Adv. Math. Econ. 14, 197–218 (2011)

Strong convergence theorems by hybrid methods for nonexpansive mappings with equilibrium problems in Banach spaces

Wataru Takahashi^{1} and Jen-Chih Yao^{3}^{2}

1 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Japan (e-mail: wataru@is.titech.ac.jp)

2 Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: yaojc@math.nsysu.edu.tw)

Received: January 11, 2010 Revised: July 9, 2010

JEL classification: C62, C68

Mathematical Subject Classification (2010): 47H05, 47H09, 47H20

Abstract.Our purpose in this paper is to prove strong convergence theorems by hybrid methods for nonexpansive mappings in a Banach space under appropriate conditions. We first prove a strong convergence theorem by the shrinking projection method for semi-positively homogeneous nonexpansive mappings with an equilibrium problem in a Banach space. Next, we obtain another strong convergence theorem by the monotone hybrid method for semi-positively homogeneous nonexpansive mappings with an equilibrium problem in a Banach space. These theorems are proved by using the concept of set convergence.

Key words:Nonexpansive mapping, fixed point, hybrid method, Mosco convergence, equilibrium problem, projection

3The research of the first author and the second author was partially supported by Grantin-Aid for Scientific Research No. 19540167 from Japan Society for the Promotion of Science and by the grant NSC 98-2115-M-110-001, respectively.