Separating Solution of a Quadratic Recurrent Equation

In this paper we consider the recurrent equation $Λ_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)Λ_{q}Λ_{p+1-q}$ for $p\ge 1$ with $f\in C[0,1]$ and $Λ_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence $Λ_p$ with $Λ_1=y^{(0)}$ tends to a finite positive limit as $p\to \infty$.

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