Convergence of iterative algorithm for G-nonexpansive mapping with digraph: application on G-variational inequality problem and signal recovery

The Halpern iteration method is among the most extensively studied approaches in the literature concerning the approximation of fixed points of G-nonexpansive mappings. In recent years, numerous extensions and modifications of this method have been proposed in various abstract settings. Among these, the formulation introduced by Tiammee et al., which adapts Halpern's iterative scheme to the context of Hilbert spaces endowed with a digraph structure, has garnered particular attention. This study investigates the convergence properties of G-nonexpansive mappings on a Hilbert space equipped with a digraph, thereby extending and enhancing the results previously obtained by Suzuki, Pang & Naraghirad and Tiammee et al. Moreover, a numerical example implemented in MATLAB R2016a is provided to illustrate and support the theoretical findings. Finally, by employing the concept of the G-variational inequality, as introduced by Kangtunyakarn in 2020, we illustrate the effectiveness of the proposed algorithm in the field of signal recovery, thereby highlighting its potential for real-world implementation.