  Given a sequence $T=(T_i)_{i\geq1}$ of nonnegative random variables, a
function f on the positive halfline can be transformed to
$\mathbb{E}\prod_{i\geq1}f(tT_i)$. We study the fixed points of this transform
within the class of decreasing functions. By exploiting the intimate
relationship with general branching processes, a full description of the set of
solutions is established without the moment conditions that figure in earlier
studies. Since the class of functions under consideration contains all Laplace
transforms of probability distributions on $[0,\infty)$, the results provide
the full description of the set of solutions to the fixed-point equation of the
smoothing transform, $X\stackrel{d}{=}\sum_{i\geq1}T_iX_i$, where
$\stackrel{d}{=}$ denotes equality of the corresponding laws, and $X_1,X_2,...$
is a sequence of i.i.d. copies of X independent of T. Further, since
left-continuous survival functions are covered as well, the results also apply
to the fixed-point equation $X\stackrel{d}{=}\inf\{X_i/T_i:i\geq1,T_i>0\}$.
Moreover, we investigate the phenomenon of endogeny in the context of the
smoothing transform and, thereby, solve an open problem posed by Aldous and
Bandyopadhyay.

Probability