Sufficient conditions are given guaranteeing that all solutions of the equation $$ x''+a(t)f(x)=0\;\;\;(xf(x)>0) $$ tend to zero as $t$ goes to infinity. The conditions contain integrals instead of maxima and minima in earlier results. Finally, a probabilistic generalization of Armellini-Tonelli-Sansone theorem is formulated.
Mathematics Subject Classification: 34D20.
Key words and phrases: Partial asymptotic stability, oscillation.