Quasi-equilibrium Problems Research Articles

Several types of well-posedness for generalized vector quasi-equilibrium problems with their relations

The conceptions of (generalized) Tykhonov well-posedness for generalized vector quasi-equilibrium problems, (generalized) Hadamard well-posedness for parametrically generalized vector quasi-equilibrium problems and (generalized) Tykhonov well-posedness for parametrical system of generalized vector quasi-equilibrium problems are introduced. The metric characterizations and/or sufficient criteria of the proposed well-posedness are presented, and the relations between (generalized) Tykhonov well-posedness for generalized vector quasi-equilibrium problems and that for constrained minimizing problems are discussed. Finally, the relations among several types of the well-posedness are exhibited in detail. MSC: Primary 49K40; 90C31; secondary 90C33

On existence and essential stability of solutions of symmetric variational relation problems

In this paper, we introduce symmetric variational relation problems and establish the existence theorem of solutions of symmetric variational relation problems. As the special cases, symmetric (vector) quasi-equilibrium problems and symmetric variational inclusion problems are obtained. Further, we study the notion of essential stability of equilibria of symmetric variational relation problems. We prove that most of symmetric variational relation problems (in the sense of Baire category) are essential and, for any symmetric variational relation problem, there exists at least one essential component of its solution set.MSC:49J53, 49J40.

Solvability of system of generalized vector quasi-equilibrium problems

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems, and establish some new existence results of a solution for the system of vector quasi-equilibrium problems.

Existence and Well-Posedness for Symmetric Vector Quasi-Equilibrium Problems

An existence result for the solution set of symmetric vector quasi-equilibrium problems that allows for discontinuities is obtained. Moreover, sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems are established.

Existence theorems of an extension for generalized strong vector quasi-equilibrium problems

In this paper, we study generalized strong vector quasi-equilibrium problems in topological vector spaces. Using the generalization of Fan-Browder fixed point theorem, we provide existence theorems for an extension of generalized strong vector quasi-equilibrium problems with and without monotonicity. The results in this paper generalize, extend and unify some well-known existence theorems in literature.

Efficiency in vector quasi-equilibrium problems and applications

In this paper, we give sufficient conditions for the existence of efficient solutions of a generalized vector quasi-equilibrium problem in topological vector spaces. The motivations for introducing this problem come from practical problems in traffic networks and the optimal control theory for discrete-time dynamical systems. The main results of the paper are proven with the help of a strongly monotonic function which can be constructed from the data of the problem under consideration. Some notions of cone-semicontinuity of set-valued maps, weaker than the usual concepts of semicontinuity, are also used in our study. As applications, we obtain existence results in vector quasi-optimization problems, Stampacchia set-valued vector quasi-variational inequality problems and Pareto vector quasi-saddle point problems. All these results are different from the corresponding ones in the literature.

Generalized vector quasi-equilibrium problems on Hadamard manifolds

In this paper, a generalized vector quasi-equilibrium problem (GVQEP) is introduced and studied on Hadamard manifolds. An existence theorem of solutions for the GVQEP is established under some suitable conditions. Some applications to a generalized vector quasi-variational inequality, a generalized vector variational-like inequality and a vector optimization problem are also presented on Hadamard manifolds.

On Hölder Calmness and Hölder Well-Posedness of Vector Quasi-Equilibrium Problems

We study stability and well-posedness of parametric vector quasi-equilibrium problems. Weak continuity of orders not greater than one around a given point, in the sense of Holder calmness of such orders, of solution maps is under consideration. Namely, we consider stability in terms of Holder calmness of solution maps at the considered point of parameter. Sufficient conditions for such Holder calmness are established for weak and strong vector quasi-equilibrium problems. When applied to the particular case of scalar equilibrium problems, our results recover recent ones appearing online first in the literature. Then we propose a Holder well-posedness notion for parametric vector quasi-equilibrium problems, based on Holder calmness of approximate solution maps, and derive sufficient conditions for Holder well-posedness of both the mentioned weak and strong vector quasi-equilibrium problems.

Erratum to “Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications”

Erratum to “Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications”

A New Maximal Theorem in Product GFC-Spaces with Application to Systems of Generalized Mixed Vector Quasiequilibrium Problems

In this paper, a new maximal element theorem is established in product GFC-spaces. As application, a new existence theorem of solutions for systems of generalized mixed vector quasi-equilibrium problems is obtained.

Topological aspects of circular metric spaces and some observations on the KKM property towards quasi-equilibrium problems

The main purpose of this paper is to study some topological nature of circular metric spaces and deduce some fixed point theorems for maps satisfying the KKM property. We also investigate the solvability of a variant of a quasi-equilibrium problem as an application.

Vector quasi-equilibrium problems: separation, saddle points and error bounds for the solution set

In this paper, we employ the image space analysis (for short, ISA) to investigate vector quasi-equilibrium problems (for short, VQEPs) with a variable ordering relation, the constrained condition of which also consists of a variable ordering relation. The quasi relatively weak VQEP (for short, qr-weak VQEP) are defined by introducing the notion of the quasi relative interior. Linear separation for VQEP (res., qr-weak VQEP) is characterized by utilizing the quasi interior of a regularization of the image and the saddle points of generalized Lagrangian functions. Lagrangian type optimality conditions for VQEP (res., qr-weak VQEP) are then presented. Gap functions for VQEP (res., qr-weak VQEP) are also provided and moreover, it is shown that an error bound holds for the solution set of VQEP (res., qr-weak VQEP) with respect to the gap function under strong monotonicity.

Existence Results in a General Equilibrium Problem

In this article, we give sufficient conditions for the existence of solutions of a general model which includes as special cases many generalized vector quasi-equilibrium problems with set-valued maps. The obtained results generalize and improve several earlier results.

Existence conditions for symmetric generalized quasi-variational inclusion problems

In this paper, we establish an existence theorem by using the Kakutani-Fan-Glicksberg fixed-point theorem for a symmetric generalized quasi-variational inclusion problem in real locally convex Hausdorff topological vector spaces. Moreover, the closedness of the solution set for this problem is obtained. As special cases, we also derive the existence results for symmetric weak and strong quasi-equilibrium problems. The results presented in the paper improve and extend the main results in the literature.

Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications

The aim of this paper is to study generalized vector quasi-equilibrium problems (GVQEPs) by scalarization method in locally convex topological vector spaces. A general nonlinear scalarization function for set-valued mappings is introduced, its main properties are established, and some results on the existence of solutions of the GVQEPs are shown by utilizing the introduced scalarization function. Finally, a vector variational inclusion problem is discussed as an application of the results of GVQEPs.

The Existence and Stability of Solutions for Vector Quasiequilibrium Problems in Topological Order Spaces

In a topological sup-semilattice, we established a new existence result for vector quasiequilibrium problems. By the analysis of essential stabilities of maximal elements in a topological sup-semilattice, we prove that for solutions of each vector quasi-equilibrium problem, there exists a connected minimal essential set which can resist the perturbation of the vector quasi-equilibrium problem.

New existence theorems for quasi-equilibrium problems and a minimax theorem on complete metric spaces

Motivated by the Suzuki's type fixed point theorems, we give several new existence theorems for scalar quasi-equilibrium problems, and vector quasi-equilibrium problem on complete metric spaces. We give important examples for our results. Note that the solution of quasi-equilibrium problem (resp. vector quasi-equilibrium problem) is unique under suitable conditions, and we can find the unique solution by the Picard iteration. Besides, we also give a new coincidence theorem on complete metric spaces. Finally, we give a new minimax theorem on complete metric spaces. Note that the solution of minimax theorem is unique under suitable conditions, and we can find the unique solution by the Picard iteration.

Levitin–Polyak well-posedness by perturbations for systems of set-valued vector quasi-equilibrium problems

This paper is devoted to the Levitin–Polyak well-posedness by perturbations for a class of general systems of set-valued vector quasi-equilibrium problems (SSVQEP) in Hausdorff topological vector spaces. Existence of solution for the system of set-valued vector quasi-equilibrium problem with respect to a parameter (PSSVQEP) and its dual problem are established. Some sufficient and necessary conditions for the Levitin–Polyak well-posedness by perturbations are derived by the method of continuous selection. We also explore the relationships among these Levitin–Polyak well-posedness by perturbations, the existence and uniqueness of solution to (SSVQEP). By virtue of the nonlinear scalarization technique, a parametric gap function g for (PSSVQEP) is introduced, which is distinct from that of Peng (J Glob Optim 52:779–795, 2012). The continuity of the parametric gap function g is proved. Finally, the relations between these Levitin–Polyak well-posedness by perturbations of (SSVQEP) and that of a corresponding minimization problem with functional constraints are also established under quite mild assumptions.

SOME ITERATIVE METHODS FOR QUASI-EQUILIBRIUM PROBLEMS

In this paper we present two new extensions of Korpelevich's extragraclient method for solving the scalar Quasi-Equilibrium Problem (QEP) in Euclidean space. In both extensions we replace the second orthogonal projection in the correction step of the extragradient method by an Armijo linesearch adapted to (QEP). We show that in both extensions we obtain, under mild conditions and without a Lipschitz condition, the global convergence of our iterative methods for computing a quasi-equilibrium solution.

A NEW SCALARIZING APPROACH AND THE SEMICONTINUITIES OF THE SOLUTION MAPPING OF A PARAMETRIC VECTOR QUASI-EQUILIBRIUM PROBLEM

This paper gives sufficient conditions for the upper and lower semicontinuities of the solution mapping of a parametric vector quasi-equilibrium problem with moving cones. We use a new scalarizing approach quite different from traditional linear scalarization approaches. The main tool of this paper is a new notion of nonlinear scalarization function which is a generalised version of the Gerstewitz function applied to set-valued maps.