Solving equilibrium and fixed-point problems in Hilbert spaces: a new strongly convergent inertial subgradient extragradient method

This article introduces a new subgradient extragradient method combined with an inertial scheme that utilizes different step size formulas to generate the iterative sequence. The study aims to find an approximate common solution to pseudomonotone equilibrium problems and fixed-point problems using a demicontractive mapping in real Hilbert spaces. The proposed methods incorporate a self-adaptive step size criterion, both monotonic and non-monotonic, which avoids the need to estimate Lipschitz-type constants. Strong convergence results for the iterative sequences generated by these methods are established under suitable conditions. Additionally, the approaches are applied to solve variational inequality and fixed-point problems. Numerical examples are provided to illustrate the effectiveness and advantages of the proposed methodologies compared to existing methods in the literature.