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

• Main Authors: Jen-Chih Yao, Sangho Kum, Lu-Chuan Ceng
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://dx.doi.org/10.1155/2010/794503

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.