On The Generalized Vector Variational Inequalities With Set-valued Mapping Problom

Generalize vector variational inequalities with set-valued mapping are very important generation of variational inequalities. They become significant foundation and tool for studying multiobjective program,equilibrium, as well as many problems in the fields of mathematics and engineering. Because generalized vector variational inequlitise touch upon many mathematical branches, such as applications in set-valued analysis,convex analysis, linear and nonlinear analysis,nonsmooth analysis, function analysis,the research for them is of much academic value. This dissertation is devoted to study the existence of solution of generalized vector variational inequalities with set-valued in Haudorff topological vector spaces by KKM-Fan theorem,through giving conditions such as lower semicontinuity and C-monotonicty. This result is also the unity and extension of a number of well-known existence theorem of solution of vector variational inequalities.When solving variational inequalities, we often transform a variational inequality problem into an optimization problem. Then we can solve the variational inequality by solving the optimization problem. Gap function plays a crucial role in this kind of transformation. Gap function is always used in the optimization problem on the initial application and get a series of good application, but the Gap function applied to the variational inequality problem is developed in recent years as a trend. So this paper will define a kind of Gap function for the set-valued generalized vector variational inequality in the Hausdorff topological vector space and will established necessary and sufficient conditions for the existence of a solution for the GVVI.