Scalar neutral delay equations and the derivative-segment shift

Use the fixed positive-delay/history convention of Retarded delay equations and their history space in dimension $d=1$. The canonical isometric coordinate map between $\mathbb{R}$ and ${\mathbb{R}}^1$ is as specified in Vector-valued functions $f:A\to {\mathbb{R}}^m$, their limits and continuity, with the dictionary to the metric notions; the formulas below write the single coordinate as a real scalar.

Let $\tau >0$, $T=K\tau$ with integer $K\ge 1$, $a\in \mathbb{R}$, and let $U$ be an open interval containing $[0,T]$. Let $f:U\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be continuous and globally Lipschitz in its last two variables: for some $L\ge 0$, $\mathrm{\mid }f(t,x,z)-f(t,\tilde{x},\tilde{z})\mathrm{\mid }\le L(\mathrm{\mid }x-\tilde{x}\mathrm{\mid }+\mathrm{\mid }z-\tilde{z}\mathrm{\mid })(t\in U)\mathrm{.}$ Let $\varphi :[-\tau ,0]\to \mathbb{R}$ be continuously differentiable, with continuity and derivative as in Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point and The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set. A scalar neutral delay initial-value problem is $y^{\mathrm{\prime }}(t)-a{y}^{\mathrm{\prime }}(t-\tau )=f(t,y(t),y(t-\tau )),0\le t\le T,$ with history $y(t)=\varphi (t)$ for $-\tau \le t\le 0$. A solution is continuous on $[-\tau ,T]$, continuously differentiable on $[0,T]$ with domain-relative endpoint derivatives, and satisfies the equation. The history is compatible at zero when ${\varphi }^{\mathrm{\prime }}(0)=a{\varphi }^{\mathrm{\prime }}(-\tau )+f(0,\varphi (0),\varphi (-\tau ))\mathrm{.}$ The parameter restriction $\mathrm{\mid }a\mathrm{\mid }<1$ is the stable derivative-shift branch used below; the equation itself is defined for every real $a$. No state-dependent delay is included.

For the finite horizon $T=K\tau$, write a derivative path as segments $u_k(s)=y^{\mathrm{\prime }}(k\tau +s)$ for $0\le k<K$ and $0\le s\le \tau$. Let ${\mathcal{X}}_K$ be the set of $K$-tuples of continuous real functions on $[0,\tau ]$, with continuity as in Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point, and define $\mathrm{\parallel }u{\mathrm{\parallel }}_{{\mathcal{X}}_K}={\max }_{0\le k<K}{\sup }_{0\le s\le \tau }\mathrm{\mid }{u}_k(s)\mathrm{\mid },$ and the delay shift $S_K$ by $(S_{K}u{)}_0(s)=0$ and $(S_{K}u{)}_k(s)=u_{k-1}(s)$ for $1\le k<K$. The known derivative history on the first segment is a forcing term; on each later segment the delayed derivative is exactly the preceding segment $S_{K}u$. Thus the neutral derivative relation has the operator form $(I-a{S}_K)u=r$, with $r$ containing the right-hand side and the known initial derivative history.

For a compatible-grid method with $h=\tau \mathrm{/}m$, $m\ge 1$, and $N=Km$, the corresponding mesh shift on arrays $v=(v_0,\dots ,v_{N-1})\in {\mathbb{R}}^N$ is $(S_{K,m}v{)}_n=0$ for $0\le n<m$ and $(S_{K,m}v{)}_n=v_{n-m}$ for $m\le n<N$. For any finite array length $M\in {\mathbb{N}}_0$ and integer $m\ge 1$, denote the same rule by $S_m^{(M)}$. If $M\ge 1$, use the max norm $\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}={\max }_{0\le n<M}\mathrm{\mid }{v}_n\mathrm{\mid }$. If $M=0$, the coordinate space ${\mathbb{R}}^0$ consists of its unique empty array, the zero vector (The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$); define its norm to be $0$ and define $S_m^{(0)}$ to be the unique map from this zero space to itself. Thus no maximum of an empty real set is used. In particular $S_{K,m}=S_m^{(Km)}$. This records the delayed discrete derivative when estimating any partial sequence of Euler steps. The known derivative values from ${\varphi }^{\mathrm{\prime }}$ occupy the first $m$ steps.