On Strong Convergence of a Halpern-Mann's Type Iteration with Perturbations for Common Fixed Point and Generalized Equilibrium Problems

Manakorn Wattanataweekul

Abstract

We establish strong convergence of a sequence generated by Halpern-Mann's type iteration with perturbation for approximating a common element of the set of fixed points of a countable family of quasi-nonexpansive mappings and the set of solutions of a generalized equilibrium problem in a real Hilbert space. With an appropriate setting, some results for solving the minimum-norm problems are also included. Finally, we consider the modified viscosity method of a countable family of nonexpansive mappings. Our results extend and improve the corresponding results due to Chuang et al., \cite{CLT2013}, Duan and He \cite{DH2014}, Nilsrakoo and Saejung \cite{NS2010}, Wang \cite{W2013}, and many others.