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.
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