  Self-consistent random phase approximation (RPA) approaches in the
relativistic framework are applied to calculate the isospin symmetry-breaking
corrections $\delta_c$ for the $0^+\to0^+$ superallowed transitions. It is
found that the corrections $\delta_c$ are sensitive to the proper treatments of
the Coulomb mean field, but not so much to specific effective interactions.
With these corrections $\delta_c$, the nucleus-independent $\mathcal{F}t$
values are obtained in combination with the experimental $ft$ values in the
most recent survey and the improved radiative corrections. It is found that the
constancy of the $\mathcal{F}t$ values is satisfied for all effective
interactions employed. Furthermore, the element $V_{ud}$ and unitarity of the
Cabibbo-Kobayashi-Maskawa matrix are discussed.
