In this paper, by combining the shrinking projection method with a modified inertial Siteration process, we introduce a new inertial hybrid iteration for two asymptotically Gnonexpansive 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. | Strong convergence of inertial hybrid iteration for two asymptotically G-nonexpansive mappings in hilbert space with graphs
