Halpern’s type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points - Chuang, Chih-Sheng; Lin, Lai-Jiu; Takahashi, Wataru (Journal of global optimization, 2013)
In this paper, we consider an iteration process of Halpern’s type for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points for a quasi-nonexpansive mapping with perturbation in a Hilbert space and then prove a strong convergence theorem for such iterations. Using this result, we obtain new strong convergence theorems in a Hilbert space. In particular, we solve partially an open problem posed by Kurokawa and Takahashi (Nonlinear Anal 73:1562–1568, 2010 ) concerning Halpern’s iterations.