  This work is concerned with boundary conditions for one-dimensional
hyperbolic relaxation systems with characteristic boundaries. We assume that
the relaxation system satisfies the structural stability condition proposed by
the second author previously and the boundary is characteristic of type I
(characteristic for the relaxation system but non-characteristic for the
corresponding equilibrium system). For this kind of characteristic
initial-boundary-value problems, we propose a modified Generalized Kreiss
condition (GKC). This extends the GKC proposed by the second author for the
non-characteristic boundaries to the present characteristic case. Under this
modified GKC, we derive the reduced boundary condition and verify its validity
by combining an energy estimate with the Laplace transform. Moreover, we show
the existence of boundary-layers for nonlinear problems.
