  This paper is devoted to the existence, uniqueness and comparison theorem on
unbounded solutions of one-dimensional backward stochastic differential
equations (BSDEs) with sub-quadratic generators, where the terminal time is
allowed to be finite or infinite. We first establish existence of the unbounded
solutions for this kind of BSDEs with generator $g$ satisfying a time-varying
one-sided linear growth in the first unknown variable $y$ and a time-varying
sub-quadratic growth in the second unknown variable $z$. Then, the uniqueness
and comparison theorem of the unbounded solutions for this kind of BSDEs are
proved under a time-varying extended convexity assumption. These results
generalized those obtained in \cite{12} to the general time interval BSDEs.
Finally, several sufficient conditions ensuring that the uniqueness holds are
put forward and verified via some innovative ideas, which are explored at the
first time even though for the case of finite time interval BSDEs.
