A Geometric S‐Iteration Method for Approximating Common Best Proximity Points of Proximally Mean Nonexpansive Mappings in Banach Spaces

ABSTRACT In this article, we introduce a new three‐step iteration method, called the geometric S‐iteration method, which combines a geometrically constructed iteration scheme with the S‐iteration method. This scheme approximates a common best proximity point of three distinct nonself‐mappings in Banach spaces. Under mild conditions, we prove that any sequence generated by the proposed method converges strongly to common best proximity points for three nonself‐mappings that form pairwise proximally mean nonexpansive pairs. As a consequence, we propose the geometric S‐iteration method for approximating a common fixed point of three self‐mappings that form pairwise mean nonexpansive pairs. Moreover, we present a numerical experiment to support our results and compare the convergence rate of the proposed method against existing algorithms. Finally, we apply our results to the solvability of a model describing the spread of a specific virus.