  We consider the Cauchy problem for the Gerdjikov-Ivanov(GI) type of the
derivative nonlinear Schr\"odinger (DNLS) equation:
$$iq_t+q_{xx}-iq^2\bar{q}_x+\frac{1}{2}|q|^4{q}=0.$$ with steplike initial
data: $q(x,0)=0$ for $x\le 0$ and $q(x,0)=Ae^{-2iBx}$ for $x>0$,where $A>0$ and
$B\in \R$ are constants.The paper aims at studying the long-time asymptotics of
the solution to this problem.We show that there are four regions in the
half-plane $-\infty<x<\infty,t>0$,where the asymptotics has qualitatively
different forms:a slowly decaying self-similar wave of Zakharov-Manakov type
for $x>-4tB$, a plane wave region:$x<-4t(B+\sqrt{2A^2(B+\frac{A^2}{4})})$, an
elliptic region:$-4t(B+\sqrt{2A^2(B+\frac{A^2}{4})})<x<-4tB$. The main tool is
the asymptotic analysis of an associated matrix Riemann-Hilbert problem.
