J. Fixed Point Theory Appl. (2019) 21:44 https://doi.org/10.1007/s11784-019-0681-3 Published online March 6, 2019 c  Springer Nature Switzerland AG 2019 Journal of Fixed Point Theory and Applications Convergence analysis for a new two-step iteration process for G-nonexpansive mappings with directed graphs Tanakit Thianwan and Damrongsak Yambangwai Abstract. In this paper, we introduce and study convergence analysis of a new two-step iteration process when applied to class of G-nonexpansive mappings. Weak and strong convergence theorems are established for the new two-step iterative scheme in a uniformly 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 numer- ical experiment for supporting our main results and comparing rate of convergence of the proposed method with the Ishikawa iteration and the modified S-iteration. Mathematics Subject Classification. 47H10, 47H09, 47E10. Keywords. G-nonexpansive mapping, weak and strong convergence, common fixed point, uniformly convex Banach space, directed graph. 1. Introduction and preliminaries At present, 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 self map on a set admits a fixed point. Among all the results in fixed point theory, the Banach contraction principle (see [6]) in metric fixed point theory is the most celebrated one due to its simplicity and ease of application in major areas of mathematics. Following the Banach con- traction principle, Boyd and Wong [9] investigated the fixed point results in nonlinear contraction mappings. Subsequently, many authors extended and generalized this fixed point theorem in different directions, in particular, by Reich [14]. In 2008, by combination of the concepts in fixed point theory and graph theory, Jachymski [10] generalized the Banach contraction prin- ciple in a complete metric space endowed with a directed graph. In 2012, Aleomraninejad et al. [5] presented some iterative scheme for G-contraction and G-nonexpansive mappings in a Banach space with a graph. In 2015, Al- furaidan and Khamsi [2] defined the concept of G-monotone nonexpansive