We introduce a new iteration algorithm for solving the Ky Fan inequality
over the fixed point set of a nonexpansive mapping, where the cost bifunction is
monotone without Lipschitz-type continuity. The algorithm is based on the idea
of the ergodic iteration method for solving multi-valued variational inequality
which is proposed by Bruck and the auxiliary problem principle for equilibrium
problems. By choosing suitable regularization parameters, we also present the
convergence analysis in detail for the algorithm and give some illustrative examples.
Keyword: Ky Fan inequalities, monotonicity, fixed points, nonexpansive mappings.