Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 291851, 10 pages
doi:10.1155/2010/291851
Research Article

On Two Iterative Methods for Mixed Monotone Variational Inequalities

Xiwen Lu,1 Hong-Kun Xu,2 and Ximing Yin1

1Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
2Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan

Received 22 September 2009; Accepted 23 November 2009

Academic Editor: Tomonari Suzuki

Copyright © 2010 Xiwen Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A mixed monotone variational inequality (MMVI) problem in a Hilbert space H is formulated to find a point u∗∈H such that 〈Tu∗,v−u∗〉+φ(v)−φ(u∗)≥0 for all v∈H, where T is a monotone operator and φ is a proper, convex, and lower semicontinuous function on H. Iterative algorithms are usually applied to find a solution of an MMVI problem. We show that the iterative algorithm introduced in the work of Wang et al., (2001) has in general weak convergence in an infinite-dimensional space, and the algorithm introduced in the paper of Noor (2001) fails in general to converge to a solution.