Open Access

An Inexact Proximal-Type Method for the Generalized Variational Inequality in Banach Spaces

Journal of Inequalities and Applications20082007:078124

https://doi.org/10.1155/2007/78124

Received: 27 July 2007

Accepted: 31 October 2007

Published: 31 January 2008

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.

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Authors’ Affiliations

(1)
Department of Mathematics, Shanghai Normal University
(2)
Department of Mathematics, University of Pisa
(3)
Department of Applied Mathematics, National Sun Yat-Sen University

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Copyright

© L. C. Ceng et al. 2007

This article is published under license to BioMed Central Ltd. 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.