A Viscosity Implicit Midpoint Rule for Reich–Suzuki-type Nonexpansive Mappings

Abstract

In this paper, we study a viscosity implicit midpoint rule for approximating fixed points of Reich–Suzuki-type nonexpansive mappings in a real Hilbert space. Motivated by the viscosity midpoint methods introduced by Vaishnav et al. and Hu for quasi-nonexpansive and nonexpansive mappings, we extend the framework to a broader class of nonlinear mappings. Under suitable assumptions on the control parameters and involved operators, we establish strong convergence results for the proposed iterative scheme. The obtained limit is characterized as the unique solution of an associated variational inequality problem.