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  • Research Article
  • Open Access

A Generalized Hybrid Steepest-Descent Method for Variational Inequalities in Banach Spaces

Fixed Point Theory and Applications20102011:754702

  • Received: 13 September 2010
  • Accepted: 9 December 2010
  • Published:


The hybrid steepest-descent method introduced by Yamada (2001) is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi (1996) introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamada's hybrid steepest-descent and Lehdili and Moudafi's algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions. Our strong convergence theorems extend and improve certain corresponding results in the recent literature.


  • Hilbert Space
  • Banach Space
  • Variational Inequality
  • Strong Convergence
  • Regularization Method

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Authors’ Affiliations

Department of Mathematics, Banaras Hindu University, Varanasi, 221005, India
Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, 804, Taiwan
Center for General Education, Kaohsiung Medical University, Kaohsiung, 807, Taiwan


© Copyright © 2011 D. R. Sahu et al. 2011

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.