Three novel two-step proximal-like methods for solving equilibrium and fixed point problems in real Hilbert spaces

In this paper, we present three new two-step proximal-like techniques for solving equilibrium problems in real Hilbert spaces. The proposed methods are equivalent to the previously described two-step extragradient method for addressing variational inequality problems in real Hilbert spaces. Rather than using Lipschitz-type constant information or any other line search process, the presented methods use a different step-size rule based on local bifunction information. Under mild constraints such as Lipschitz continuity and pseudomonotonicity, strong convergence findings for the given techniques are well demonstrated. The primary results are used to generate solutions for variational inequalities and fixed point problems. Such findings are useful and contribute to the generalization of previous findings in the literature. Finally, we present a set of numerical experiments to demonstrate the performance and efficacy of the proposed methodologies.