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

Staffans, Olof J.

doi:10.1090/S0002-9947-1976-0422936-9

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

Author: Olof J. Staffans
Journal: Trans. Amer. Math. Soc. 218 (1976), 239-259
MSC: Primary 40E05; Secondary 45D05
MathSciNet review: 0422936
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Abstract: We study the relation between the condition

$\displaystyle \mathop {\sup }\limits_{T > 0} \int_{[0,T]} {\bar \varphi (t)} \int_{[t - T,t]} {\varphi (t - s)\;dv (s)\;dt < \infty }$

and the asymptotic behavior of the bounded function $\varphi$ when $\nu$ is a positive definite measure. Earlier we have proved that if $\nu$ is strictly positive definite and $\varphi$ satisfies a tauberian condition, then $\varphi (t) \to 0$ as $t \to \infty$ . Here we characterize the spectrum of the limit set of $\varphi$ in the case when $\nu$ 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.

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