Optimal Superconvergence Orders of Iterated Collocation Solutions for Volterra Integral Equations with Vanishing Delays

In this paper we analyze the optimal convergence properties of collocation approximations to solutions of Volterra integral equations of the second kind with vanishing variable delays. The focus of the analysis is on the superconvergence of the (iterated) collocation approximation corresponding to collocation in the space of (discontinuous) piecewise polynomials of degree $m-1 \geq 0$. We show that on uniform meshes the iterated collocation solution possesses the global superconvergence order $m+1$, and that the solution's order of local superconvergence at the nodes of the underlying mesh cannot exceed $m+2$, in sharp contrast to problems with nonvanishing delays. Moreover, the optimal order $p^{\ast } = m+2$ can be attained only under suitable assumptions. This result also resolves a conjecture of 1997 regarding the attainable order of local superconvergence for Volterra integral equations containing a proportional delay $qt$ with $q \in (0,1)$.