Quasi-equilibrium Problems Research Articles

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

Existence of quasiequilibria in metric vector spaces

This paper provides sufficient conditions for the existence of solutions for quasiequilibrium problems and generalized game problems in the setting of infinite-dimensional metrizable spaces. To this purpose, we prove a modified version of a selection theorem due to Michael [15] by exploiting the fact that any compact set in a metric space is both complete and separable. Thereafter, by a fixed point technique which is based on the notion of inside point of a convex set, we provide some existence results without requiring the upper semicontinuity and the closed-valuedness of the feasibility maps.

Existence Results for Systems of Quasi-Variational Relations

We study the existence of solutions for a system of variational relations, in a general form, using a fixed point result for contractions in metric spaces. As a particular case, we give sufficient conditions for the existence of solutions of a system of quasi-equilibrium problems.

Optimal Control of Problems Governed by Mixed Quasi-Equilibrium Problems Under Monotonicity-Type Conditions with Applications

Optimal Control of Problems Governed by Mixed Quasi-Equilibrium Problems Under Monotonicity-Type Conditions with Applications

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.

Quasi-Intersection Problems and Fixed Point Theorems Concerning Separately Scalar Weakly l.s.c and u.s.c Mappings

In this paper, we first prove a new result on quasi-variational inequality problems; then, we use it to show the existence of solutions to quasi-intersection problems. In particular, we establish several results on the existence of solutions to generalized quasi-equilibrium problems and fixed point problems concerning separately scalar weakly lower and upper semi-continuous mappings. The results in this paper generalize some well-known existence theorems from the literature.

A coercivity condition for nonmonotone quasiequilibria on finite-dimensional spaces

New existence results for quasiequilibrium problems on unbounded feasible sets in a finite-dimensional space and without any assumption of monotonicity are established. The key point behind these results is a weak coercivity condition for a generalized game which extends a recent one proposed in Konnov and Dyabilkin (J Glob Optim 49:575–587, 2011) for equilibrium problems and an older one given in Cubiotti (Comput Math Appl 30:11–22, 1995) for quasiequilibrium problems. Some examples are also given.

Application of the penalty method to limit quasi-equilibrium problems

We solve a general quasi-equilibrium problem in a finite-dimensional setting, where only approximation sequences are used instead of exact values of the bi-function and feasible map. We suggest to utilize a sequence of solutions of auxiliary problems based on a penalty method. Its convergence is attained without concordance of penalty and approximation parameters under mild coercivity type conditions.

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.

Equilibrium formulations of relative optimization problems

We consider relative or subjective optimization problems where the goal function and feasible set are dependent of the current state of the system under consideration. We propose equilibrium formulations of the corresponding problems that lead to general (quasi-)equilibrium problems. We propose to apply a regularized version of the penalty method for the general quasi-equilibrium problem, which enables us to establish existence results under weak coercivity conditions and replace the quasi-equilibrium problem with a sequence of the usual equilibrium problems. We describe several examples of applications and show that the subjective approach can be extended to non-cooperative game problems.

Quasi-equilibrium problems with non-self constraint map

In 2016 Aussel, Sultana and Vetrivel developed the concept of projected solution for Nash equilibria. The purpose of this work is to study the same concept of solution, but for quasi-equilibrium problems. Our results recover several existence theorems for quasi-equilibrium problems in the literature. Additionally, we show the existence of projected solutions for quasi-optimization problems, quasi-variational inequality problems, and generalized Nash equilibrium problems.

Stability for Parametric Vector Quasi-Equilibrium Problems With Variable Cones

This article is devoted to the study of stability conditions for a class of quasi-equilibrium problems with variable cones in normed spaces. We introduce concepts of upper and lower semicontinuity of vector-valued mappings involving variable cones and their properties, we also propose a key hypothesis. Employing this hypothesis and such concepts, we investigate sufficient/necessary conditions of the Hausdorff semicontinuity/continuity for solution mappings to such problems. We also discuss characterizations for the hypothesis which do not contain information regarding solution sets. As an application, we consider the special case of traffic network problems. Our results are new or improve the existing ones.

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.

Common fixed point theorems for set-valued mappings in normed spaces

Let \(\Phi \) be the class of all real functions \(\varphi : [0, \infty [ \times [0, \infty [ \rightarrow [0, \infty [\) that satisfy the following condition: there exists \(\alpha \in ]0, 1[ \text { such that } \varphi ((1 - \alpha ) r, \alpha r) 0\). In this paper, we show that if X is a nonempty compact convex subset of a real normed vector space, any two closed set-valued mappings \(T, S: X \rightrightarrows X\), with nonempty and convex values, have a common fixed point whenver there exists a function \(\varphi \in \Phi \) such that $$\begin{aligned} \Vert y - u\Vert \le \varphi (\Vert y - x\Vert , \Vert u - x\Vert ), \; \text { for all } x\in X, y\in T(x), u\in S(x). \end{aligned}$$ Next, we prove that the same conclusion holds when at least one of the set-valued mappings is lower semicontinuous with nonempty closed and convex values. Our common fixed point theorems turn out to be useful for a unitary treatment of several problems from optimization and nonlinear analysis (quasi-equilibrium problems, quasi-optimization problems, constrained fixed point problems, quasi-variational inequalities).

Time-Dependent Generalized Nash Equilibrium Problem

We prove an existence result for the time-dependent generalized Nash equilibrium problem under generalized convexity without neither a quasi-variational inequality reformulation nor a quasi-equilibrium problem reformulation. Furthermore, an application to the time-dependent abstract economy is considered.

Regularized gap functions and error bounds for generalized mixed strong vector quasiequilibrium problems

In this paper, we consider generalized mixed strong vector quasiequilibrium problems in infinite-dimensional spaces. Using the nonlinear scalarization method, we first propose gap functions for such problems. Then, based on the Moreau–Yosida regularization of these functions, we introduce regularized gap functions for the reference problems. Next, we study sufficient conditions for the mentioned functions to be continuous. Finally, error bounds for the concerning problems in terms of regularized gap functions are investigated. Some examples are provided to illustrate our results.

A Ky Fan Minimax Inequality for Quasiequilibria on Finite-Dimensional Spaces

Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite-dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued maps which holds in finite-dimensional spaces. Furthermore, this result allows one to locate the position of a solution. Sufficient conditions, which are easier to verify, may be obtained by imposing restrictions either on the domain or on the bifunction. These facts make it possible to yield various existence results which reduce to the well-known Ky Fan minimax inequality when the constraint map is constant and the quasiequilibrium problem coincides with an equilibrium problem. Lastly, a comparison with other results from the literature is discussed.