  Among the potential applications of topological insulators, we theoretically
study the coexistence of proximity-induced ferromagnetic and superconducting
orders in the surface states of a 3-dimensional topological insulator. The
superconducting electron-hole excitations can be significantly affected by the
magnetic order induced by a ferromagnet. In one hand, the surface state of the
topological insulator, protected by the time-reversal symmetry, creates a
spin-triplet and, on the other hand, magnetic order causes to renormalize the
effective superconducting gap. We find Majorana mode energy along the
ferromagnet/superconductor interface to sensitively depend on the magnitude of
magnetization $m_{zfs}$ from superconductor region, and its slope around
perpendicular incidence is steep with very low dependency on $m_{zfs}$. The
superconducting effective gap is renormalized by a factor $\eta(m_{zfs})$, and
Andreev bound state in
ferromagnet-superconductor/ferromagnet/ferromagnet-superconductor (FS/F/FS)
Josephson junction is more sensitive to the magnitude of magnetizations of FS
and F regions. In particular, we show that the presence of $m_{zfs}$ has a
noticeable impact on the gap opening in Andreev bound state, which occurs in
finite angle of incidence. This directly results in zero-energy Andreev state
being dominant. By introducing the proper form of corresponding Dirac spinors
for FS electron-hole states, we find that via the inclusion of $m_{zfs}$, the
Josephson supercurrent is enhanced and exhibits almost abrupt crossover curve,
featuring the dominant zero-energy Majorana bound states.
