A new iterative algorithm with errors for maximal monotone operators and its applications

Finding zero points for maximal monotone operator is a very active topic in different branches of mathematical and engineering sciences since many physically significant problems can be ultimately converted to it. Consequently, considerable research efforts have been devoted, especially within the past 20 years or so, to the study of iterative algorithms of zero points for maximal monotone operators. By now, there already exist some algorithms, such as proximal point algorithm, hybrid algorithm and regularity algorithm, etc. But all those methods are not quite enough to deal with problems defined in a more general space. In this paper, a new iterative algorithm with errors is introduced which is proved to be weakly convergent to zero point of maximal monotone operator by using some techniques of Lyapunov functional and generalized projection operator, etc. Moreover, some applications of the new algorithm are demonstrated. One of it is to solve a kind of variational inequalities which play a significant role in economics, finance, transportation, elasticity, optimization and structural analysis, etc. The other is to find a minimizer of a given convex function, which is also a very important topic in applied mathematics.