Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 794503, 22 pages
doi:10.1155/2010/794503
Research Article

Algorithm for Solving a Generalized Mixed Equilibrium Problem with Perturbation in a Banach Space

Lu-Chuan Ceng,1 Sangho Kum,2 and Jen-Chih Yao3

1Department of Mathematics, Shanghai Normal University, Scientific Computing Key Laboratory of Shanghai Universities, Shanghai 200234, China
2Department of Mathematics Education, Chungbuk National University, Cheongju 361763, Republic of Korea
3Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

Received 22 February 2010; Accepted 11 April 2010

Academic Editor: Wataru Takahashi

Copyright © 2010 Lu-Chuan 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

Let B be a real Banach space with the dual space B*. Let ϕ:B→R∪{+∞} be a proper functional and let Θ:B×B→R be a bifunction. In this paper, a new concept of η-proximal mapping of ϕ with respect to Θ is introduced. The existence and Lipschitz continuity of the η-proximal mapping of ϕ with respect to Θ are proved. By using properties of the η-proximal mapping of ϕ with respect to Θ, a generalized mixed equilibrium problem with perturbation (for short, GMEPP) is introduced and studied in Banach space B. An existence theorem of solutions of the GMEPP is established and a new iterative algorithm for computing approximate solutions of the GMEPP is suggested. The strong convergence criteria of the iterative sequence generated by the new algorithm are established in a uniformly smooth Banach space B, and the weak convergence criteria of the iterative sequence generated by this new algorithm are also derived in B=H a Hilbert space.