Very recently, Moudafi (2011) introduced an algorithm with weak convergence for the split common fixed‐point problem. In this paper, we will continue to consider the split common fixed‐point problem. We discuss the strong convergence of the viscosity approximation method for solving the split common fixed‐point problem for the class of quasi‐nonexpansive mappings in Hilbert spaces. Our results improve and extend the corresponding results announced by many others.

In this paper, we propose a viscosity approximation method to solve the split common fixed point problem and consider the bounded perturbation resilience of the proposed method in general Hilbert spaces. Under some mild conditions, we prove that our algorithms strongly converge to a solution of the split common fixed point problem, which is also the unique solution of the variational inequality problem. Finally, we show the convergence and effectiveness of the algorithms by two numerical examples.