Abstract We introduce a new iteration method and prove strong convergence the-
orems for finding a common element of the set of fixed points of a nonexpansive
mapping and the solution set of monotone and Lipschitz-type continuous Ky Fan
inequality. Under certain conditions on parameters, we show that the iteration se-
quences generated by this method converge strongly to the common element in a real
Hilbert space. Some preliminary computational experiences are reported.