On a separating solution of a recurrent equation

We consider the recurrent equation Λ _p = 1/{p - 1}sumlimits_{p_1 = 1}^{p - 1} f left( {{p_1 }/p} right)Λ _{p_1 } \cdot Λ _{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.