Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 973028, 23 pages
doi:10.1155/2011/973028
Research Article

Hybrid Proximal-Type Algorithms for Generalized Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings

Lu-Chuan Zeng,1,2 Q. H. Ansari,3 and S. Al-Homidan3

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
3Department of Mathematics and Statistics, King Fahd University of Petroleum & Minerals (KFUPM), P.O. Box 119, Dhahran 31261, Saudi Arabia

Received 24 September 2010; Accepted 18 October 2010

Academic Editor: Jen Chih Yao

Copyright © 2011 Lu-Chuan Zeng 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

The purpose of this paper is to introduce and consider new hybrid proximal-type algorithms for finding a common element of the set E P of solutions of a generalized equilibrium problem, the set 𝐹 ( 𝑆 ) of fixed points of a relatively nonexpansive mapping 𝑆 , and the set 𝑇 − 1 0 of zeros of a maximal monotone operator 𝑇 in a uniformly smooth and uniformly convex Banach space. Strong convergence theorems for these hybrid proximal-type algorithms are established; that is, under appropriate conditions, the sequences generated by these various algorithms converge strongly to the same point in E P ∩ 𝐹 ( 𝑆 ) ∩ 𝑇 − 1 0 . These new results represent the improvement, generalization, and development of the previously known ones in the literature.