Strong convergence theorems for firmly nonexpansive-type mappings and equilibrium problems in Banach spaces

Abstract

In this article, we propose and investigate several iterative methods for approximating fixed points of a firmly nonexpansive-type mapping and for finding a common element of the set of fixed points of the firmly nonexpansive-type mapping and the set of solutions of an equilibrium problem in Banach space. By using the conception of generalized projection, strong convergence theorems for firmly nonexpansive-type mappings and equilibrium problems in Banach space are established under suitable assumptions, which extend and modify some known results in the literature.

Keywords:

• strong convergence theorem
• firmly nonexpansive-type mapping
• equilibrium problem
• fixed point
• generalized projection operator

AMS Subject Classifications::

• 47H09
• 47J05
• 47J25

Acknowledgements

The authors are grateful to four anonymous referees for making some useful suggestions and comments leading to the improvements on an earlier version of this study. The first author thanks Professor Jun Li, School of Mathematics and Information, China West Normal University for providing constructive suggestions and financial support. This study also was supported by the Natural Science Foundation of China (Nos 71171150, 60804065), the Academic Award for Excellent Ph.D. Candidates funded by Wuhan University and the Fundamental Research Fund for the Central Universities.