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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0422936-9
PII: S 0002-9947(1976)0422936-9
Keywords: Tauberian theorem, positive definite, quadratic form, asymptotic behavior, limit set, asymptotic spectrum, removable zeros, nonlinear Volterra equation
Article copyright: © Copyright 1976 American Mathematical Society