Modified proximal-like extragradient methods for two classes of equilibrium problems in Hilbert spaces with applications

Abstract

The objective of this study is to introduce two new extragradient methods to solve equilibrium problems involving two distinct classes of bifunction. Based on the pseudomonotonicity hypothesis and the specific Lipschitz-type cost bifunction condition, we have shown a weak convergence theorem for the first proposed method. The bifunction is strongly pseudomonotone in the second method, but the step-size rule does not depend on the strongly pseudomonotone constant and Lipschitz-type constants. In contrast, the first convergence result, we prove a strong convergence theorem with the use of a particular class of variable step-size rule sequences. To confirm the validity of the proposed convergence results, we examine two well-known Nash–Cournot equilibrium models for the numerical experiment and show that the competitive advantage of our proposed methods is based on time of execution and number of iterations.