The Existence Theorem Of The Solution Of Set-valued Equilibrium Problems And The Gap Function Of Variational Inequalities

Vector optimization theory is one of the main research fields of optimization theory and application. The study of this topic involves many disciplines, such as convex analysis, nonlinear analysis, nonsmooth analysis, partial ordering theory, and so on. One of important problems in the vector optimization problems is the equilibrium problem. The study of equilibrium problems has been generalized by many researchers from single objective problems to generalized vector equilibrium problems and generalized vector quasi-equilibrium problems involving set-valued mappings. The existence of the solutions of the vector equilibrium problems is an important topic of the research of the equilibrium problem. However, the ordering cones in the equilibrium problems which studied by many researchers are fixed. In this thesis, we study the set-valued equilibrium problem with varying ordering cone. Another important problem of the vector optimization problems is vector variational inequality. Gap functions play an important role in solving solution and calculation of the variational inequalities. Recently, some researchers study the gap function of the weakly minimal solution of the generalized variational inequality. In this thesis, we study the gap function of the strongly minimal solution of the generalized variational inequality.In this thesis, we first give some existence theorem of the set-valued equilib-rium problems by applying Ky Fan’s Lemma. Then, we study some properties of the solutions of the set-valued equilibrium problems by applying the Tikhonov regu-larization methods. Finally, we give a gap function of the strongly minimal solution of the generalized variational inequality.