  For an abstract elementary class $\mathbf{K}$ and a cardinal $\lambda \geq
LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions,
categoricity in two succesive cardinals, almost stability for
$\lambda^+$-minimal types and continuity of splitting in $\lambda$, that
stability in $\lambda$ is equivalent to the existence of a model in
$\lambda^{++}$. The forward direction holds without any cardinal or
categoricity assumptions, this result improves both [Vas18b, 12.1] and [MaYa24,
3.14].
  Moreover, we prove a categoricity theorem for abstract elementary classes
with weak amalgamation and tameness under mild structural assumptions in
$\lambda$. A key feature of this result is that we do not assume amalgamation
or arbitrarily large models.
