We present an implicit method of steps for differential equations with state-dependent delays and validated numerics to rigorously enclose solutions of initial-value problems. Our approach uses a combination of contraction mapping arguments based on a Newton–Kantorovich type theorem and piecewise polynomial interpolation. Completing multiple steps of integration is challenging, and we resolve it by smooth interpolation of the previous solution, resulting in an interval-valued polynomial initial condition for the subsequent step. A set of examples is provided.

中文翻译：

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微分方程与状态相关延迟的验证积分

我们提出了一种用于具有状态相关延迟和经过验证的数值的微分方程的隐式步骤方法，以严格包含初始值问题的解决方案。我们的方法结合使用基于 Newton-Kantorovich 类型定理和分段多项式插值的收缩映射参数。完成多个积分步骤是具有挑战性的，我们通过对先前解的平滑插值来解决它，从而为后续步骤生成区间值多项式初始条件。提供了一组示例。