On the existence of solutions of the system of nonlinear quasi-mixed equilibrium problems and their stability of the procedure scheme

In this paper, we provide a simplified proof of the existence of solutions for the system of nonlinear quasi-mixed equilibrium problems studied by Suantai and Petrot (Appl. Math. Lett. 24:308–313, 2011), utilizing Banach’s fixed point theorem. This result remains valid under weaker assumptions through the product space approach. Moreover, we show that the stability analysis of the iterative algorithm proposed in their work (and in (J. Inequal. Appl. 2010:437976, 2010)) contains a flaw, which we address using Ostrowski’s classical result from 1967 (Z. Angew. Math. Mech. 47:77–81, 1967). Finally, we examine and improve the convergence theorem for solving a system of variational inequalities as established by Chang et al. (Appl. Math. Lett. 20:329–334, 2007).