Convergence to common solutions of various problems for nonexpansive mappings in Hilbert spaces
Graduate School of Education, Mathematics Education, Kyungnam University, Changwon, Kyungnam, 631-701, Republic of Korea
Fixed Point Theory and Applications 2012, 2012:185 doi:10.1186/1687-1812-2012-185Published: 24 October 2012
In this paper, motivated and inspired by Ceng and Yao (J. Comput. Appl. Math. 214(1):186-201, 2008), Iiduka and Takahashi (Nonlinear Anal. 61(3):341-350, 2005), Jaiboon and Kumam (Nonlinear Anal. 73(5):1180-1202, 2010), Kim (Nonlinear Anal. 73:3413-3419, 2010), Marino and Xu (J. Math. Anal. Appl. 318:43-52, 2006) and Saeidi (Nonlinear Anal. 70:4195-4208, 2009), we introduce a new iterative scheme for finding a common element of the set of solutions of a mixed equilibrium problem for an equilibrium bifunction, the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of some variational inequality problem, and the set of fixed points of a left amenable semigroup of nonexpansive mappings with respect to W-mappings and a left regular sequence of means defined on an appropriate space of bounded real-valued functions of the semigroup S. Furthermore, we prove that the iterative scheme converges strongly to a common element of the above four sets. Our results extend and improve the corresponding results of many others.
MSC: 43A65, 47H05, 47H09, 47H10, 47J20, 47J25, 74G40.