In the standard theory of delay equations, the fundamental solution does not ‘live’ in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with “retarded” equations, and in Part III we show how the Stieltjes integral enables incorporation of unbounded perturbations corresponding to neutral delay equations.

在延迟方程的标准理论中，基本解并不存在于状态空间中。为了消除这种古老的异常，我们扩大了状态空间。结果，我们失去了解算子的强连续性，这又导致Riemann积分不再足以为常数变式赋予含义。为了进行补偿，我们在规范双对空间的设置中开发了Stieltjes-Pettis积分。第一部分提供了一般理论，第二部分介绍了“延迟”方程，第三部分展示了Stieltjes积分如何使与中性延迟方程相对应的无穷微扰动并入。