Quasi-equilibrium Problems Research Articles

Approximate duality for vector quasi-equilibrium problems and applications

In this paper, we introduce new versions of ϵ -dual problems of a vector quasi-equilibrium problem with set-valued maps, and we give an ϵ -duality result between approximate solutions of the primal and dual problems. As the first application of the main result, we obtain an ϵ -duality for a vector quasi-equilibrium problem whose ϵ -solutions are understood in the sense of proper efficiency. The second application is devoted to an ϵ -duality for a vector optimization problem with set-valued maps.

Upper semicontinuity result for the solution mapping of a mixed parametric generalized vector quasiequilibrium problem with moving cones

In this paper, we give sufficient conditions for the upper semicontinuity property of the solution mapping of a parametric generalized vector quasiequilibrium problem with mixed relations and moving cones. The main result is proven under the assumption that moving cones have local openness/local closedness properties and set-valued maps are cone-semicontinuous in a sense weaker than the usual sense of semicontinuity. The nonemptiness and the compactness of the solution set are also investigated.

Mixed quasi nonconvex equilibrium problems

In this paper, we introduce and study a new class of equilibrium problems, known as mixed quasi nonconvex equilibrium problems. We use the auxiliary principle technique to suggest and analyze some iterative schemes for solving nonconvex equilibrium problems. We prove that the convergence of these iterative methods requires either pseudomonotonicity or partially relaxed strongly monotonicity. Our proofs of convergence are very simple. As special cases, we obtain earlier known results for solving equilibrium problems and variational inequalities involving the convex sets.

Existence of solutions for generalized vector quasi-equilibrium problems

Existence of solutions for generalized vector quasi-equilibrium problems

Upper semicontinuity in a parametric general variational problem and application

In this paper, we give sufficient conditions for the upper semicontinuity property of the solution mapping of a general model which includes many generalized vector quasi-equilibrium problems with set-valued maps as special cases. The main result generalizes and improves several recent results. An example is given to illustrate such generalization and improvement. The main result is also applied to a model which can be interpreted as a system of generalized vector quasi-equilibrium problems with moving cones. The main tools of the paper are some new notions of cone-semicontinuity properties and openness/closedness properties of families of set-valued maps.

Systems of quasi-variational relations with applications

In this paper, we introduce a system of quasi-variational relations (in short, SQVR) and present several examples which show that it is a very general and unified model of several problems. We establish the existence of solutions of SQVP, in general, and several other problems, in particular. As an application of our results, we derive maximal element theorems and a collectively fixed point theorem for a family of multivalued maps. As further applications, we study Ky Fan type inequality / inclusion problem for vector valued bifunctions which includes constrained Nash equilibrium problem as a special case. We also present a common fixed point theorem for a family of multivalued maps. The results of this paper improve and generalize several known results on (system of) quasi-equilibrium problems, (system of) quasi-variational inclusions, constrained Nash equilibrium problem, collectively fixed point theorem and KKM type theorems for a family of multivalued maps. Our results also contain several results which appeared in recent literature.

Hybrid inclusion and disclusion systems with applications to equilibria and parametric optimization

In this paper, we first establish the existence theorems of the solution of hybrid inclusion and disclusion systems, from which we study mixed types of systems of generalized quasivariational inclusion and disclusion problems and systems of generalized vector quasiequilibrium problems. Some applications of existence theorems to feasible points for various mathematical programs with variational constraints or equilibrium constraints, system of vector saddle point and system of minimax theorem are also given.

Nonempty intersection theorems in topological spaces with applications

In this paper, we establish some new nonempty intersection theorems for generalized L-KKM mappings and prove some new fixed point theorems for set-valued mappings under suitable conditions in topological spaces. As applications, an existence theorem for an equilibrium problem with lower and upper bounds and two existence theorems for a quasi-equilibrium problem with lower and upper bounds are obtained in topological spaces. Our results generalize some known results in the literature.

Some results on systems of quasi-variational inclusion problems and systems of generalized quasi-variational inclusion problems

In this paper, we study systems of quasi-variational inclusion problem and systems of quasi-variational disclusion problem. From the existence theorems of solution for these two types of problems, we study various types of systems of quasi-variational inclusion problems, systems of quasi-equilibrium problems, systems of quasi-KKM theorem, abstract economics and system of KKM theorem. We also show their equivalent relations. We study further existence theorems of solution for generalized quasi-variational inclusion problem. Our results are different from any existence result in the literature.

Continuity of solution maps of parametric quasiequilibrium problems

In this article, we consider a parametric vector quasiequilibrium problem in topological vector spaces. Sufficient conditions for solution maps to be lower and Hausdorff lower semicontinuous, upper semicontinuous and continuous are established. Our results improve recent existing ones in the literature.

Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems

In this paper, we first introduce the parametric form of the generalized operator equilibrium problem and generalized operator quasi-equilibrium problem due to Kum and Kim [S. Kum, W.K. Kim, On generalized operator quasi-equilibrium problems, J. Math. Anal. Appl. 345 (2008) 559–565]. Next, we consider the upper and lower semicontinuity properties for the parametric generalized operator equilibrium problem (simply, PGOEP) and the parametric generalized operator quasi-equilibrium problem (simply, PGOQEP). Our semicontinuity properties of the solution sets for (PGOEP) and (PGOQEP) can be suitable for the stability properties of many kinds of nonlinear problems, e.g., the operator equilibrium problem, operator quasi-equilibrium problem, operator variational-like inequality problem, operator variational inequality problem, operator minimization problem and vector equilibrium problem.

A SEPARATION APPROACH TO VECTOR QUASI-EQUILIBRIUM PROBLEMS: SADDLE POINT AND GAP FUNCTION

The image space approach is applied to the study of vector quasiequilibrium problems. Exploiting separation arguments in the image space, Lagrangian-type optimality conditions and gap functions for vector quasiequilibrium problems are derived.

ON THE SOLUTION EXISTENCE OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH DISCONTINUOUS MULTIFUNCTIONS

In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set $K$ in a normed space $X$, a subset $D$ in a Hausdorff topological vector space $Y$, and a closed convex cone $C$ in $R^n$. Let $\Gamma: K\to 2^K$, $\Phi : K\rightarrow 2^{D}$ be two multifunctions and $f : K\times D\times K\to R^n$ be a single-valued mapping. Find a point $(\hat x, \hat y)\in K\times D$ such that \begin{gather} (\hat x, \hat y)\in \Gamma(\hat x)\times\Phi(\hat x),\,\, {\rm and}\,\, \{f(\hat x, \hat y, z): z\in\Gamma(\hat x)\}\cap (-{\rm Int }C)=\emptyset. \notag \end{gather} We prove some existence theorems for the problem in which $\Phi$ can be discontinuous and $K$ can be unbounded.

New systems of generalized vector quasi-equilibrium problems in product FC-spaces

The notions of C i (x)-FC-diagonally quasiconvex, C i (x)-FC-quasiconvex and C i (x)-FC-quasiconvex-like for set-valued mappings are introduced in FC-spaces without convexity structure. By applying these notions and a maximal element theorem for a family of set-valued mappings on product FC-space due to author, some new existence theorems of solutions for four new classes of systems of generalized vector quasi-equilibrium problems are proved in noncompact FC-spaces. These results improve and generalize some recent known results in literature to noncompact FC-spaces.

On the existence of solutions to quasivariational inclusion problems

We establish sufficient existence conditions for general quasivariational inclusion problems, which contain most of variational inclusion problems and quasiequilibrium problems considered in the literature. These conditions are shown to extend recent existing results and sharpen some of them even for particular cases.

SYSTEMS OF GENERALIZED VECTOR QUASI-VARIATIONAL INCLUSION PROBLEMS AND APPLICATION TO MATHEMATICAL

In this paper, we introduce and study some new systems of generalized vector quasi-variational inclusion problems involving condensing mappings in locally $FC$-uniform spaces. These systems contain many known systems of generalized vector quasi-variational inclusion problems, systems of generalized vector quasi-equilibrium problems and systems of vector quasi-optimization problems as special cases. By applying an existence theorem of maximal elements of a family of set-valued mappings involving condensing mapping due to author, we prove some new existence theorems of solutions for the systems of generalized quasi-variational inclusion problems. As applications, some existence results of solutions of the mathematical programs with systems of generalized vector quasi-variational inclusion constraints are established in noncompact locally $FC$-uniform

ON THE SOLUTION EXISTENCE OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH DISCONTINUOUS MULTIFUNCTIONS

In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set $K$ in a normed space $X$, a subset $D$ in a Hausdorff topological vector space $Y$, and a closed convex cone $C$ in $R^n$. Let $\Gamma: K\to 2^K$, $\Phi : K\rightarrow 2^{D}$ be two multifunctions and $f : K\times D\times K\to R^n$ be a single-valued mapping. Find a point $(\hat x, \hat y)\in K\times D$ such that \begin{gather} (\hat x, \hat y)\in \Gamma(\hat x)\times\Phi(\hat x),\,\, {\rm and}\,\, \{f(\hat x, \hat y, z): z\in\Gamma(\hat x)\}\cap (-{\rm Int }C)=\emptyset. \notag \end{gather} We prove some existence theorems for the problem in which $\Phi$ can be discontinuous and $K$ can be unbounded.

A SEPARATION APPROACH TO VECTOR QUASI-EQUILIBRIUM PROBLEMS: SADDLE POINT AND GAP FUNCTION

The image space approach is applied to the study of vector quasiequilibrium problems. Exploiting separation arguments in the image space, Lagrangian-type optimality conditions and gap functions for vector quasiequilibrium problems are derived.

Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems

In this paper, Levitin-Polyak well-posedness for two classes ofgeneralized vector quasi-equilibrium problems is introduced.Criteria and characterizations of the Levitin-Polyak well-posednessare investigated. By virtue of gap functions for the generalizedvector quasi-equilibrium problems, some equivalent relations areobtained between the Levitin-Polyak well-posedness for optimizationproblems and the Levitin-Polyak well-posedness for generalizedvector quasi-equilibrium problems. Finally, a set-valued version ofEkeland's variational principle is derived and applied to establisha sufficient condition for Levitin-Polyak well-posedness of a classof generalized vector quasi-equilibrium problems.

On System of Generalized Vector Quasiequilibrium Problems with Applications

We introduce a new system of generalized vector quasiequilibrium problems which includes system of vector quasiequilibrium problems, system of vector equilibrium problems, and vector equilibrium problems, and so forth in literature as special cases. We prove the existence of solutions for this system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of generalized quasi-equilibrium problems and the generalized Debreu-type equilibrium problem for both vector-valued functions and scalar-valued functions.