MiKM: multi-step inertial Krasnosel’skiǐ–Mann algorithm and its applications

Abstract

In this paper, we first introduce a multi-step inertial Krasnosel’skiǐ–Mann algorithm (MiKM) for nonexpansive operators in real Hilbert spaces. We give the convergence of the MiKM by investigating the convergence of the Krasnosel’skiǐ–Mann algorithm with perturbations. We also establish global pointwise and ergodic iteration complexity bounds of the Krasnosel’skiǐ–Mann algorithm with perturbations. Based on the MiKM, we construct some multi-step inertial splitting methods, including the multi-step inertial Douglas–Rachford splitting method (MiDRS), the multi-step inertial forward–backward splitting method, multi-step inertial backward–forward splitting method and and the multi-step inertial Davis–Yin splitting method. Numerical experiments are provided to illustrate the advantage of the MiDRS over the one-step inertial DRS and the original DRS.