Existence Of Solutions For System Of Generalized Mixed Equilibrium Problems In Banach Spaces

Equilibrium problem in many fields such as economy, optimization, control theory, engineering technology and operator research et.al have a wide range of applications, which have been paid close attention by many scholars. The systems of generalized mixed equilibrium problems are generalization of equilibrium problems, which involve control theory, physics, economics and so on. They become an important foundation for fixed points problems, saddle point problems and other problems, also is one of focal point problems in this field,research on which touch upon such mathematical branches as convex, functional analysis, set-valued analysis. Therefore, the research for them has important learning value and appling value.In this thesis, the existence of solutions for two classes systems of generalized mixed equilibrium problems in Banach spaces and existence results for extend vector variational-like inequality are researched. These results generalize the corresponding results in related literatures.In chapter 2, under the different conditions, the existence of solutions for two classes systems of generalized mixed equilibrium problems are studied in Banach spaces. First, two classes systems of auxiliary generalized mixed equilibrium problems are introduced. Without any coercive conditions, the existence and uniqueness of solutions for two classes systems of auxiliary generalized mixed equilibrium problems are proved. Next, by using the auxiliary principle technique, two new iterative algorithms are constructed. Under suitable conditions, we prove that two new iterative sequences generated by iterative algorithm converge strongly to the solutions of two classes systems of generalized mixed equilibrium problems. These results generalize some recent results in this field.In chapter 3 is devoted to the existence of solutions for extended vector variational-like inequality. With the mapping satisfying the conditions of properly hg??? ? quasimonotone of Stampacchia types and Minty types, hg??? ? pseudomonotone, the existence of solutions for extended vector variational-like inequality is proved by the known KKM theorem.