Accelerated non-monotonic explicit proximal-type method for solving equilibrium programming with convex constraints and its applications

The main objective of this study is to introduce a new two-step proximal-type method to solve equilibrium problems in a real Hilbert space. This problem is a general mathematical model and includes a number of mathematical problems as a special case, such as optimization problems, variational inequalities, fixed point problems, saddle time problems and Nash equilibrium point problems. A new method is analogous to the famous two-step extragradient method that was used to solve variational inequality problems in a real Hilbert space established previously. The proposed iterative method uses an inertial scheme and a new non-monotone stepsize rule based on local bifunctional values rather than any line search method. A strong convergence theorem for the constructed method is proven by letting mild conditions on a bifunction. These results are being used to solve fixed point problems as well as variational inequalities. Finally, we considered two test problems, and the computational performance was presented to show the performance and efficiency of the proposed method