Positive definite measures with applications to a Volterra equation

Staffans, Olof J.

doi:10.1090/S0002-9947-1976-0458086-5

Positive definite measures with applications to a Volterra equation
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by Olof J. Staffans PDF

Trans. Amer. Math. Soc. 218 (1976), 219-237 Request permission

Abstract:

We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation \begin{equation}\tag {$\ast $} x’(t) + \int _0^t {g(x(t - \tau ))\;d\mu (\tau ) = f(t),}\end{equation} 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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