  By incorporating a new matrix splitting and the momentum acceleration into
the relaxed-based matrix splitting (RMS) method \cite{soso2023}, a
generalization of the RMS (GRMS) iterative method for solving the generalized
absolute value equations (GAVEs) is proposed. Unlike some existing methods, by
using the Cauchy's convergence principle, we give some sufficient conditions
for the existence and uniqueness of the solution to the GAVEs and prove that
our method can converge to the unique solution of the GAVEs. Moreover, we can
obtain a few new and weaker convergence conditions for some existing methods.
Preliminary numerical experiments show that the proposed method is efficient.
