Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 275208, 17 pages
doi:10.1155/2009/275208
Research Article

Hybrid Approximate Proximal Point Algorithms for Variational Inequalities in Banach Spaces

L. C. Ceng,1,2 S. M. Guu,3 and J. C. Yao4

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3Department of Business Administration, College of Management, Yuan-Ze University, Chung-Li, Taoyuan Hsien 330, Taiwan
4Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan

Received 10 March 2009; Accepted 7 June 2009

Academic Editor: Vy Khoi Le

Copyright © 2009 L. C. Ceng 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

Let C be a nonempty closed convex subset of a Banach space E with the dual E∗, let T:C→E∗ be a continuous mapping, and let S:C→C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator we study the variational inequality (for short, VI(T−f,C)): find x∈C such that 〈y−x,Tx−f〉≥0 for all y∈C, where f∈E∗ is a given element. By combining the approximate proximal point scheme both with the modified Ishikawa iteration and with the modified Halpern iteration for relatively nonexpansive mappings, respectively, we propose two modified versions of the approximate proximal point scheme L. C. Ceng and J. C. Yao (2008) for finding approximate solutions of the VI(T−f,C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of the VI(T−f,C), which is also a fixed point of S.