  We show that the {\it gapped} triplet superconductivity, i.e., a triplet
superconductor with triplet order parameter, can be realized in strong
spin-orbit-coupled quantum wells in proximity to $s$-wave superconductor. It is
revealed that with the singlet order parameter induced from the superconducting
proximity effect, in quantum wells, not only can the triplet pairings arise due
to the spin-orbit coupling, but also the triplet order parameter can be induced
due to the repulsive effective electron-electron interaction, including the
electron-electron Coulomb and electron-phonon interactions. This is a natural
extension of the work of de Gennes, in which the repulsive-interaction-induced
singlet order parameter arises in the normal metal in proximity to $s$-wave
superconductor [Rev. Mod. Phys. {\bf 36}, 225 (1964)]. Specifically, we derive
the effective Bogoliubov-de Gennes equation, in which the self-energies due to
the effective electron-electron interactions contribute to the singlet and
triplet order parameters. It is further shown that for the singlet order
parameter, it is efficiently suppressed due to this self-energy
renormalization; whereas for the triplet order parameter, it is the $p$-wave
($p_x\pm ip_y$) one with the ${\bf d}$-vector parallel to the effective
magnetic field due to the spin-orbit coupling. Finally, we perform the
numerical calculation in InSb (100) quantum wells. Specifically, we reveal that
the Coulomb interaction is much more important than the electron-phonon
interaction at low temperature. Moreover, it shows that with proper electron
density, the minimum of the renormalized singlet and the maximum of the induced
triplet order parameters are comparable, and hence can be experimentally
distinguished.
