Viscosity approximations by the shrinking projection method in Hilbert spaces

We consider viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, and the modified shrinking projection method proposed by Qin, Cho, Kang, and Zhou, for finding a common fixed point of countably many nonlinear mappings, and we prove strong convergence theorems which extend some known results. We also consider semigroups of nonlinear mappings and obtain strong convergence of iterative schemes which approximate a common fixed point of the semigroup under certain conditions.