  We consider the nonlinear Schr\"odinger equation on the half-line with a
given Dirichlet (Neumann) boundary datum which for large $t$ tends to the
periodic function $g_0^b(t)$ ($g_1^b(t)$). Assuming that the unknown Neumann
(Dirichlet) boundary value tends for large $t$ to a periodic function
$g_1^b(t)$ ($g_0^b(t)$), we derive an easily verifiable condition that the
functions $g_0^b(t)$ and $g_1^b(t)$ must satisfy. Furthermore, we introduce two
different methods, one based on the formulation of a Riemann-Hilbert problem,
and one based on a perturbative approach, for constructing $g_1^b(t)$
($g_0^b(t)$) in terms of $g_0^b(t)$ ($g_1^b(t)$).
