Solving equilibrium and fixed-point problems in Hilbert spaces: A class of strongly convergent Mann-type dual-inertial subgradient extragradient methods

This paper aims to enhance the convergence rate of the extragradient method by carefully selecting inertial parameters and employing an adaptive step-size rule. To achieve this, we introduce a new class of Mann-type subgradient extragradient methods that utilize a dual- inertial framework, applying distinct step-size formulas to generate the iterative sequence. Our main objective is to approximate a common solution to pseudomonotone equilibrium and fixed-point problems involving demicontractive mappings in real Hilbert spaces. The proposed methods integrate self-adaptive, monotone, and non-monotone step-size criteria, thereby eliminating the need to estimate Lipschitz-type constants. Under suitable conditions, we establish strong convergence theorems for the resulting iterative sequences. Moreover, we demonstrate the applicability of the proposed methods to both variational inequality and fixedpoint problems. Numerical experiments confirm that these methods offer improved efficiency and performance compared to existing approaches in the literature.