Well-posedness of initial value problems for functional differential and algebraic equations of mixed type

Abstract

We study the well-posedness of initial value problems for scalar functional algebraic and differential functional equations of mixed type. We provide a practical way to determine whether such problems admit unique solutions that grow at a specified rate. In particular, we exploit the fact that the answer to such questions is encoded in an integer n]. We show how this number can be tracked as a problem is transformed to a reference problem for which a Wiener-Hopf splitting can be computed. Once such a splitting is available, results due to Mallet-Paret and Verduyn-Lunel can be used to compute n]. We illustrate our techniques by analytically studying the wellposedness of two macro-economic overlapping generations models for which Wiener-Hopf splittings are not readily available.