  At each time $n\in\mathbb{N}$, let
$\bar{Y}^{(n)}=(y_{1}^{(n)},y_{2}^{(n)},\cdots)$ be a random sequence of
non-negative numbers that are ultimately zero in a random environment
$\xi=(\xi_{n})_{n\in\mathbb{N}}$ in time, which satisfies for each
$n\in\mathbb{N}$ and a.e.
$\xi,~E_{\xi}[\sum_{i\in\mathbb{N}_{+}}y_{i}^{(n)}(\xi)]=1.$ The existence and
uniqueness of the non-negative fixed points of the associated smoothing
transform in random environments is considered. These fixed points are
solutions of the distributional equation for
$a.e.~\xi,~Z(\xi)\overset{d}{=}\sum_{i\in\mathbb{N}_{+}}y_{i}^{(0)}(\xi)Z_{i}(T\xi),$
where when given the environment $\xi$, $Z_{i}(T\xi)~(i\in\mathbb{N}_{+})$ are
$i.i.d.$ non-negative random variables, and distributed the same as $Z(\xi)$.
As an application, the martingale convergence of the branching random walk in
random environments is given as well. The classical results by Biggins (1977)
has been extended to the random environment situation.
