We consider a second order dynamical system for solving equilibrium problems in Hilbert spaces. Under mild conditions, we prove existence and uniqueness of strong global solution of the proposed dynamical system. We establish the exponential convergence of trajectories under strong pseudo monotonicity and Lipschitz-type conditions. We then investigate a discrete version of the second order dynamical system, which leads to a fixed point type algorithm with inertial effect and relaxation. The linear convergence of this algorithm is established under suitable conditions on parameters. Finally some numerical experiments are reported confirming the theoretical results.

We prove the finite convergence of the sequences generated by some extragradient-type methods solving variational inequalities under the weakly sharp condition of the solution set. In addition, we provide estimations for the number of iterations to guarantee the sequence converges to a point in the solution set and prove that these estimations are optimal. Numerical examples are presented to illustrate the theoretical results.