  We have measured mesoscopic superconducting Au$_{0.7}$In$_{0.3}$ rings
prepared by e-beam lithography and sequential deposition of Au and In at room
temperature followed by a standard lift-off procedure. In samples showing no
Little-Parks resistance oscillations, highly unusual double resistance
anomalies, two resistance peaks found near the onset of superconductivity, were
observed. Although resistance anomaly featuring a single resistance peak has
been seen in various mesoscopic superconducting samples, double resistance
anomalies have never been observed previously. The dynamical resistance
measurements suggest that there are two critical currents in these samples. In
addition, the two resistance peaks were found to be suppressed at different
magnetic fields. We attribute the observed double resistance anomalies to an
underlying phase separation in which In-rich grains of intermetallic compound
of AuIn precipitate in a uniform In-dilute matrix of Au$_{0.9}$In$_{0.1}$. The
intrinsic superconducting transition temperature of the In-rich grains is
substantially higher than that of the In-dilute matrix. The suppression of the
conventional Little-Parks resistance oscillation is explained in the same
picture by taking into consideration a strong variation in the $T_c$ of the
In-rich grains. We also report the observation of an unusual
magnetic-field-induced metallic state with its resistance higher than the
normal-state resistance, referred to here as excessive resistance, and an h/2e
resistance oscillation with the amplitude of oscillation depends extremely
weakly on temperature.
