A faster iterative scheme for common fixed points of $G$-nonexpansive mappings via directed graphs: application in split feasibility problems

We have suggested a new modified iterative scheme for approximating a common fixed point of two $G$-nonexpansive mappings. Our approach was based on an iterative scheme in the context of Banach spaces via directed graphs. First, we proved a weak convergence theorem using the Opial's property of the underlying space. A weak convergence result without the Opial's property was also given. After this, we established several strong convergence theorems using various mild conditions. We also carried out some numerical simulations to examine the main techniques. Eventually, we obtained an application of our result to solve split feasibility problems (SFP) in the context of $G$-nonexpansive mappings.