  This paper studies the initial-boundary-value problem (IBVP) of a nonlinear
Schr\"odinger equation posed on a strip domain $\mathbb{R}\times[0,1]$ with
non-homogeneous Dirichlet boundary conditions. For any $s\ge0$, if the initial
data $\varphi(x,y)$ is in Sobolev space $H^s(\mathbb{R}\times[0,1])$ and the
boundary data $h(x,t)$ is in $$ {\cal H}^s (\mathbb{R} ) = \left \{ h (x, t)
\in L^2 ( \mathbb{R}^2 ) \ \big | \ ( 1 + |\lambda | + |\xi|)^{\frac12} ( 1+
|\lambda | + |\xi |^2 )^{\frac{s}{2}}\hat h ( \lambda, \xi ) \in L^2
(\mathbb{R}^2 ) \right \} $$ where $\hat h $ is the Fourier transform of $h$
with respect to $t$ and $ x$, the local well-posedness of the IBVP in $C([0,T];
H^s(\mathbb{R} \times [0,1]))$ is proved. The global well-posedness is also
obtained for $s = 1$. The basic idea used here relies on the derivation of an
integral operator for the non-homogeneous boundary data and the proof of the
series version of Strichartz's estimates for this operator. After the problem
is transformed to finding a fixed point of an integral operator, the
contraction mapping argument then yields a fixed point using the Strichartz's
estimates for initial and boundary operators. The global well-posedness is
proved using {\it a-priori} estimates of the solutions.
