Vector Ekeland's Variational Principle And Generalized Symmetric Vector Quasi-equilibrium Problems

In this thesis, we study three problems: Ekeland's variational principle for set-valued mappings, an equivalent result between a variant of Ekeland's variational principles and a minimization theorem for set-valued mappings, and the existence results for the solutions of a generalized symmetric vector quasi-equilibrium problem. It is organized as follows:Firstly, we use a nonlinear scalarization function and the generalized Ekeland's variational principle for a half distance vector valued function to verify a new vector Ekeland's variational principle for set-valued mappings. As a corollary of the main result, we obtain a generalized Ekeland's variational principle for K -upper semi-continuous mappings. Simultaneously, we use an example to explain that the corresponding result of [13] is a special case of the main result.Secondly, we prove that the lower semi-continuous and lower bounded function, of which the domain does not need to be compact, has an exact optimal solution on whole space under the meaning of set optimization criterion. By using a variant of Ekeland's variational principle for set optimization introduced by T. X. D. Ha [20], we obtain a minimization theorem for a set-valued mapping and prove that the minimization theorem is equivalent to the Ekeland's variational principle obtained by T. X. D. Ha [20]. Simultaneously, we show that our minimization theorem is a generalization of Takahashi theorem.Finally, we introduce a generalized symmetric vector quasi-equilibrium problem (GSVQEP). By virtue of the Kakutani-Fan-Glicksberg fixed point theorem and a nonlinear scalarization function, existence results for the solutions of generalized symmetric vector quasi-equilibrium problem are obtained in locally convex Hausdorff topological vector spaces. Moreover, by virtue of the nonlinear scalarization function and a symmetric quasi-equilibrium theorem, an existence result is also established in real Hausdorff topological vector spaces. Our existence theorems extend the main results in [22] and [23] from vector-valued mappings to set-valued mappings.