Fixed points of the smoothing transform: The boundary case

Let A = (A₁,A₂, A₃,...) be a random sequence of non-negative numbers that are ultimately zero with E[? A _i] = 1 and E[?A_i log A_i= 0. The uniqueness of the non-negative fixed points of the associated smoothing transform is considered. These fixed points are solutions to the functional equation f(?) = E[?_if(?A_i)], where f is the Laplace transform of a non-negative random variable. The study complements, and extends, existing results on the case when E[?A _ilog A_i 0. New results on the asymptotic behaviour of the solutions near zero in the boundary case, where E[?A_ilog A _i 0, are obtained.