Two generalized non-monotone explicit strongly convergent extragradient methods for solving pseudomonotone equilibrium problems and applications

The main objective of this paper is to introduce two new proximal-like methods to solve the equilibrium problem in a real Hilbert space. The equilibrium problem is a general mathematical problem that unites several useful mathematical problems, including optimization problems, variational inequalities, fixed-point problems, saddle point problems, complementary problems, and Nash equilibrium problems. Both new methods are analogous to the well-known extragradient method, which has been used in the literature to solve variational inequality problems. The proposed methods make use of a non-monotone variable step size rule that is revised for each iteration and is determined mainly by previous iterations. The advantage of these methods is that they can be used without prior knowledge of Lipschitz-type constants or any line-search method. By allowing for some mild condition, the strong convergence of both methods is established. Numerical studies are presented to demonstrate the computational behavior of new methods and to compare them to other existing methods.