On a separating solution of a recurrent equation

We consider the recurrent equation $$\Lambda _p = \frac{1}{{p - 1}}\sum\limits_{p_1 = 1}^{p - 1} f \left( {\frac{{p_1 }}{p}} \right)\Lambda _{p_1 } \cdot \Lambda _{p - p_1 } $$ . which depends on the initial condition Λ₁ = x. Under some conditions on f we show that there exists the value of x for which Λ_p tends to a constant as p tends to infinity.