具有状态依赖时滞的泛函微分方程的边值问题
Boundary-value problems of functional differential equations with state-dependent delays
AI总结：
本文针对具有状态依赖时滞的泛函微分方程周期边值问题，证明了分段多项式配置方法的收敛性，通过引入弱可微性克服了非局部Lipschitz连续非线性带来的理论框架应用困难，并证明了离散化问题解能以预期阶数近似局部唯一解。
摘要：
我们证明了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分段多项式配置方法的收敛性。时滞的状态依赖性导致了非局部Lipschitz连续的非线性性，从而阻碍了抽象离散化理论框架的直接应用。我们采用一种称为弱可微性的较弱可微性形式，证明泛函微分方程的局部唯一解被离散化问题的解以预期阶数近似。
英文摘要：
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.
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