  We study the Cauchy problem for a generalized derivative nonlinear
Schr\"odinger equation with the Dirichlet boundary condition. We establish the
local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions
are constructed as a limit of approximate solutions by a method independent of
a compactness argument. We also discuss the global existence of solutions in
the energy space $H^1$.
