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

An algorithm based on resolvent operators for solving positively semidefinite variational inequalities

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.