On the attainable order of collocation methods for delay differential equations with proportional delay

To analyze the attainable order of m-stage implicit ( collocation-based) Runge Kutta methods for the delay differential equation (DDE) y' (t) = by (qt), 0 < q less than or equal to 1 with y (0) = 1, and the delay Volterra integral equation (DVIE) y (t) = 1 + b/q integral(o)(qt) y(s) ds with proportional delay qt, 0 < q less than or equal to 1, our particular interest lies in the approximations (and their orders) at the first mesh point t = h for the collocation solution v(t) of the DDE and the iterated collocation solution u(it) (t) of the DVIE to the solution y(t). Recently, H. Brunner proposed the following open problem: For m greater than or equal to 3, do there exist collocation points c(i) = c(i) (q), i = 1, 2,..., m in [0, 1] such that the rational approximant v(h) is the (m, m)-Pade approximant to y(h)? If these exist, then \v(h) - y(h)\ = O(h(2m+1)) but what is the collocation polynomial M-m(t; q) = K Pi(i=1)(m) (t - c(i)) of v(th), t is an element of [0, 1]?" In this paper, we solve this question affirmatively, and give the related results between the collocation solution v( t) of the DDE and the iterated collocation solution u(it)(t) of the DVIE. We also answer to Brunner s second open question in the case that one collocation point is fixed at the right end point of the interval.