Separating Solution of a Quadratic Recurrent Equation

In this paper we consider the recurrent equation

$Lambda_{p+1}=frac{1}{p}sum_{q=1}^pfbigg(frac{q}{p+1}bigg)Lambda _{q}Lambda_{p+1-q}$

for p≥1 with f∈C[0,1] and Λ₁=y>0 given. We give conditions on f that guarantee the existence of y ⁽⁰⁾ such that the sequence Λ _p with Λ₁=y ⁽⁰⁾ tends to a finite positive limit as p→∞.