  Motivated by the recent experiment in NiGa$_2$S$_4$, the spin-1 Heisenberg
model on a triangular lattice with the ferromagnetic nearest- and
antiferromagnetic third-nearest-neighbor exchange interactions, $J_1 = -(1-p)J$
and $J_3 = pJ, J > 0$, is studied in the range of the parameter $0 \leq p \leq
1$. Mori's projection operator technique is used as a method, which retains the
rotation symmetry of spin components and does not anticipate any magnetic
ordering. For zero temperature several phase transitions are observed. At $ p
\approx 0.2$ the ground state is transformed from the ferromagnetic order into
a disordered state, which in its turn is changed to an antiferromagnetic
long-range ordered state with the incommensurate ordering vector at $p \approx
0.31$. With growing $p$ the ordering vector moves along the line to the
commensurate point $Q_c = (2 \pi /3, 0)$, which is reached at $p = 1$. The
final state with the antiferromagnetic long-range order can be conceived as
four interpenetrating sublattices with the $120\deg$ spin structure on each of
them. Obtained results offer a satisfactory explanation for the experimental
data in NiGa$_2$S$_4$.
