Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 673932, 16 pages
doi:10.1155/2010/673932
Research Article

Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

Songnian He and Xiao-Lan Liang

College of Science, Civil Aviation University of China, Tianjin 300300, China

Received 30 September 2009; Accepted 13 January 2010

Academic Editor: Tomonari Suzuki

Copyright © 2010 Songnian He and Xiao-Lan Liang. 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

Let H be a real Hilbert space and let F:H→H be a boundedly Lipschitzian and strongly monotone operator. We design three hybrid steepest descent algorithms for solving variational inequality VI(C,F) of finding a point x∗∈C such that 〈Fx∗,x−x∗〉≥0, for all x∈C, where C is the set of fixed points of a strict pseudocontraction, or the set of common fixed points of finite strict pseudocontractions. Strong convergence of the algorithms is proved.