Abstract: Under the non-uniform Lipschitz condition (in $t$) of the
generator $g$ with respect to $(y,z)$ for backward stochastic
differential equations (BSDEs), a representation theorem of
generators and a converse theorem of solutions were established for
BSDEs with a finite or an infinite time intervals, which extend some
existing results.

Key words: backward stochastic differential equation (BSDE), non-uniform Lipschitz condition, representation theorem, converse comparison theorems