  The recent discovery of high-temperature superconductivity in single-layer
iron selenide has generated significant experimental interest for optimizing
the superconducting properties of iron-based superconductors through the
lattice modification. For simulating the similar effect by changing the
chemical composition due to S doping, we investigate the superconducting
properties of high-quality single crystals of FeSe$_{1-x}$S$_{x}$ ($x$=0, 0.04,
0.09, and 0.11) using magnetization, resistivity, the London penetration depth,
and low temperature specific heat measurements. We show that the introduction
of S to FeSe enhances the superconducting transition temperature $T_{c}$,
anisotropy, upper critical field $H_{c2}$, and critical current density
$J_{c}$. The upper critical field $H_{c2}(T)$ and its anisotropy are strongly
temperature dependent, indicating a multiband superconductivity in this system.
Through the measurements and analysis of the London penetration depth $\lambda
_{ab}(T)$ and specific heat, we show clear evidence for strong coupling two-gap
$s$-wave superconductivity. The temperature-dependence of $\lambda _{ab}(T)$
calculated from the lower critical field and electronic specific heat can be
well described by using a two-band model with $s$-wave-like gaps. We find that
a $d$-wave and single-gap BCS theory under the weak-coupling approach can not
describe our experiments. The change of specific heat induced by the magnetic
field can be understood only in terms of multiband superconductivity.
