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

 

Positive definite measures with applications to a Volterra equation


Author: Olof J. Staffans
Journal: Trans. Amer. Math. Soc. 218 (1976), 219-237
MSC: Primary 45D05; Secondary 45M05, 42-XX
MathSciNet review: 0458086
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Abstract: We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation

$\displaystyle x'(t) + \int_0^t {g(x(t - \tau ))\;d\mu (\tau ) = f(t),}$ ($ \ast$)

with a positive definite kernel $ \mu $. In particular, we give new sufficient conditions on the kernel $ \mu $, which together with standard assumptions on f and g yield results on boundedness and asymptotic behavior of the solutions of $ (\ast)$. Our proofs are based on the theory of distribution Fourier transforms.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0458086-5
PII: S 0002-9947(1976)0458086-5
Keywords: Volterra equation, nonlinear integrodifferential equation, asymptotic stability, positive definiteness, strict positive definiteness, Fourier transform, distributions, $ {\text{weak}^\ast}$ convergence
Article copyright: © Copyright 1976 American Mathematical Society