Before revision:  The existence of the equilibria in the case of the sufficiently large number of banks  is guaranteed by the solvability for the coupled Riccati equations as the number of banks goes to infinity in each group.
After revision:  The existence of the equilibria in the case of the sufficiently large number of banks are sufficiently large is guaranteed by the solvability for the coupled Riccati equations as the number of banks goes to infinity in each group.