Retarded delay equations and their history space

Let $d\ge 1$, $\tau >0$, and $T>0$. The history space for delay $\tau$ is ${\mathcal{C}}_{\tau }:=C([-\tau ,0],{\mathbb{R}}^d),\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_{-\tau \le s\le 0}\mathrm{\parallel }\varphi (s){\mathrm{\parallel }}_2,$ where continuity of vector-valued functions and the Euclidean norm are as in Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions and The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Its completeness, so that it is a Banach space with this norm, is established in The continuous delay-history space is Banach ↗.

A retarded delay initial-value problem on $[0,T]$ consists of a history $\varphi \in {\mathcal{C}}_{\tau }$ and a vector field $f:U\times {\mathbb{R}}^d\times {\mathbb{R}}^d\to {\mathbb{R}}^d$, where $U$ is an open interval containing $[0,T]$, such that $f$ is continuous and there is an $L\ge 0$ giving the following global Lipschitz bound (in the convention of Lipschitz map, $\alpha$-Hölder map for rational $0<\alpha \le 1$, and contraction): $\mathrm{\parallel }f(t,u,v)-f(t,\tilde{u},\tilde{v}){\mathrm{\parallel }}_2\le L(\mathrm{\parallel }u-\tilde{u}{\mathrm{\parallel }}_2+\mathrm{\parallel }v-\tilde{v}{\mathrm{\parallel }}_2)$ for all $t\in U$ and all $u,v,\tilde{u},\tilde{v}\in {\mathbb{R}}^d$. A solution is a continuous function $y:[-\tau ,T]\to {\mathbb{R}}^d$ whose restriction to $[0,T]$ is continuously differentiable, for which $y(s)=\varphi (s)$ on $[-\tau ,0]$, and which satisfies the compatibility condition $y(0)=\varphi (0)$ and the retarded equation $y^{\mathrm{\prime }}(t)=f(t,y(t),y(t-\tau )),0\le t\le T\mathrm{.}$ At $t=0$ the derivative is the right derivative. The equation is an ordinary first-order IVP on each method-of-steps interval once the delayed history on that interval is known.

The definition concerns a fixed positive constant delay. It does not include state-dependent delays, neutral equations containing a delayed derivative, or integrals over a delay window; those are distinct interfaces.