Fixed Point Theory and Applications 
Volume 2007 (2007), Article ID 76040, 15 pages
doi:10.1155/2007/76040
Research Article

An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities

Juhe Sun, Shaowu Zhang, and Liwei Zhang

 Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning 116024, China
Received 16 June 2007; Accepted 19 September 2007

Recommended by Nan-Jing Huang

Abstract

A new monotonicity, M-monotonicity, is introduced, and the resolvant operator of an M-monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality (VI) problem VI (S+n,F+G) is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method is given for calculating ε-solutions to the sequence of fixed point problems, enabling the proximal point algorithm to be implementable.