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 fixed-point problems. Numerical experiments confirm that these methods offer improved efficiency and performance compared to existing approaches in the literature.