Let F (·) be a c.d.f. on [0,∞), f ( s ) = ∑ ∞ 0 p j s i a p.g.f. with p 0 = 0, < 1 < m = Σ j p j < ∞ and 1 < ρ <∞. For the functional equation for a c.d.f. H (·) on [0,∞] we establish that if 1 – F ( x ) = O ( x – θ ) for some θ > α =(log m )/(log p) there exists a unique solution H (·) to (∗) in the class C of c.d.f.’s satisfying 1 – H ( x ) = o ( x – α ). We give a probabilistic construction of this solution via branching random walks with discounting. We also show non-uniqueness if the condition 1 – H ( x ) = o ( x – α ) is relaxed.