Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 78124, 14 pages
doi:10.1155/2007/78124
Research Article
An Inexact Proximal-Type Method for the Generalized Variational Inequality in Banach Spaces
1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, Pisa 56100, Italy
3Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
Received 27 July 2007; Accepted 31 October 2007
Academic Editor: Charles Ejike Chidume
Copyright © 2007 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
We investigate an inexact proximal-type method, applied to the generalized variational inequality problem with maximal monotone operator in reflexive Banach spaces. Solodov and Svaiter (2000) first introduced a new proximal-type method for generating a strongly convergent sequence to the zero of maximal monotone operator in Hilbert spaces, and subsequently Kamimura and Takahashi (2003) extended Solodov and Svaiter algorithm and strong convergence result to the setting of uniformly convex and uniformly smooth Banach spaces. In this paper Kamimura and Takahashi's algorithm is extended to develop a generic inexact proximal point algorithm, and their convergence analysis is extended to develop a generic convergence analysis which unifies a wide class of proximal-type methods applied to finding the zeroes of maximal monotone operators in the setting of Hilbert spaces or Banach spaces.