The solution existence of general variational inclusion problems

Abstract

We propose general variational inclusion problems which are slightly different from corresponding problems considered in several recent papers in the literature and show that they are advantageous. Sufficient conditions for the solution existence are established. As applications we derive consequences for several special cases of variational inclusion problems, quasioptimization problems, equilibrium problems and implicit variational inequalities and show that they improve the results of some recent existing papers.

About Semicontinuity of Set-valued Maps and Stability of Quasivariational Inclusions

“…Let P , Q : X × X × → 2 Z . In [8][9][10] and [11], the following inclusion problem was investigated…”

“…Notice, as seen in [8][9][10] and [11], that for the mentioned problems, but with other constraints or other types of the inclusions, analogous study methods can be applied.…”

Well-posedness for systems of generalized mixed quasivariational inclusion problems and optimization problems with constraints

“…For more details, we refer the reader to [4,11,12,24,27,28,30,31] and the references therein. In 2009, Long and Huang [18] generalized the concept of α-well-posedness to symmetric quasiequilibrium problems in Banach spaces, which include equilibrium problems, Nash equilibrium problems, quasivariational inequalities, variational inequalities, and fixed-point problems as special cases.…”

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.…”