TẠP CHÍ KHOA HỌC TRƯỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINH Tập 17, Số 6 (2020): 1137-1149 HO CHI MINH CITY UNIVERSITY OF EDUCATION JOURNAL OF SCIENCE Vol. 17, No. 6 (2020): 1137-1149 ISSN: 1859-3100 Website: http://journal.hcmue.edu.vn 1137 Research Article * STRONG CONVERGENCE OF INERTIAL HYBRID ITERATION FOR TWO ASYMPTOTICALLY G-NONEXPANSIVE MAPPINGS IN HILBERT SPACE WITH GRAPHS Nguyen Trung Hieu * , Cao Pham Cam Tu Faculty of Mathematics Teacher Education, Dong Thap University, Cao Lanh City, Viet Nam * Corresponding author: Nguyen Trung Hieu – Email: ngtrunghieu@dthu.edu.vn Received: April 07, 2020; Revised: May 08, 2020; Accepted: June 24, 2020 ABSTRACT In this paper, by combining the shrinking projection method with a modified inertial S- iteration process, we introduce a new inertial hybrid iteration for two asymptotically G- nonexpansive mappings and a new inertial hybrid iteration for two G-nonexpansive mappings in Hilbert spaces with graphs. We establish a sufficient condition for the closedness and convexity of the set of fixed points of asymptotically G-nonexpansive mappings in Hilbert spaces with graphs. We then prove a strong convergence theorem for finding a common fixed point of two asymptotically G-nonexpansive mappings in Hilbert spaces with graphs. By this theorem, we obtain a strong convergence result for two G-nonexpansive mappings in Hilbert spaces with graphs. These results are generalizations and extensions of some convergence results in the literature, where the convexity of the set of edges of a graph is replaced by coordinate-convexity. In addition, we provide a numerical example to illustrate the convergence of the proposed iteration processes. Keywords: asymptotically G-nonexpansive mapping; Hilbert space with graphs; inertial hybrid iteration 1. Introduction and preliminaries In 2012, by using the combination concepts between the fixed point theory and the graph theory, Aleomraninejad, Rezapour, and Shahzad (2012) introduced the notions of G- contractive mapping and G-nonexpansive mapping in a metric space with directed graphs and stated the convergence for these mappings. After that, there were many convergence results for G-nonexpansive mappings by some iteration processes established in Hilbert spaces and Banach spaces with graphs. In 2018, Sangago, Hunde, and Hailu (2018) introduced the notion of an asymptotically G-nonexpansive mapping and proved the weak and strong convergence of a modified Noor iteration process to common fixed points of a finite family of asymptotically G-nonexpansive mappings in Banach spaces with graphs. After that some authors proposed a two-step iteration process for two asymptotically G- nonexpansive mappings 1 2 , : TT  (Wattanataweekul, 2018) and a three-step iteration process for three asymptotically G-nonexpansive mappings 1 2 3 , , : TTT  (Wattanataweekul, 2019) as follows: Cite this article as: Nguyen Trung Hieu, & Cao Pham Cam Tu (2020). Strong convergence of inertial hybrid iteration for two asymptotically G-nonexpansive mappings in Hilbert space with graphs. Ho Chi Minh City University of Education Journal of Science, 17(6), 1137-1149.