Boundary-value problems of functional differential equations with state-dependent delays

TL;DR

This paper proves the convergence of collocation methods for periodic boundary value problems involving functional differential equations with state-dependent delays, overcoming challenges posed by non-Lipschitz nonlinearities.

Contribution

It introduces the concept of mild differentiability to establish convergence of discretization methods for complex functional differential equations.

Findings

Piecewise polynomial collocation methods converge with expected order.

Mild differentiability enables analysis despite non-Lipschitz nonlinearities.

Locally unique solutions are effectively approximated.

Abstract

We prove convergence of piecewise polynomial collocation methods applied to periodic boundary value problems for functional differential equations with state-dependent delays. The state dependence of the delays leads to nonlinearities that are not locally Lipschitz continuous preventing the direct application of general abstract discretization theoretic frameworks. We employ a weaker form of differentiability, which we call mild differentiability, to prove that a locally unique solution of the functional differential equation is approximated by the solution of the discretized problem with the expected order.