Separating Solution of a Quadratic Recurrent Equation

Details Separating Solution of a Quadratic Recurrent Equation Separating Solution of a Quadratic Recurrent Equation

2008-12-03

arxiv.org/abs/0812.0776v1

Journal of Statistical Physics, Volume 136, Number 4 / August, 2009

Mathematics

Dynamical Systems

13 pages, 6 figures, submitted to J. Stat. Phys

Scientific paper

10.1007/s10955-009-9801-6

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 $p\to \infty$.

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