Two new extragradient methods for solving pseudomonotone the equilibrium problem in Hilbert spaces

. This paper introduces two new extragradient methods designed to solve pseudomonotone equilibrium problems subject to a Lipschitz-type condition. These methods incorporate a variable stepsize criterion that dynamically adjusts with each iteration based on prior iterations. A distinguishing feature of these methods is their independence from prior knowledge of Lipschitz-type constants or any line-search method. The convergence theorems for the proposed methods are established under mild conditions, without requiring the knowledge of Lipschitz-type constants. Additionally, the paper includes several investigations demonstrating the numerical efficacy of the methods and facilitating comparisons with other approaches. This paper contributes to the advancement of computational methods for addressing pseudomonotone equilibrium problems across various applications.