Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 754702, 28 pages
Research Article

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

1Department of Mathematics, Banaras Hindu University, Varanasi 221005, India
2Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan
3Center for General Education, Kaohsiung Medical University, Kaohsiung 807, Taiwan

Received 13 September 2010; Accepted 9 December 2010

Academic Editor: S. Al-Homidan

Copyright © 2011 D. R. Sahu 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.


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.