Generalized Vector Quasi-equilibrium Problems Research Articles

Maximal elements of [formula omitted]-majorized mappings in product [formula omitted]-spaces and applications (II)

This paper is a continuation of the preceding paper of the author. Four classes of systems of generalized vector quasi-equilibrium problems are introduced and studied in FC -spaces. The notions of C i ( x ) – FC -quasiconvexity, C i ( x ) -quasiconvexity and C i ( x ) -quasiconvexity-like for set-valued mappings are introduced in FC -spaces. By applying these notions and the equilibrium existence theorems for generalized games obtained in the author’s preceding paper, some new existence theorems of solutions for these systems of generalized vector quasi-equilibrium problems are established in noncompact product FC -spaces. These results improve and generalize some recent results in literature to product FC -spaces without any convexity structure.

Generalized abstract economy and systems of generalized vector quasi-equilibrium problems

The present paper is in two-fold. The first fold is devoted to the existence theory of equilibria for generalized abstract economy with a lower semicontinuous constraint correspondence and a fuzzy constraint correspondence defined on a noncompact/nonparacompact strategy set. In the second fold, we consider systems of generalized vector quasi-equilibrium problems for multivalued maps (for short, SGVQEPs) which contain systems of vector quasi-equilibrium problems, systems of generalized mixed vector quasi-variational inequalities and Debreu-type equilibrium problems for vector valued functions as special cases. By using the results of first fold, we establish some existence results for solutions of SGVQEPs.

Some existence results for solutions of generalized vector quasi-equilibrium problems

In this paper, we consider more general forms of generalized vector quasi-equilibrium problems for multivalued maps which include many known vector quasi-equilibrium problems and generalized vector quasi-variational inequality problems as special cases. We establish some existence results for solutions of these problems under pseudomonotonicity and u-hemicontinuity/l-hemicontinuity assumptions.

Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems

In this paper, we study an existence theorem of systems of generalized quasivariational inclusions problem. By this result, we establish the existence theorems of solutions of systems of generalized equations, systems of generalized vector quasiequilibrium problem, collective variational fixed point, systems of generalized quasiloose saddle point, systems of minimax theorem, mathematical program with systems of variational inclusions constraints, mathematical program with systems of equilibrium constraints and systems of bilevel problem and semi-infinite problem with systems of equilibrium problem constraints.

Maximal elements of [formula omitted]-majorized mappings in product [formula omitted]-spaces and applications (I)

Let I be any index set. Some new families of G KKM -mappings and G KKM -majorized mappings from a topological space X into finite continuous topological spaces ( Y i , φ N i ) (in short, FC -spaces) involving a set-valued mapping T ∈ KKM ( Y , X ) with KKM property are introduced where Y = ∏ i ∈ I Y i . Some existence theorems of maximal elements for the families of G KKM -majorized mappings and G KKM -mappings are proved under noncompact setting of product FC -spaces. As applications, some new equilibrium existence theorems of generalized games are established in noncompact product FC -spaces. These theorems improve and generalize some recent results in the literature. Some further applications to the system of generalized vector quasi-equilibrium problems will be given in a follow-up paper.

The Study of the System of Generalized Vector Quasi-equilibrium Problems

In this paper, we study the system of generalized vector quasi-equilibrium problems, which includes as special cases the system of vector quasi-equilibrium problems and the system of generalized vector equilibrium problems, and establish the existence and essential components of the solution set under perturbations of its best-reply map. Moreover, we also derive a new existence theorem of Ky Fan's points for a set-valued map.

Stampacchia Generalized Vector Quasiequilibrium Problems and Vector Saddle Points

In this paper, Stampacchia generalized vector quasiequilibrium problem and generalized vector loose saddle points for set-valued mappings are introduced. By using the scalarization method and the fixed-point theorem, existence theorems are established.

System of Generalized Vector Quasi–Equilibrium Problems in Locally FC–Spaces

A new class of locally finite continuous topological spaces (for short, locally FC–spaces) and a class of system of generalized vector quasi–equilibrium problems are introduced. By applying a generalized Himmelberg type fixed point theorem for a set–valued mapping with KKM–property due to the author, a collectively fixed point and an equilibrium existence theorem of generalized game are first proved in locally FC–spaces. By applying our equilibrium existence theorem of generalized game, some new existence theorems of equilibrium points for the system of generalized vector quasi–equilibrium problems are proved in locally FC–spaces. These theorems improve, unify and generalize many known results in the literatures.

On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.

Generalized vector quasi-equilibrium problems with applications to common fixed point theorems and optimization problems

In this paper, we first establish the existence theorems of generalized vector quasi-equilibrium problems. From these results, we establish the existence theorems of common fixed point theorems for two multivalued maps and mathematical programs with an equilibrium constraint as applications.

Gap Functions and Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems

This paper deals with generalized vector quasi-equilibrium problems. By virtue of a nonlinear scalarization function, the gap functions for two classes of generalized vector quasi-equilibrium problems are obtained. Then, from an existence theorem for a generalized quasi-equilibrium problem and a minimax inequality, existence theorems for two classes of generalized vector quasi-equilibrium problems are established.

Existence of Solutions to Generalized Vector Quasi–Equilibrium Problems with Discontinuous Mappings

Let X, Y be two finite–dimensional topological vector spaces, Z a Hausdorff topological vector space, K ⊂ X and D ⊂ Z be two nonempty sets, C be a pointed, closed, and convex cone in Y with int C ≠ O. Let S : K → 2K and T : K → 2D be two multivalued mappings, and φ : K×D×K → Y be a trifunction. In this paper, we consider the generalized vector quasi–equilibrium problem, which is formulated by finding \(\hat x\) ∈ K and ŷ ∈ T(\(\hat x\)) such that \(\hat x\) ∈ S(\(\hat x\)) and φ(\(\hat x\), ŷ, u) /∈ −int C for all u ∈ S(\(\hat x\)). We establish an existence result in which T is not supposed to have any continuity property. Our results extend and improve the corresponding results of Cubiotti, Yao and Guo.

Generalized vector quasi-equilibrium problems with set-valued mappings

A new mathematical model of generalized vector quasiequilibrium problem with set-valued mappings is introduced, and several existence results of a solution for the generalized vector quasiequilibrium problem with and without -condensing mapping are shown. The results in this paper extend and unify those results in the literature.

System of Generalized Vector Quasi-Equilibrium Problems with Applications to Fixed Point Theorems for a Family of Nonexpansive Multivalued Mappings

In this paper, we establish the existence theorems of the generalized vector quasi-equilibrium problems. From some existence theorem, we establish fixed point theorems for a family of lower semicontinuous or nonexpansive multivalued mappings. We also obtain the existence theorems of system of mixed generalized vector variational-like inequalities and existence theorems of the Debreu vector equilibrium problems and the Nash vector equilibrium problems.

Systems of Simultaneous Generalized Vector Quasiequilibrium Problems and their Applications

In this paper, we introduce systems of simultaneous generalized vector equilibrium problems and prove the existence of their solutions. As application of our results, we derive the existence theorems for solutions of systems of vector saddle–point problems. Consequently, we prove the existence of a solution of systems of generalized minimax inequalities. Further application of our results is also given to establish the existence of a solution of a Debreu-type equilibrium problem for vector-valued functions.

Maximal Element Theorems with Applications to Generalized Games and Systems of Generalized Vector Quasi-Equilibrium Problems in G-Convex Spaces

By applying the maximal element theorems on product of G-convex spaces due to the first author, some equilibrium existence theorems for generalized games with fuzzy constraint correspondences are proved in G-convex spaces. As applications, some existence theorems of solutions for the system of generalized vector quasiequilibrium problem are established in noncompact product of G-convex spaces. Our results improve and generalize some recent results in the literature to product of G-convex spaces.

Generalized vector quasi-equilibrium problems

This paper deals with generalized vector quasi-equilibrium problems. Using a so-called nonlinear scalarization function and a fixed point theorem, existence theorems for two classes of generalized vector quasi-equilibrium problems are established.

Generalized vector quasi-equilibrium problems in locally G-convex spaces

Some classes of generalized vector quasi-equilibrium problems (in short, GVQEP) are introduced and studied in locally G-convex spaces which includes most of generalized vector equilibrium problems, generalized vector variational inequality problems, quasi-equilibrium problems and quasi-variational inequality problems as special cases. First, an equilibrium existence theorem for one person games is proved in locally G-convex spaces. As applications, some new existence theorems of solutions for the GVQEP are established in noncompact locally G-convex spaces. These results and argument methods are new and completely different from that in recent literature.

The existence of solutions for the system of generalized vector quasi-equilibrium problems

In this paper, we study the system of generalized vector quasi-equilibrium problems which includes as special cases the system of generalized vector equilibrium problems and the system of vector quasi-equilibrium problems, and establish the existence result for its solutions by means of the Kakutani–Fan–Glicksberg fixed points theorem. As special cases, we also derive the existence results for weakly Pareto–Nash equilibrium points for multiobjective generalized game problems and multiobjective game problems.

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

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