  We study two types of linear instabilities in Einstein-Born-Infeld-scalar
field theory and show that small Born-Infeld (BI) black holes suffer from
superradiant instability within a particular range of frequencies; the larger
the BI coupling parameter, the slower the growth rate of the instability. It is
predicted that a small BI black hole decays into a small hairy black hole that
has a charged scalar condensate floating near the horizon. We numerically show
that there is a phase transition between hairy and BI black holes. The metric
solution shows the existence of a hairy black hole for $T<T_{c}$ and a BI black
hole for $T>T_{c}$. In addition, the BI black hole will suffer from tachyonic
mass, resulting in a near horizon scalar condensation instability, which is
related to a planar hairy black hole. We show that the BI coupling parameter
has a direct relation to the critical temperature in the small hairy black hole
framework. This is not true for planar hairy black holes where a larger BI
coupling parameter leads to a smaller critical temperature. Finally, it is
shown that a small hairy black hole is stable at the critical temperature so it
can be considered as the end point of superradiant instability.
