Delay differential equation

Definition

Let $n, k \ge 1$, let $0 < \tau_1 < \cdots < \tau_k$ be constants, and let $f \colon \R \times (\R^n)^{k+1} \to \R^n$ be a function. A delay differential equation with constant delays $\tau_1, \ldots, \tau_k$ is an equation $$x'(t) = f\bigl(t, x(t), x(t - \tau_1), \ldots, x(t - \tau_k)\bigr)$$ for an unknown function $x$ with values in $\R^n$. Let $t_0 \in \R$ and let $\phi \colon [-\tau_k, 0] \to \R^n$ be continuous, the history. A solution of the initial value problem is a continuous $x \colon [t_0 - \tau_k, t_0 + T) \to \R^n$, for some $T > 0$, with these properties. It equals the history: $x(t_0 + \theta) = \phi(\theta)$ for $\theta \in [-\tau_k, 0]$. It is differentiable on $(t_0, t_0 + T)$, and the equation holds there.