The Iterative Algorithm And Its Convergence Of Solutions For A Class Of Variational Inequalities In Hilbert Space

The variational inequality and complementary theory was founded by Stampacchia, Lions, Browder, Ky Fan, Cottle, Dantizig, Duvaut, Lewy and, Brezis in the1960s. After that, many researchers made meticulous research, and achieved a good deal of excellent results. Now, variational inequality theory get great promotion along different directions, and is appled widely in Mechanics, Cybernetics, Economic mathematics, Countermeasures theory, Differential equation and Optimization theory. Variational inequality is a powerful Mathematical tool, and it is very meaningful to research this kind of problems from different aspects.In this paper, we introduces the development history, the research methods and the applications in practice of variational inequality. An important and valuable method solving variational inequality is projection algorithm, many researchers make detailed research to the solving problems of various variational inequality problems using projection algorithms. However, when variational inequality has a nonlinear term, projection algorithms is limited. Therefore, scholars introduce resolvent operators to replace projection operator to solve this kind of variational inequality. The main method is:establishing the equivalence between the variational inequality and the fixed point equation using the resolvent-type of maximal monotone operator,constructing iterative algorithm, and then forming iterative sequences, finally proving the convergence of iterative sequences.Now, in this paper, we propose a class of new iterative algorithm aiming at a class of variational inequality problems. Afterwards, we prove the convergence of sequences which produced by algorithm.