Two new algorithms are proposed in this paper for solving an equilibrium problem whose associated bifunction is monotone and satisfies a Lipschitz-type condition in a Hilbert space. In the first algorithm, it is assumed that the value of the Lipschitz constant of the bifunction is known while in the second one the prior knowledge of this constant is not explicitly requested. The proposed algorithms are constructed around the proximal-like mapping and the regularized method and use some new variable stepsize rules. Strong convergence theorems are established under some mild conditions imposed on bifunction and control parameters. Finally several numerical results are provided to illustrate the behavior of the new algorithms and to compare them to well-known algorithms.