Existence conditions in variational inclusions with constraints

Abstract

In this article we study a general variational inclusion problem with constraints. An existence theorem is given for the scalar problem which allows us to derive several results on existence of solutions of variational inequality and variational inclusion models in which the set-valued data are present.

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…”

“…It encompasses a large class of problems of applied mathematics including quasi-optimization problems, quasi-variational inclusion, quasi-equilibrium problems, quasi-variational relation problems etc. Typical instances of (GEP) are shown in [3,[11][12][13][14].…”

“…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.…”

Bilevel generalized mixed equilibrium problems involving generalized mixed variational-like inequality problems in reflexive Banach spaces

“…It is well-known that various bilevel problems, equilibrium constraint optimization problems, bilevel decision problems, and mathematical program problems with equilibrium constraints represent important classes of optimization problems that have been wildly investigated (see [1][2][3][4][5][6][7][8][9][10][11][12]). However, these papers deal mainly with the existence of solutions of the problems mentioned above.…”

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

“…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.…”