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

Abstract

In this article, we introduce two hybrid proximal-type algorithms and two hybrid shrinking projection algorithms by using the hybrid proximal-type method and the hybrid shrinking projection method, respectively, for finding a common element of the set of solutions of an equilibrium problem, the set of fixed points of a relatively nonexpansive mapping, and the set of solutions to the equation 0 ∈ Tx for a maximal monotone operator T defined on a uniformly smooth and uniformly convex Banach space. The strong convergence of the sequences generated by the proposed algorithms is established. Our results improve and generalize several known results in the literature.

Keywords:

• Equilibrium problems
• Generalized projection
• Hybrid proximal-type algorithms
• Hybrid shrinking projection algorithms
• Maximal monotone operators
• Relatively nonexpansive mappings
• Strong convergence
• Uniformly smooth and uniformly convex Banach spaces

AMS Subject Classification:

• 49J40
• 47J20
• 47H05
• 65K10

ACKNOWLEDGMENTS

In this research, L.-C. Ceng was partially supported by the Leading Academic Discipline Project of Shanghai Normal University (DZL707), Innovation Program of Shanghai Municipal Education Commission Grant (09ZZ133), National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). J.-C. Yao was partially supported by a research grant number NSC 98-2923-E-110-003-MY3 of National Science Council of Taiwan.