  We consider the inhomogeneous version of the fixed-point equation of the
smoothing transformation, that is, the equation $X \stackrel{d}{=} C + \sum_{i
\geq 1} T_i X_i$, where $\stackrel{d}{=}$ means equality in distribution,
$(C,T_1,T_2,...)$ is a given sequence of non-negative random variables and
$X_1,X_2,...$ is a sequence of i.i.d.\ copies of the non-negative random
variable $X$ independent of $(C,T_1,T_2,...)$. In this situation, $X$ (or, more
precisely, the distribution of $X$) is said to be a fixed point of the
(inhomogeneous) smoothing transform. In the present paper, we give a necessary
and sufficient condition for the existence of a fixed point. Further, we
establish an explicit one-to-one correspondence with the solutions to the
corresponding homogeneous equation with C=0. Using this correspondence, we
present a full characterization of the set of fixed points under mild
assumptions.
