Numerical Algorithms https://doi.org/10.1007/s11075-019-00768-w ORIGINAL PAPER A new modified three-step iteration method for G-nonexpansive mappings in Banach spaces with a graph Damrongsak Yambangwai 1 · Sukanya Aunruean 1 · Tanakit Thianwan 1 Received: 3 April 2018 / Accepted: 27 June 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019 Abstract In the present article, we establish weak and strong convergence theorems of a new modified three-step iteration method for three G-nonexpansive mappings in a uni- formly convex Banach space with a directed graph. Moreover, weak convergence theorem without making use of the Opial’s condition is proved. We also show the numerical experiment for supporting our main results and comparing rate of conver- gence of the new modified three-step iteration with the three-step Noor iteration and the SP iteration. We also provide some numerical examples to illustrate the conver- gence behavior and advantages of the proposed method. Furthermore, we apply our results to find solutions of constrained minimization problems and split feasibility problems. Keywords G-nonexpansive mapping · Three-step Noor iteration · SP iteration · Uniformly convex Banach space · Directed graph Mathematics Subject Classification (2010) 47H10 · 47H09 · 47E10 1 Introduction Fixed point theory is an immensely active area of research due to its applications in multiple fields. It addresses the results which state that, under certain conditions, a  Tanakit Thianwan tanakit.th@up.ac.th Damrongsak Yambangwai damrong.sut@gmail.com Sukanya Aunruean lynn.sukanya@gmail.com 1 Department of Mathematics, School of Science, University of Phayao, Phayao, 56000, Thailand