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.

翻译：我们证明了分段多项式配置方法应用于具有状态依赖时滞的泛函微分方程周期边值问题的收敛性。时滞的状态依赖性导致了非局部Lipschitz连续的非线性项，阻碍了直接应用一般抽象离散化理论框架。我们采用一种较弱的可微性形式，称之为温和可微性，证明了泛函微分方程的局部唯一解能够以预期阶数被离散化问题的解所逼近。