Abstract
In this paper, we suggest and analyze an iteration method studied by Takahashi and Toyoda (J. Optim. Theory Appl. 118(2):417-428, 2003) for finding a common element of a set of fixed points of nonexpansive mappings and a set of solutions of variational inequalities with inverse strongly monotone mappings under some mild conditions. The strong convergence is proved and positive answer is given to an open question posed by Takahashi and Toyoda.