On a separating solution of a recurrent equation

We consider the recurrent equation
$$\Lambda_p = \frac{1}{p-1} \sum \limits_{p_1=1}^{p-1} f\biggl(\frac{p_1}{p}\biggr) \Lambda_{p_1}\Lambda_{p-p_1}$$
which depends on the initial condition $ \Lambda_1 = x$. Under some conditions on $f$ we show that there exists the value of x for which $\Lambda_p$ tends to a constant as p tends to infinity.