Analysis of continuous collocation solutions for a kind of Volterra functional integral equations with proportional delay (Q651908)

The authors prove that, under usual smoothness conditions on the given data, the delay Volterra equation \[ y(t) = b(t, y(qt)) + f(t) + \int_0^t K_0(t,s) y(s) ds + \int_0^{qt} K_1(t,s) y(s) ds \] with some \(q \in (0,1)\) has a uniquely determined continuous solution. Sufficient conditions for the solution to be in \(C^\nu\) with some \(\nu \in \mathbb N\) are also given. For the numerical solution of this equation, a piecewise linear collocation method is suggested. The convergence properties of this method are investigated.