A new algorithm for approximating solutions of the common variational inclusion

Abstract

This paper studies the common variational inclusion problem in real Hilbert spaces. To solve this problem, we propose a new accelerated approach with two initial parameter steps and establish a strong convergence theorem. Our scheme combines the viscosity approximation method with Tseng’s forward backward-forward splitting method and uses self-adaptive step sizes. We simultaneously compute the inertial extrapolation and viscosity approximation at the first step of each iteration. We show that the iterative method converges strongly under conventional and appropriate assumptions. We also study some applications to the common minimum point problems, split feasibility problems, and to the least absolute selection and shrinkage operators (LASSO). Finally, we present two numerical results in Hilbert space and an application to the LASSO problem in order to illustrate the convergence analysis of the considered methods as well as compare our results to the related ones introduced by Cholamjiak et al. (J. Sci. Comput., 88(85), 2021) and Gibali and Thong (Calcolo, 55(49), 2018).