Some iterative algorithms for Reich-Suzuki nonexpansive mappings and relaxed $(\alpha,k)$-cocoercive mapping with applications to a fixed point and optimization problems

In this paper, we propose an iterative method for finding the common element of the set of fixed points of Reich-Suzuki nonexpansive mappings and the set of solutions of the variational inequalities problems in the framework of Hilbert spaces. In addition, we establish convergence results for these proposed iterative methods under some mild conditions. Furthermore, we establish analytically and numerically that our newly proposed iterative method converges to a common element of the set of fixed points of a Reich-Suzuki nonexpansive mapping and the set of solutions of the variational inequalities problems faster compared to some well-known iterative methods in the literature. Finally, we apply our proposed iterative method to approximate the solution of a convex minimization problem. The results obtained in this paper improve, extend and unify some related results in the literature.