Solution Existence Of Equilibrium Problems

For the research of solution existence of equilibrium problems, authors at home and abroad mainly use four methods as follows: KKM Theorem, Fixed-Point Theorem, Maximal Element Theorem and Variational Principle. In the process of solving problems, four methods are different in space limit and request for conditions of functions or mappings. In this paper, we make comparison between them to discuss their distinction and connection with studying solution existence of same or similar equilibrium problems, while finding the conditions under which they transform each other. Concrete content of this paper is as follows: In the first chapter, we outline the development and actuality of the research at home and abroad for solution existence of equilibrium problems recently. In the second chapter, we recall concepts and results we will use in the sequel. In the third chapter, we discuss the distinction and connection between KKM Theorem and Fixed-Point Theorem, with studying solution existence of generalized vector equilibrium problems and that between KKM Theorem and Maximal Element Theorem with studying generalized vector quasi-equilibrium problems. In the fourth chapter, we discuss the distinction and connection between Fixed-Point Theorem and Maximal Element Theorem with studying systems of generalized vector quasi-equilibrium problems. In the fifth chapter, with research of solution existence of scalar equilibrium problems, we discuss the distinction and connection between Ekeland's Variational Principle and KKM Theorem and that between Ekeland's Variational Principle and Maximal Element Theorem. In the sixth chapter, we compare Ekeland's Variational Principle with Fixed-Point Theorem with studying vector equilibrium problems.