Boundary-value problems of functional differential equations with state-dependent delays
<|reference_start|>Boundary-value problems of functional differential equations with state-dependent delays: 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.   An additional difficulty is that linearizations required for solving the discretized nonlinear problem with Newton iterations are not well defined. We show that Newton iterations still converge if one uses the linearization in regularized solutions. The Newton iterations' asymptotic convergence ratio is limited by the numerical discretization error. Thus, Newton iterations should show better convergence for approximations on finer meshes.<|reference_end|>
@article{ando'2024boundary-value,
    title={Boundary-value problems of functional differential equations with
  state-dependent delays},
    author={Alessia Ando', Jan Sieber},
    journal={arXiv preprint arXiv:2410.07375},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.07375},
    primaryClass={math.NA cs.NA}
}