Separating Solution of a Quadratic Recurrent Equation

TL;DR

This paper analyzes a quadratic recurrence relation involving a continuous function and establishes conditions under which the sequence converges to a finite positive limit, providing insights into its long-term behavior.

Contribution

It introduces conditions on the function f that ensure the existence of a specific initial value leading to convergence of the recurrence sequence.

Findings

Identifies conditions on f for sequence convergence.

Provides a method to determine initial value y^{(0)}.

Shows the sequence tends to a finite positive limit under these conditions.

Abstract

In this paper we consider the recurrent equation $$\Lambda_{p+1}=\frac1p\sum_{q=1}^pf\bigg(\frac{q}{p+1}\bigg)\Lambda_{q}\Lambda_{p+1-q}$$ for $p\ge 1$ with $f\in C[0,1]$ and $\Lambda_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence $\Lambda_p$ with $\Lambda_1=y^{(0)}$ tends to a finite positive limit as $p\to \infty$.