Regularization iterative method of bilevel form for equilibrium problems in Hilbert spaces

We study the regularization method for a monotone equilibrium problem and propose a new iterative method for solving the problem with a Lipschitz-type condition in a Hilbert space. The method is designed by the proximal mapping incorporated with regularization terms. The method uses variable stepsizes which are taken by simple rules without a linesearch procedure. The strong convergence of iterative sequences generated by the method is established under some suitable conditions imposed on control parameters. It turns out that the obtained asymptotic solution by the method is the solution of an equilibrium problem whose constraint is the solution set of the considered equilibrium problem. The computational effectiveness of the method is illustrated by several numerical examples.