An iterative method for amenable semigroup and infinite family of non expansive mappings in Hilbert spaces

begin{abstract} In this paper, we introduce an iterative method for amenable semigroup of non expansive mappings and infinite family of non expansive mappings in the frame work of Hilbert spaces. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. The results presented in this paper mainly extend the corresponding results announced by Qin et al. [X. Qin, Y. J. Cho, and S. M. Kang, An iterative method for an infinite family of non-expansive mappings in Hilbert spaces, Bull. Malays. Math. Sci. Soc. 32 (2009) 161-171] and many others. end{abstract}