  Starting from the relativistic form of the Bonn potential as a bare
nucleon-nucleon interaction, the full Relativistic Brueckner-Hartree-Fock
(RBHF) equations are solved for finite nuclei in a fully self-consistent basis.
This provides a relativistic ab initio calculation of the ground state
properties of finite nuclei without any free parameters and without three-body
forces. The convergence properties for the solutions of these coupled equations
are discussed in detail at the example of the nucleus $^{16}$O. The binding
energies, radii, and spin-orbit splittings of the doubly magic nuclei $^{4}$He,
$^{16}$O, and $^{40}$Ca are calculated and compared with the earlier RBHF
calculated results in a fixed Dirac Woods-Saxon basis and other
non-relativistic ab initio calculated results based on pure two-body forces.
