Modified Combined Relaxation Method for General Monotone Equilibrium Problems in Hilbert Spaces

In this paper, we study a class of general monotone equilibrium problems in a real Hilbert space which involves a monotone differentiable bifunction. For such a bifunction, a skew-symmetric type property with respect to the partial gradients is established. We suggest to solve this class of equilibrium problems with the modified combined relaxation method involving an auxiliary procedure. We prove the existence and uniqueness of the solution to the auxiliary variational inequality in the auxiliary procedure. Further, we prove also the weak convergence of the modified combined relaxation method by virtue of the monotonicity and the skew-symmetric type property.Communicated by F. GiannessiHis research was partially supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE, China and by the Dawn Program Foundation in Shanghai.His research was partially supported by a grant from the National Science Council of Taiwan.