Viscosity S-iteration method with inertial technique and self-adaptive step size for split variational inclusion, equilibrium and fixed point problems

Several efficient methods have been developed in the literature for approximating solutions of fixed point and optimization problems. However, the S-iteration process has been shown to outperform many of these existing methods. In this paper, we study the problem of finding the common solution of split variational inclusion problem, equilibrium problem and common fixed point of nonexpansive mappings. We introduce an improved S-iteration method, which combines inertial and viscosity techniques with self-adaptive step size for approximating the solution of the problem in the framework of Hilbert spaces. Moreover, under some mild conditions we prove strong convergence theorem for the proposed algorithm without the knowledge of the operator norm and we apply our result to study split minimization problem, split feasibility problem and relaxed split feasibility problem. Finally, we present some numerical experiments with graphical illustrations to demonstrate the implementability and efficiency of our proposed method in comparison with some existing state of the art methods in the literature.