Analysis of collocation solutions for a class of functional equations with vanishing delays

We study the existence, uniqueness and regularity properties of solutions for the functional equation y(t) = b(t)y(θ(t)) + f(t), t ∈ [0, T], where the delay function θ(t) vanishes at t = 0. Functional equations corresponding to the linear delay function θ(t) = qt (0 < q < 1) represent an important special case. We then analyse the optimal order of convergence of piecewise polynomial collocation approximations to solutions of these functional equations. The theoretical results are illustrated by extensive numerical examples.