Communications of the
Korean Mathematical Society CKMS

The purpose of this paper is to prove strong convergence theorems for common fixed points of two families of weak relatively nonexpansive mappings and a family of equilibrium problems by a new monotone hybrid method in Banach spaces. Because the hybrid method presented in this paper is monotone, so that the method of the proof is different from the original one. We shall give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping in Banach space $l^{2}$. Our results improve and extend the corresponding results announced in [W. Takahashi and K. Zembayashi, {\it Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings}, Fixed Point Theory Appl. (2008), Article ID 528476, 11 pages; doi:10.1155/2008/528476] and [Y. Su, Z. Wang, and H. Xu, {\it Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings}, Nonlinear Anal. {\bf 71} (2009), no. 11, 5616--5628] and some other papers.

Keywords: relatively nonexpansive, weak relatively nonexpansive, common fixed point, generalized projection, equilibrium problem, monotone hybrid algorithm, maximal monotone operator

MSC numbers: 47H05, 47H09, 47H10