  We investigate the excitonic fluctuation and its mediated superconductivity
in the quasi one-dimensional three-chain Hubbard model for Ta$_2$NiSe$_5$ known
as a candidate material for the excitonic insulator. In the semiconducting case
and the semimetallic case with a small band-overlapping where one conduction
($c$) band and one valence ($f$) band cross the Fermi level, the excitonic
fluctuation with $\bm{q}=\bm{0}$ is enhanced due to the $c$-$f$ Coulomb
interaction and diverges towards the uniform excitonic order corresponding to
the excitonic insulator. On the other hands, in the semimetallic case with a
large band-overlapping where two $c$ bands and one $f$ band cross the Fermi
level, the non-uniform excitonic fluctuation with $\bm{q}\neq \bm{0}$
corresponding to the nesting vector between the $c$ and $f$ Fermi-surfaces
(FSs) becomes dominant and results in the Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) excitonic order characterized by the condensation of excitons with
finite center-of-mass momentum $\bm{q}$. Near the instability, the largely
enhanced excitonic fluctuations mediate the $c$-$f$ interband Cooper pairs with
finite center-of-mass momentum resulting in the FFLO superconductivity, which
is expected to be realized in the semimetallic Ta$_2$NiSe$_5$ under high
pressure.
