Strong and Weak Convergence Theorems for Generalized Mixed Equilibrium Problem with Perturbation and Fixed Pointed Problem of Infinitely Many Nonexpansive Mappings

Very recently, Plubtieng and Kumam [S. Plubtieng, P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math. 224 (2009) 614-621] proposed an iterative algorithm for finding a common solution of a variational inequality problem for an inverse-strongly monotone mapping and a fixed point problem of a countable family of nonexpansive mappings, and obtained a weak convergence theorem. In this paper, based on Plubtieng-Kumam's iterative algorithm we introduce a new iterative algorithm for finding a common solution of a generalized mixed equilibrium problem with perturbation and a fixed point problem of a countable family of nonexpansive mappings in a Hilbert space. We first derive a strong convergence theorem for this new algorithm under appropriate assumptions and then consider a special case of this new algorithm. Moreover, we establish a weak convergence theorem for this special case under some weaker assumptions. Such a weak convergence theorem unifies, improves and extends Plubtieng-Kumam's weak convergence theorem. It is worth pointing out that the proof method of strong convergence theorem is very different from the one of weak convergence theorem.