  Mass, metallicity, and star formation rate (SFR) of a galaxy are crucial
parameters in understanding galaxy formation and evolution. However, the
relation among these is still a matter of debate for luminous infrared
galaxies, which carry a bulk of SFR budget of the universe at $z\sim1$. We have
investigated the relation among stellar mass, gas-phase oxygen abundance, and
SFR of AKARI-detected mid-IR galaxies at $z\sim0.88$ in the AKARI NEP deep
field. We observed about 350 AKARI sources with Subaru/FMOS NIR spectrograph,
and detected secure and expected H$\alpha$ emission lines from 25 and 44
galaxies, respectively. The SFR of our sample is almost constant ($\sim
25M_{\odot}/yr$) over the stellar mass range of our sample. Compared with
main-sequence (MS) galaxies at a similar redshift range, the average SFR of our
detected sample is comparable for massive galaxies
($\sim10^{10.58}~M_{\odot}$), while higher by $\sim$0.6dex for less massive
galaxies ($\sim 10^{10.05}~M_{\odot}$). We measure metallicities from the
[NII]/H$\alpha$ emission line ratio.
  We find that the mass-metallicity relation of our individually measured
sources agrees with that for optical-selected star-forming galaxies at
$z\sim0.1$, while metallicities of stacked spectra agree with that of MS
galaxies at $z\sim0.78$. Considering high SFR of individually measured sources,
FMR of the IR galaxies is different from that at $z\sim0.1$. However, on the
mass-metallicity plane, they are consistent with the MS galaxies, highlighting
higher SFR of the IR galaxies. This suggests the evolutionary path of our IR
galaxies is different from that of MS galaxies. A possible physical
interpretation includes that the star-formation activities of IR galaxies at
$z\sim0.88$ in our sample are enhanced by interaction and/or merger of
galaxies, but the inflow of metal-poor gas is not yet induced, keeping the
metallicity intact.
