Positive Definite Measures with Applications to a Volterra Equation
Olof J. Staffans · Transactions of the American Mathematical Society · 1976
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.