Strong Convergence Results for Asymptotically $G$-Nonexpansive Mappings in Hilbert Spaces with Graphs: Hieu Trung Nguyen, Cao Pham Cam Tu

In this paper, by combining two modified Ishikawa iteration processes and two modified $S$-iteration processes with the shrinking projection method, we propose four new hybrid iteration processes for two asymptotically $G$-nonexpansive mappings. We also prove some strong convergence theorems for common fixed points of  two asymptotically $G$-nonexpansive mappings in Hilbert spaces with graphs. These  theorems  are the extension  and  improvement of  certain main results  in [H.A. Hammad, W. Cholamjiak, D. Yambangwai, and H. Dutta, A modified shrinking projection methods for numerical reckoning fixed points of $G$-nonexpansive mappings in Hilbert spaces with graphs, Miskolc Math. Notes 20(2) (2019) 941-- 956].  In addition, we provide a  numerical example for supporting obtained results.