Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 40E05, 45D05

Retrieve articles in all journals with MSC: 40E05, 45D05

Additional Information

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

American Mathematical Society