A Strongly Convergent Hybrid Method to Unify Split Generalized Mixed Equilibrium and Fixed Point Problems

The aim of this paper is twofold, we propose an extension to the split generalized mixed equilibrium problem firstly and introduce an iterative method based on a hybrid extragradient method. The goal is to efficiently find a common solution for both the split generalized mixed equilibrium problem and the fixed point problem concerning a nonexpansive mapping within the context of real Hilbert spaces. We conduct a thorough analysis of the proposed iterative method and establish a strong convergence theorem under certain mild conditions. Moreover, we present various implications derived from our main result and conduct numerical experiments to validate our findings. Our outcomes represent a substantial expansion and generalization of existing iterative methods and results within this field.