization techniques, we obtain existence criteria of the solutions for vector equilibrium problems. The last section of the article is devoted to the scalar and vector quasi-equilibrium problems.

Solutions of system of generalized vector quasi-equilibrium problems in locally FC-uniform spaces

In this paper, we establish a collectively fixed point theorem and an equilibrium existence theorem for generalized games in product locally FC-uniform spaces. As applications, some new existence theorems of solutions for the system of generalized vector quasi-equilibrium problems are derived in product locally FC-uniform spaces. These theorems are new and generalize some known results in the literature.

Path connectedness of the efficient solution set for generalized vector quasi-equilibrium problems

Path connectedness of the efficient solution set for generalized vector quasi-equilibrium problems

Existence results for quasi-equilibrium problems under a weaker equilibrium condition

By a quasi-equilibrium problem is understood an equilibrium problem with a constraint set depending on the current point. In this paper, we establish existence results for scalar and vector quasi-equilibrium problems, when the standard equilibrium condition is replaced by a weaker one.

Existence conditions for symmetric strong vector quasi-equilibrium problems

In the following paper, the symmetric strong vector quasi-equilibrium problems will be studied thoroughly. Afterward, the existence conditions of solution sets for these problems has been established. The results which are presented in this paper improve and extend the main results mentioned in the literature. Our results can be illustrated by some interesting examples. In 1994, Noor and Oettli introduced the following the symmetric scalar quasi-equilibrium problem. This problem is one of the generalization of the symmetric scalar quasi-equilibrium problem which is presented by Noor and Oettli. Since then, the symmetric vector quasi-equilibrium problem has been investigated by a huge number of authors in different ways. The research works mentioned above are one of our motivation to improve and extend the problem. So, in this paper, we will introduce the vector quasi-equilibrium problems. Afterward, some existence conditions of solution sets for these problems will be established. The symmetric vector quasi-equilibrium problems consist of many optimization - related models namely symmetric vector quasi-variational inequality problems, fixed point problems, coincidence-point problems and complementarity problems, etc. In recent years, a lot of results for existence of solutions for symmetric vector quasi-equilibrium problems, vector quasi-equilibrium problems, vector quasivariational inequality problems and optimization problems have been established by many authors in different ways. We will present our work in the following steps. In the first section of our paper, we will introduce the model of symmetric vector quasi-equilibrium problems. In the following section, we recall definitions, lemmas which can be used for the main results. In the last section, we will establish some conditions for existence and closedness of the solutions set by applying fixed-point theorem for symmetric vector quasi-equilibrium problems. The results presented in this paper improve and extend the main results in the literature. Some examples are given to illustrate our results. Hence our results, Theorem 3.1 and Theorem 3.6 have significant improvements.

Set-Valued Symmetric Generalized Strong Vector Quasi-Equilibrium Problems with Variable Ordering Structures

In this paper, two types of set-valued symmetric generalized strong vector quasi-equilibrium problems with variable ordering structures are discussed. By using the concept of cosmically upper continuity rather than the one of upper semicontinuity for cone-valued mapping, some existence theorems of solutions are established under suitable assumptions of cone-continuity and cone-convexity for the equilibrium mappings. Moreover, the results of compactness for solution sets are proven. As applications, some existence results of strong saddle points are obtained. The main results obtained in this paper unify and improve some recent works in the literature.

On the stability of solutions for the generalized vector quasi-equilibrium problems via free-disposal set

<p style='text-indent:20px;'>In this paper, we mainly discuss the stability of generalized vector quasi-equilibrium problems (GVQEPs) where the ordering relations are defined by free-disposal set. Firstly, by virtue of the oriented distance function <inline-formula><tex-math id="M1">\begin{document}$ (\triangle) $\end{document}</tex-math></inline-formula>, gap functions for (GVQEPs) are given and some properties of them are studied. Then, under some types of continuity assumption, the sufficient conditions of the upper semicontinuity and the upper Painlevé-Kuratowski convergence of solutions for (GVQEPs) are talked about. Moreover, sufficient and necessary conditions of the lower semicontinuity and the lower Painlevé-Kuratowski convergence of solutions for (GVQEPs) are obtained in normed linear spaces. Some examples are given to illustrate the results, and our results are new and extend some known results in the literature.

Existence results for generalized vector quasi-equilibrium problems

Existence results for generalized vector quasi-equilibrium problems

The generic stability of solutions for vector quasi-equilibrium problems on Hadamard manifolds

The generic stability of solutions for vector quasi-equilibrium problems on Hadamard manifolds

Error bounds for generalized mixed weak vector quasiequilibrium problems via regularized gap functions

This paper introduces regularized gap functions for a class of generalized mixed weak vector quasiequilibrium problems. Then, error boundsfor the concerning problems via regularized gap functions are established. Someexamples are provided to illustrate the results.

On the existence of solutions for vector quasiequilibrium problems

In this paper, we establish some existence theorems for vector quasiequilibrium problems in real locally convex Hausdorff topological vector spaces by using Kakutani-Fan-Glicksberg fixed-point theorem. Moreover, we also discuss the closedness of the solution sets for these problems. The results presented in the paper are new and improve some main results in the literature.

Unified vector quasiequilibrium problems via improvement sets and nonlinear scalarization with stability analysis

This paper has two objectives. The first one is to propose a new vector quasiequilibrium problem where the ordering relation is defined via an improvement set $ D $, and its weak version, also their Minty-type dual problems and the corresponding set-valued cases. These models provide unified frameworks to deal with well-known exact and approximate vector quasiequilibrium problems with vector-valued or set-valued mappings. The second one is to study solution stability in the sense of Hölder continuity of the unique solution to parametric unified (resp. weak) vector quasiequilibrium problems, by employing the Gerstewitz scalarization techniques. In particular, we deduce a new stability result for the typical vector optimization problem related with (resp. weak) $ D $-optimality, by considering perturbations of both the objective function and the feasible set.

Generalized strong vector quasi-equilibrium problems with variable ordering structure

In this paper, two kinds of generalized strong vector quasi-equilibrium problems with variable ordering structure are considered by using the concept of cosmically upper continuity rather than upper semi-continuity for cone-valued mapping. Firstly, a key local property of cosmically upper continuity for cone-valued mapping is discussed. Next, under suitable conditions of cone-continuity and cone-convexity for equilibrium mapping, several existence theorems of solutions and closedness of solution sets are established for these two kinds of generalized strong vector quasi-equilibrium problems with variable ordering structure. Moreover, an example is given to illustrate the validity of our theorems. These results obtained in this paper extend and develop some recent works in this field.

Vectorial form of Ekeland variational principle with applications to vector equilibrium problems

ABSTRACTWe first derive an Ekeland type variational principle for vector-valued bifunctions without assuming triangular inequality for bifunctions. Then, by using such variational principle, we derive some existence results for solutions of vector equilibrium problems and vector quasi-equilibrium problems.

Well-Posedness and Structural Stability on a System of Simultaneous Generalized Vector Quasi-Equilibrium Problems

In this paper, we establish the stable results for a system of simultaneous generalized vector quasi-equilibrium problems (SSGVQEP) by using its bounded rationality model. Under the abstract frame, a unified well-posedness on Hadamard types and Tikhonov types well-posedness for SSGVQEP is introduced. Moreover, sufficient condition for the well-posedness of SSGVQEP is given. Finally, we prove that the majority (in Baire category sense) of SSGVQEP is structural stability.