On the existence of solutions to quasivariational inclusion problems

Abstract

Quasivariational inclusion problems, Quasiequilibrium problems, Implicit variational inequalities, The solution existence, Fixed points, Nash equilibria, Traffic networks, 49J40, 91B50, 90B20, 90C29,

Applications of some basic theorems in the KKM theory

“…(X) In 2009, Hai et al [25] established sufficient existence conditions for general quasi-variational inclusion problems, which contain most of variational inclusion problems and quasi-equilibrium problems considered in the literature. The authors made use of Proposition 5.1 and noted that "it turns out that this theorem is suitable to our problem setting and helps to get new results or to sharpen some recent existing ones.…”

Semicontinuity of Solution Sets to Parametric Quasivariational Inclusions with Applications to Traffic Networks I: Upper Semicontinuities

“…For the sake of simplicity of presentation, throughout the paper, we assume that S rα (λ, μ, η) = ∅ for all (λ, μ, η) in a neighborhood of (λ 0 , μ 0 , η 0 ) ∈ Λ × M × N. (About conditions for the solution existence in quasivariational inclusion problems see Hai and Khanh [11,12], Hai et al [13].) Theorem 3.1 Assume for problem (P rα ) that, for ∅ = U ⊆ X,…”

“…A natural extension of a quasivariational inequality and quasiequilibrium is a quasivariational inclusion, see Hai and Khanh [11,12], Hai et al [13], Luc and Tan [24,25], Tan [35], and Tuan and Sach [36]. It should be noted here that the term "variational inclusion" is understood in different ways in several recent papers.…”

“…Variational inclusion problems in Chidume et al [6], Ding [7], Fang and Huang [10], Peng [31], Salahuddin [32], Shi and Liu [34], and Verma [37,38] are problems of finding zeroes of maximal monotone mappings. In this note the terminology is similar to Hai and Khanh [11,12], Hai et al [13], Luc and Tan [24,25], Tan [35], and Tuan and Sach [36]. The main efforts have been devoted to the solution existence.…”

Parametric Set-Valued Vector Quasi-Equilibrium Problems

“…Up to now, the main efforts for equilibrium problems have been made for the solution existence; see for example 1-6 and the references therein. A few results have been obtained for properties of solution sets, see 7-12 . Motivated and inspired by works in 1,5,[8][9][10][11][12] , in this paper, we will introduce two kinds of parametric set-valued vector quasi-equilibrium problems and study the solution existence of these problems. In addition, we will investigate the upper and lower semicontinuities of their solution maps with respect to the parameters.…”