  We present a well-designed sample of more than 1000 type 1 quasars at
$3.5<z<5$ and derive UV quasar luminosity functions (QLFs) in this redshift
range. These quasars were selected using the Sloan Digital Sky Survey (SDSS)
imaging data in SDSS Stripe 82 and overlap regions with repeat imaging
observations. They are about one magnitude fainter than those found using the
SDSS single-epoch data. The spectroscopic observations were conducted by the
SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) as one of the BOSS
ancillary programs. This quasar sample reaches $i\sim21.5$ mag and bridges
previous samples from brighter surveys and deeper surveys. We use a
$1/V_\mathrm{a}$ method to derive binned QLFs at $3.6<z<4.0$, $4.0<z<4.5$, and
$4.5<z<4.9$, and use a double-power law model to parameterize the QLFs. We also
combine our data with those in the literature to better constrain the QLFs in
the context of a much wider luminosity baseline. We find that the faint-end and
bright-end slopes of the QLFs in this redshift range are around $-1.7$ and
$-3.7$, respectively, with uncertainties from 0.2$-$0.3 to $>0.5$. The
evolution of the QLFs from $z\sim5$ to $3.5$ can be described by a pure density
evolution model ($\propto10^{kz}$) and the parameter $k$ is similar to that at
$5<z<7$, suggesting a nearly uniform evolution of the quasar density at
$z=3.5-7$.
