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We study the relation between the condition  \[    sup T &gt; 0 ∫  [ 0 , T ] φ  ¯  ( t ) ∫  [ t −  T , t ] φ  ( t −  s ) d v ( s ) d t &gt; ∞  \sup \limits _{T &gt; 0} \int _{[0,T]} {\bar \varphi (t)} \int _{[t - T,t]} {\varphi (t - s)\;dv (s)\;dt &gt; \infty }     \]  and the asymptotic behavior of the bounded function      φ  \varphi       when      ν  \nu       is a positive definite measure. Earlier we have proved that if      ν  \nu       is strictly positive definite and      φ  \varphi       satisfies a tauberian condition, then      φ  ( t ) →  0 \varphi (t) \to 0       as      t →  ∞  t \to \infty      . Here we characterize the spectrum of the limit set of      φ  \varphi       in the case when      ν  \nu       is not strictly positive definite. Applying this theory to a nonlinear Volterra equation we get some new results on the asymptotic behavior of its bounded solutions.

Tauberian theorems for a positive definite form, with applications to a Volterra equation