Abstract
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.
Highlights
The so-called equilibrium problem, which is called generalized Ky Fan minimax inequality, was firstly studied by Blum and Oettli [1] in 1994 in finite-dimensional Euclidean spaces
In order to deal with cone-valued mapping, we need the following modified concept of semi-continuity for cone-valued mapping, which is called cosmically upper continuity and was proposed by Luc and Penot in [35]: A cone-valued mapping C is said to be cosmically upper continuous at x0 ∈ K if the mapping x → C(x) ∩ cl(B1) is u.s.c. at x0 ∈ K
4 Conclusions The aim of this paper is to investigate generalized strong vector quasi-equilibrium problems with variable ordering structure, which has already been studied in the previous literature
Summary
The so-called equilibrium problem, which is called generalized Ky Fan minimax inequality, was firstly studied by Blum and Oettli [1] in 1994 in finite-dimensional Euclidean spaces It provides a brief and unified framework for modeling many problems originating from practice and theory, such as mathematical economics, Nash equilibrium problem, fixed pointed problem, saddle point problem, optimization problem, complementary problem and variational inequality problem, etc. We can see that there are many meaningful works on the existence of solutions for (GVQEP) (see, e.g., [15,16,17,18,19,20,21,22] and the references therein) As applications of these results, they are applied to study the existence of mathematical programs with equilibrium constraints (MPEC), generalized Nash equilibrium, generalized semi-infinite programs, generalized quasi-variational inequality, and so on. These studies of (GVQEP) can be applied to
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