Set-valued Variational Inclusions Iterative Solution Of The Equilibrium Problem

Two hundreds of years ago, variational principle appeared as a very important branch of mathematics. It plays an irreplaceable role in both theoretics and applications. In 1960's, Lions, Browder, Stampacchia, Ky Fan, Frichera and Hartman etc. proposed and studied variational inequalities theory, which has become perfectly so far thanking for the study of many mathematicians [1,2, 4-6]. Variational inequalities theory has applied in a large number of areas such as mechanics, differential equations, cybernetics, economics, optimization, mathematic programming and so on.Variational inclusions are useful and important generalizations of variational inequalities. The problems were first proposed by A. Hassouni and A. Moudafi[3] in 1994. In recent years, many scholars have studied the problems and obtained lots of results and algorithms, which based on various type of spaces and operators, all these helped the variational inclusions developing fast. In the first part of this article, a new class of set-valued variational inclusion problem in Banach spaces was introduced and some new type of iterative algorithms for solving the problems are studied.Inspired by the idea of the algorithms above, we discussed a class of equilibrium problems in the second part of this article. Equilibrium is a momentous notion of economics and finance. A lot of mathematicians studied the problems deeply by using set-valued analysis, the fruitful method therein is variational inequalities[7, 8, 21, 22]. There are a substantial number of papers on existence results for solving equilibrium problems based on different monotonicity and various compactness assumptions, but up to now only a few of iterative methods to solve such problems have been done[9-l 1]. Then, we investigate iterative methods in Hilbert spaces for solving such problems in the second part.