On a class of strongly contractive quadratic recurrent systems

We consider a class of nonlinear recurrent systems of the form $\Lambda_p=\frac1p\sum_{p_1=1}^{p-1} f(\frac {p_1}p)\Lambda_{p_1}\Lambda_{p-p_1}$, $p>1$, where f is a given function on the interval $[0,1]$ and $\Lambda_1=x$ is an adjustable real-valued parameter. Under some suitable assumptions on the function $f$, we show that there exists an initial value $x^*$ for which $\Lambda_p=\Lambda_p(x^*)\to\mathrm{const}$ as $p\to\infty$. More precise asymptotics of $\Lambda_p$ is also derived.