  In this paper an explicit algorithm is proposed for solving an equilibrium
problem whose associated bifunction is pseudomonotone and satisfies a
Lipschitz-type condition. Contrary to many algorithms, our algorithm is done
without using explicitly the Lipschitz constants of bifunction although its
convergence is obtained under such that condition. The introduced method is a
form of proximal-like method whose steplengths are explicitly generated at each
iteration without using any linesearch procedure. First we prove the
convergence of the algorithm, and after we establish its $R$-linear rate of
convergence under the assumption of strong pseudomonotonicity of the
bifunction. Afterwards several numerical results are displayed to illustrate
and to compare the behavior of the new algorithm with other ones.
