Generalized Semi-continuity Of Vector-Valued Functions And Vector Equilbrium Problems

The semi-continuity of vector-valued functions and vector equilibrium problems are significant details of research in nonlinear analysis.The vector optimization problems have been extensively applied and studied by many scholars in recent decades,and the semi-continuity of vector functions as a powerful research tool are used to vector optimization problems.It is studied for the minimax theorems,Ekeland's variational principle,equilibrium problems and fixed point theorem and so on.However,vector equilibrium problems encompass complementarity problems,variational problems,Nash equilibrium problems and fixed point problem and so on,at the same time,one is studied by Ekeland's variational principle as a important tool.In this paper,we mainly study generalized semi-continuity of vector-valued functions,and weaker than the definition given above and its applications.We presented not only a class of minimax inequalities and Ekeland's variational principle for generalized semi-continuity of vector-valued functions,but also give the existence of solutions for vector equilibrium problems.This paper consists of three chapters.The full paper is divided into three chapters.The first chapter is introduction.In this chapter,we introduce the research significance and current situation of generalized semi-continuity of vector-valued functions.Secondly,we introduce the research significance and current situation of vector equilibrium problems.The second chapter mainly studies the generalized semi-continuity of vector-valued functions.We first prove that weak order lower semi-continuity is weak lower semi-continuity from above.Secondly,the well-known results of Browder are extended to vector form by using weak lower semi-continuity from above,with the partial order induced by completely regular cone on Banach space.Finally,the partial order induced by strongly minihedral cone on Banach space,and we establish a new kind of vector form of minimax inequalities for using lower semi-continuity from above and upper semi-continuous from below.The third chapter mainly studies the existence solutions of vector equilibrium problems.We establish generalized vector Ekeland's variational principles and W-distance of Ekeland's variational principles for lower semi-continuity from above by using linear scalar functions and nonlinear scalar functions,and can get the existence of solutions for vector equilibrium problems by using it.On the basis of the second chapter,we do not rely on the vector Ekeland's variational principles,but can use the generalized Weierstrass theorem toobtain the nonempty solution of the vector equilibrium problem.