On Generalized Quasi-Vector Equilibrium Problems via Scalarization Method

Abstract

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 quasiequilibrium 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 quasiequilibrium 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.

Generalized strong vector quasi-equilibrium problems with variable ordering structure

“…In 1992, Chen and Huang [26] generalized it to variational inequality. Afterwards, there have been many works devoted to the study of various equilibrium, optimization, and variational inequality problems with variable ordering structure (see, for example, [27][28][29][30][31][32] and the references therein). It is worth pointing out that, in these literature sources, the condition of upper semi-continuity for cone-valued mapping has been widely used.…”

Generalized Vector Quasi-Equilibrium Problems Involving Relaxed Continuity of the Set-Valued Maps

“…This explains the increasing attention paid to it by many researchers nowadays. For further information on this topic, the reader is referred to the following list of selected publications: [1][2][3][4][5], [7], [8], [10], [12], [14][15][16][17].…”