Vector Equilibrium Problems And Ekeland's Variational Principle For Set-Valued Mappings

In this paper, we mainly study three themes, vector variational inequality and set-valued optimization problems, generalized vector quasi-equilibrium problems and Ekeland's variational principles for set-valued mappings.Firstly, we study the relationships between the solutions of a vector variational inequality and the weak minimizers of a set-valued optimization problem. For this purpose,we introduce a definition of weak contingent generalized gradient and discuss the existences of its solutions. Then, we define a vector variational-like inequality by virtue of this weak contingent generalized gradient and establish the relationships between the solutions of the vector variational inequality and the weak minimizers of a set-valued optimization problem.Secondly, we use the fixed point theorem in [1] to study existence results for two classes of generalized vector quasi-equilibrium problems ((GVQEP1) and (GVQEP2), for short) respectively. Then, we give an example to explain that the existence result for (GVQEP1) is different from the one in [2]. Simultaneously, an example is given to show that the existence results for (GVQEP2) in this paper are different from ones in [3]. Finally, we obtain an existence result of solutions for vector quasi optimization problems by using the existence result for (GVQEP1).Finally, we establish a set form of the Ekeland's variational principle, through which we can obtain existence results for a general vector equilibrium problems (GVEP, for short) in both compact and noncompact domains, but without any convexity request of domain. Moreover, we also give some equivalent results to our variational principle.