Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 291851, 10 pages
doi:10.1155/2010/291851
On two iterative methods for mixed monotone variational inequalities
Xiwen Lu1
, Hong-Kun Xu2
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
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.