A novel iterative algorithm for solving a class of variational inequality problems: revisions to ‘On the intermixed method for mixed variational inequality problems: another look and some corrections’

Abstract

In this paper, we investigate the combination of mixed variational inequality problems and propose a novel algorithm that integrates the subgradient extragradient method with the Krasnoselskii-Mann-type method. This algorithm aims to find a common solution for the combination of mixed variational inequality problems and two sets of variational inequality problems in a real Hilbert space. Importantly, our work addresses and corrects certain errors in Saejung’s study (J. Inequal. Appl. 2024:42, 2024) concerning mixed variational inequality problems, which were initially introduced in Khuangsatung and Kangtunyakarn’s research (J. Inequal. Appl. 2023:1, 2023). We establish the weak and strong convergence result for the proposed algorithm under suitable conditions. As an application, we apply our main results to the split feasibility problem and the constrained convex minimization problem. Finally, a numerical example is provided to demonstrate the convergence behavior of the algorithm.