On the attainable order of collocation methods for the neutral functional-differential equations with proportional delays

In this paper, we extend the recent results of H. Brunner in BIT (1997) for the DDE y'(t) = by(qt), y(0) = 1 and the DVIE y(t) = 1+ integral(0)(t) by(qs)ds with proportional delay qt, 0 < q less than or equal to 1, to the neutral functional-differential equation (NFDE): [GRAPHICS] and the delay Volterra integro-differential equation (DVIDE) : [GRAPHICS] with proportional delays p(i)t and g(i)t, 0 < p(i), q(i) less than or equal to 1 and complex numbers a, b(i) and c(i). We analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods at the first mesh point t = h for the collocation solution vc(t) of the NFDE and the 'iterated collocation solution u(it)(t)' of the DVIDE to the solution y(t), and investigate the existence of the collocation polynomials M-m(t) of v(th) or (M) over cap(m)(t) of u(it)(th), t is an element of [0, 1] such that the rational approximant v(h) or u(it)(h) is the (m, m)-Pade approximant to y(h) and satisfies \v(h) - y(h)\ = O(h(2m+1)). If they exist, then we actually give the conditions of M-m(t) and (M) over cap(m)(t), respectively.