Generalized 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 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.

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

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

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.

Abstract generalized vector quasi-equilibrium problems in noncompact Hadamard manifolds

This paper deals with the abstract generalized vector quasi-equilibrium problem in noncompact Hadamard manifolds. We prove the existence of solutions to the abstract generalized vector quasi-equilibrium problem under suitable conditions and provide applications to an abstract vector quasi-equilibrium problem, a generalized scalar equilibrium problem, a scalar equilibrium problem, and a perturbed saddle point problem. Finally, as an application of the existence of solutions to the generalized scalar equilibrium problem, we obtain a weakly mixed variational inequality and two mixed variational inequalities. The results presented in this paper unify and generalize many known results in the literature.

Existence and Hadamard well-posedness of a system of simultaneous generalized vector quasi-equilibrium problems

An existence result for the solution set of a system of simultaneous generalized vector quasi-equilibrium problems (for short, (SSGVQEP)) is obtained, which improves Theorem 3.1 of the work of Ansari et al. (J. Optim. Theory Appl. 127:27-44, 2005). Moreover, a definition of Hadamard-type well-posedness for (SSGVQEP) is introduced and sufficient conditions for Hadamard well-posedness of (SSGVQEP) are established.

Existence and Connectedness of Solutions for Generalized Vector Quasi-Equilibrium Problems

In this paper, we establish three existence theorems for strongly efficient solutions, weakly efficient solutions and efficient solutions of generalized vector quasi-equilibrium problems by using the Fan-KKM theorem and the scalarization method. Moreover, we investigate the connectedness of the sets of weakly efficient solutions and efficient solutions for generalized vector quasi-equilibrium problems and make a new attempt to establish the path connectedness of the set of weakly efficient solutions for generalized vector quasi-equilibrium problems.

On Generalized Quasi-Vector Equilibrium Problems via Scalarization Method

In this paper, we consider the nonlinear scalarization function in the setting of topological vector spaces and present some properties of it. Moreover, using the nonlinear scalarization function and Fan---Glicksberg---Kakutani's fixed point theorem, we obtain an existence result of a solution for a generalized vector quasi-equilibrium problem without using any monotonicity and upper semi-continuity (or continuity) on the given maps. Our result can be considered as an improvement of the known corresponding result. After that, we introduce a system of generalized vector quasi-equilibrium problem which contains Nash equilibrium and Debreu-type equilibrium problem as well as the system of vector equilibrium problem posed previously. We provide two existence theorems for a solution of a system of generalized vector quasi-equilibrium problem. In the first one, our multi-valued maps have closed graphs and the maps are continuous, while in the second one, we do not use any continuity on the maps. Moreover, the method used for the existence theorem of a solution of a system of generalized vector quasi-equilibrium problem is not based upon a maximal element theorem. Finally, as an application, we apply the main results to study a system of vector optimization problem and vector variational inequality problem.

Existence of solutions for generalized vector quasi-equilibrium problems in abstract convex spaces with applications

In this paper, several kinds of generalized vector quasi-equilibrium problems are introduced and studied in abstract convex spaces. Using the properties of Γ-convex and \(\mathfrak{KC}\)-maps, some sufficient conditions are given to guarantee the existence of solutions in connection with these generalized vector quasi-equilibrium problems. As applications, some existence theorems of solutions for the generalized semi-infinite programs with vector quasi-equilibrium constraints are also given.

Upper Semicontinuity of Solution Mappings to Parametric Generalized Vector Quasiequilibrium Problems

We establish the upper semicontinuity of solution mappings for a class of parametric generalized vector quasiequilibrium problems. As applications, we obtain the upper semicontinuity of solution mappings to several problems, such as parametric optimization problem, parametric saddle point problem, parametric Nash equilibria problem, parametric variational inequality, and parametric equilibrium problem.

Hölder continuity results for nonconvex parametric generalized vector quasiequilibrium problems via nonlinear scalarizing functions

In this paper, new results for Hölder continuity of the unique solution to a parametric generalized vector quasiequilibrium problem are established via nonlinear scalarization, with and without using the free-disposal condition. Especially, a new kind of monotonicity hypothesis is proposed. The globally Lipschitz property together with other useful properties of the well-known Gerstewitz nonlinear scalarization function are fully exploited for proving. Moreover, our approach does not impose any convexity condition on the considered model. The oriented distance function is also employed for studying Hölder continuity.

On the Hölder continuity of solution maps to parametric generalized vector quasi-equilibrium problems via nonlinear scalarization

In this paper, by using a nonlinear scalarization technique, we obtain sufficient conditions for Holder continuity of the solution mapping for a parametric generalized vector quasi-equilibrium problem with set-valued mappings. The results are different from the recent ones in the literature.

New Results on Henig Proper Generalized Vector Quasiequilibrium Problems

New sufficient conditions are given for the existence of solutions of a Henig proper generalized vector quasiequilibrium problem with moving cones. They are established by a new scalarizing approach, which is based on a suitable nonlinear scalarization function, proposed recently for set-valued maps in Sach and Tuan (J. Optim. Theory Appl. 157:347–364 (2013)). Examples are given to illustrate our main results.

Erratum to: Generalized vector quasi-equilibrium problems on Hadamard manifolds

Erratum to: Generalized vector quasi-equilibrium problems on Hadamard manifolds

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

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.

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.