Several Kinds Of Vector Variational Inequalities

The theory of vector variational inequalities, which is an important mathematics tool to study optimization, differential equation, mechanics, game theory, control theory, equilibrium, as well as linear or nonlinear problems in the field of mathematics and engineering, is one of focus problems in the mathematical branches and finds wide applications in set-valued analysis, convex analysis, linear and nonlinear analysis, nonsmooth analysis, fixed point theory. Therefore, the research has important learning value.The solution existence for vector variational inequalities and the relationships between the solutions to variational inequalities and the solutions to vector optimization problems, as well as the relationships between Stampacchia vector variational inequalities and Minty vector variational inequalities, play important roles in the theory of variational inequalities. In this paper, we focus on these three problems and extend first two problems to Riemannian manifolds. The main points of this paper is as follows:Minty vector variational inequalities are extended to Minty vector variational-like inequalities. The relationships between Minty vector variational-like inequalities and Stampacchia vector variational-like inequality problems, as well as Minty vector variational-like inequalities and vector optimization problems, are studied. Some relationships between the solutions to Minty vector variational-like inequalities with lower Dini directional derivative and the solutions to vector optimization problems, as well as radial decrease properties of vector valued functions, are investigated. Moreover, the affine solution sets of Minty vector variational-like inequalities are presented. Last, a sufficient condition is give fora kind of Minty weak vector variational-like inequalities.The sufficient conditions for generalized quasi-variational-like inequalities are presented. Four kinds of generalized vector quasi-equilibrium problems are studied. By using fixed point theorem, the solution existence problems for these four kinds of generalized vector quasi-equilibrium problems are investi- gated. Moreover, four kinds of generalized vector quasi-variational-like inequalities, which are the generalization of vector variational inequalities, are defined and studied. Last, a kind of generalized nonlinear vector variational-like inequalities is presented and the solution existence for this kind of variational inequalities is given.The definitions of generalized directional derivative and generalized gradient of Lipschitz functions defined on Riemannian manifolds are presented. Some properties of the directional derivative and the gradient are investigated. By applying Ekeland variational principle, the necessary conditions for mathematical programming problems are derived. The concepts of invex functions and vector variational inequalities on Riemannian manifolds are given. Some properties of geodesic convex and invex functions are studied. The relationships between vector optimization problems involve geodesic convex or invex functions and vector variational inequalities are investigated. Moreover, the solution existence for vector variational inequalities on Riemannian manifolds is given.