  A two-dimensional superconductor (SC) on surfaces of topological insulators
(TIs) is a mixture of s-wave and helical p-wave components when induced by
s-wave interactions, since spin and momentum are correlated. On the basis of
the Abrikosov-Gor'kov theory, we reveal that unconventional SCs on the surfaces
of TIs are stable against time-reversal symmetric (TRS) impurities within a
region of small impurity concentration. Moreover, we analyze the stability of
the SC on the surfaces of TIs against impurities beyond the perturbation theory
by solving the real-space Bogoliubov-de Gennes equation for an effective
tight-binding model of a TI. We find that the SC is stable against strong TRS
impurities. The behaviors of bound states around an impurity suggest that the
SC on the surfaces of TIs is not a topological SC.
