In this paper, we present a new hybrid extragradient algorithm for finding a common element of the fixed point problem for a demicontractive mapping and the split equilibrium problem for a pseu-domonotone and Lipschitz-type continuous bifunction. By using a new technique of choosing the step size of the proposed method, our algorithms do not need any prior information of the operator norm. In fact, we propose an inertial type algorithm in order to accelerate its convergence rate and then prove strong convergence theorem of our proposed method under some control conditions. Moreover, we give some numerical experiments to support our main results.