Elementary fixed points of the BRW smoothing transforms with infinite number of summands

The branching random walk (BRW) smoothing transform $T$ is defined as $T:\text{distr}(U_{1})\mapsto \text{distr} (\sum_{i=1}^{L}X_{i}U_{i})$, where given realizations $\{X_{i}\}_{i=1}^{L}$ of a point process, $U_{1},U_{2},...$ are conditionally independent identically distributed random variables, and $0\leq \text{Prob}\{L=\infty \}\leq 1$. Given $\alpha \in (0,1]$, $\alpha$-\emph{elementary} fixed points are fixed points of $T$ whose Laplace-Stieltjes transforms $\phi$ satisfy $\underset{s\to +0}{\lim}\dfrac{1-\phi(s)}{s^{\alpha}}=m$, where $m$ is any given positive number. If $\alpha=1$, these are the fixed points with finite mean. We show exactly when elementary fixed points exist. In this case these are the only fixed points of $T$ and are unique up to a multiplicative constant. These results do not need any moment conditions. In particular, Biggins' martingale convergence theorem is proved in full generality. Essentially we apply recent results due to Lyons (1997) and Goldie and Maller (2000) as the key point of our approach is a close connection between fixed points with finite mean and perpetuities. As a by-product, we lift from our general results the solution to a Pitman-Yor problem. Finally, we study the tail behaviour of some fixed points with finite mean.