Two-step inertial Tseng's extragradient method for variational inequality and hierarchical fixed-point problems

We developed novel inertial Tseng-type extragradient methods, in both two-step and one-step formulations based on the Krasnosel'skiǐ-Mann iteration framework. The proposed schemes were designed to solve variational inequality problems (VIPs) and hierarchical fixed-point problems (HFPPs) governed by nonexpansive and quasi-nonexpansive operators in Hilbert spaces. The obtained sequences were shown to converge weakly to a shared solution, together with additional supporting results, and its applicability was demonstrated through an example. This formulation generalizes and brings together several known methods and outcomes in the literature.