Existence of solutions for generalized vector quasi-equilibrium problems

Abstract

This paper deals with three classes of generalized vector quasi-equilibrium problems with or without compact assumptions. Using the well-known Fan-KKM theorems, their existence theorems for them are established. Some examples are given to illustrate our results.Keywords Generalized vector quasi-equilibrium problem · Weak type C x -diagonal quasi-convex and strong type C x -diagonal quasi-convex · Fan-KKM theorem Mathematics Subject Classification (2000) 49K30 · 90C29

On the existence of solutions to generalized quasi-equilibrium problems

“…2 to two problems of variational inclusions that are extensively studied in the recent literature under different forms (see for examples, in [3,5,7,8,[11][12][13][14][15][16][17][18],…). Namely, we consider the problem of findingx ∈ D such thatx…”

“…7 We assume that D, P are as in Corollary 3.6. In addition, suppose that G : D × D → 2 Y is a multivalued mapping with nonempty values and C : D → 2 Y is a cone multivalued mapping with G(x, x) ⊆ −intC(x) for any x ∈ D, satisfying the following conditions:…”

“…For the last decade there have been a number of generalizations of these problems to different directions such as quasi-equilibrium problems with constraint sets depending on parameters, quasi-variational and quasi-equilibrium inclusion problems with multi-valued data (see, for examples, in [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). Problem (GEP) described above is quite general.…”

A Note on Continuity of Solution Set for Parametric Weak Vector Equilibrium Problems

“…There are many papers which have discussed the existence results for different types of vector equilibrium problems (see [1][2][3] and references therein).…”

Semicontinuity of the Solution Mapping to a Parametric Generalized Weak Ky Fan Inequality

“…In the literature, existence results for various types of Ky Fan inequalities have been investigated intensively; for example, see [3][4][5] and the references therein. The stability analysis of solution mappings for parametric Ky Fan inequalities is another important topic in the vector optimization theories and applications.…”