# State-dependent Delay Differential Equations on $H^{1}$

Johanna Frohberg and Marcus Waurick Institut für Angewandte Analysis, TU
Bergakademie Freiberg, GermanyCorresponding Author, Institut für Angewandte
Analysis, TU Bergakademie Freiberg, Germany

###### Abstract

Abstract Classically, solution theories for state-dependent delay equations
are developed in spaces of continuous or continuously differentiable
functions. In applications, these two approaches naturally offer some
limitations. The former can be technically challenging to apply in as much as
suitably Lipschitz continuous extensions of mappings onto (open subsets of)
the space of continuous functions are required; whereas the latter approach
leads to potentially undue restrictions on the class of initial pre-histories.
Here, we establish a solution theory (i.e., unique existence) for state-
dependent delay equations for arbitrary Lipschitz continuous pre-histories and
suitably Lipschitz continuous right-hand sides on the Sobolev space $H^{1}$.
The provided solution theory is independent of previous ones and is based on
the contraction mapping principle on exponentially weighted spaces yielding
global in time well-posedness right away. In particular, initial pre-histories
are not required to belong to certain solution manifolds and the generality of
the approach permits the consideration of a large class of functional
differential equations even when the continuity of the right-hand side has
constraints on the derivative. The solution theory provided implies the
classical Picard–Lindelöf theorem as a special case.

Keywords State-dependent delay equations, solution manifold, functional
differential equation
MSC2020  Primary 34K05 Secondary 34K30

###### Contents

  1. 1 Introduction
  2. 2 Functional analytic prerequisites and the pre-history map
  3. 3 A key estimate for the solution theory
  4. 4 Solution theory for state-dependent delay equations
  5. 5 Local solution theory for functional differential equations
  6. 6 Applications
    1. 6.1 On a problem from mathematical biology
    2. 6.2 A positioning problem
  7. 7 Conclusion

## Acknowledgements

We thank Prof Petr Hájek and Prof Friedrich Martin Schneider for providing
some insights on (Lipschitz) retractions on the space of continuous functions.
We thank Prof Schneider also for making us aware of Kirszbraun’s Theorem.

##  1 Introduction

State-dependent delay equations are delay equations where the retarded
argument of the right-hand side is dependent on the solution itself. This fact
necessitates a different solution theory compared to other (delay)
differential equations.

There are several approaches and notions to address unique existence of state-
dependent delay equations in the literature. In order to sketch their main
properties, we consider the following particular state-dependent delay
equation. Note that in Section 4, we will provide a solution theory for
Lipschitz continuous solutions for equations of the form (1).

Given $h,T>0$ and a Lipschitz continuous pre-history
$\phi\colon(-h,0)\to\mathbb{R}^{n}$, we consider

$\begin{cases}x^{\prime}(t)=g(t,x(t),x(t+r(x_{(t)}))),&t\in(0,T);\\\
x_{(0)}=\phi,\end{cases}$ (1)

where we used the usual notation $x_{(t)}(s)\coloneqq x(t+s)$; $r$, taking
values in $[-h,0]$, is a suitable delay functional defined on a set of
functions defined on $[-h,0]$ and
$g\colon[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is
continuous and Lipschitz continuous uniformly in $t$; that is, there exists
$L\geq 0$ such that for all $t\in[0,T]$ and $u,v,w,z\in\mathbb{R}^{n}$, we
have

$\|g(t,u,w)-g(t,v,z)\|\leq L(\|u-v\|+\|w-z\|).$

If $r$ was constant, the equation at hand is a (state-independent) delay
differential equation, which can be dealt with classically. Indeed,
integrating the equation, a fixed point of the integral equation is a solution
of the differential equation. The contraction mapping principle can be
employed either using a weighted norm or by a suitable time-stepping method
analysing the Picard iterates, see, e.g., [16, 24], [25, Chapter 4].

The theory is somewhat less elementary if one considers non-constant $r$. In
this situation, one needs to involve also the derivative of $x$: a counter-
example for uniqueness in spaces of continuous functions is already given in
[28]. We shall sketch a technical reason for the involvement of the derivative
of the unknown in any fixed point statement. If one wants to use a similar
approach as in the $r$-is-constant case, formal integration of (1) (assuming
everything is well-defined) leads to

$x(t)=\phi(0)+\int_{0}^{t}g(s,x(s),x(s+r(x_{(s)})))\,\mathrm{d}s.$

Then, a possible map $\Phi$ for any fixed point argument reads

$\Phi\colon
x\mapsto\phi(0)+\int_{0}^{t}g(s,x(s),x(s+r(x_{(s)})))\,\mathrm{d}s.$

In order for $\Phi$ to be contractive on some function space and at the same
time using Lipschitz continuity of $g$, one is required to estimate a term of
the following form

$\int_{0}^{t}\|x(s+r(x_{(s)})-y(s+r(y_{(s)})\|\,\mathrm{d}s\\\
\leq\int_{0}^{t}\|x(s+r(x_{(s)})-y(s+r(x_{(s)})\|\,\mathrm{d}s+\int_{0}^{t}\|y(s+r(x_{(s)})-y(s+r(y_{(s)})\|\,\mathrm{d}s.$

The first term on the right-hand side can easily be dealt with in several
function spaces and associated norms, the second term requires knowledge of
the size of the derivative of $y$ _a priori_. Besides, note that in certain
cases, the knowledge of the size of the derivative is also necessary for
suitable Lipschitz estimates for $r$, see [26, Section 3] or Section 6.2
below.

There are several ways in the literature of addressing well-posedness of
state-dependent delay equations. For a general introduction to delay and
functional differential equations, we refer to [17, 7]. We note in passing
that further generalisations of functional differential equations to infinite-
dimensional state spaces are structurally more complicated, see, e.g., [10].
In any case, concerning state-dependent delay equations, we refer to the
fundamental contributions [13, 26] for many examples and a guide to the
literature. In order to position our approach to the ones in the literature,
we briefly mention four main sources:

  1. (i)

the solution manifold approach introduced by [26],

  2. (ii)

the retraction approach introduced at least in [18] and used quite recently in
[4],

  3. (iii)

the mixed norm approach in [14], and

  4. (iv)

the prolongation approach in [21, 22].

In (i) and (ii), the equation (1) is considered as functional differential
equation. In [26] solutions in the space of continuously differentiable
functions are looked out for. As a consequence, the equation itself dictates a
necessary condition for the initial value namely that
$\phi^{\prime}(0)=g(\phi(0),\phi(r(\phi)))$ (note that in [26] the autonomous
case is considered; hence the first argument of $g$ is skipped). This
condition defines a submanifold $X_{g}$ of $C^{1}$—the so-called solution
manifold. It can be shown that given suitable differentiability assumptions on
the right-hand side, local well-posedness for the equation in question
follows; with the constructed semiflow associated to the differential equation
having $C^{1}$-trajectories on $\mathbb{R}\times X_{g}$, see [27]. Ultimately,
the well-posedness theorem is based on the contraction mapping principle in
$C^{1}$ with a sensible analysis of the constants involved. In applications,
checking the individual conditions can be quite technical, see, e.g., [9].

The approach in [18] is based on an existence result in [12, Theorem 2.2.1]
(which in turn uses Schauder’s fixed point theorem) and an a posteriori
uniqueness analysis for ‘almost locally Lipschitz continuous functionals’,
i.e., functionals that are locally Lipschitz continuous with respect to the
uniform norm if additionally the set of functions is constrained to be
Lipschitz continuous. Hence, the underlying space for unique existence is the
space of continuous functions subject to a Lipschitz condition. Identifying
certain continuity conditions for the right-hand sides only on a specific set
of continuous functions, the general theory is applied using certain
(Lipschitz) retractions; that is, (Lipschitz) continuous mappings defined on
the whole of $C$ projecting onto the said specific set of functions. Given
that the underlying space is a space of continuous functions such retractions
need to be considered specifically for each set in turn. We particularly refer
to the convex and compact subset of $C$ constructed in [11, Section 5], that
is not a Lipschitz retract; see also Remark 6.5 for more technicalities
arising in the retraction approach for continuous functions. In any case,
using Lipschitz continuous functions as potential solutions only, one realises
that the condition to lie on the solution manifold need not be met here (let
alone that under the general assumptions considered in [18] the mere existence
of the manifold is in question). The argument in [4] for global solutions of
the particular equation treated there requires arguments involving positivity.

The approach in (iii) was developed in order to study differentiability in
certain Sobolev type norms. Similar to the rationale developed here—in [14], a
contraction mapping principle is applied namely a variation of the so-called
Uniform Contraction Principle. The norms studied are mixed norm type spaces
looking at the Lipschitz norm of the initial pre-history combined with a
$W^{1,p}$-norm for the solution for positive times. Note that the contraction
principle can be applied, if the final time horizon, $T$, is chosen small
enough, see [14, Lemma 4.2].

In (iv), the author addresses well-posedness on a more fundamental level
introducing the concept of prolongations. This leads to sufficient and
necessary conditions for well-posedness, see [21]. Also the role of a
potential jump in the continuity of the derivative is considered. In any case,
however, the pre-history space is a space of continuous or continuously
differentiable functions and the contraction mapping theorem is involved using
a time-stepping approach for the integrated differential equation and metrics
related to the uniform norm (possibly combined with its derivative), see [21,
p 3515] or [22, Proof of Lemma 5.11].

In this note we develop a solution theory of equations of the type in (1) in
spaces of weakly differentiable functions; more precisely on
$H^{1}(-h,T;\mathbb{R}^{n})$; that is, the space of square integrable
functions with square integrable derivative. More so, we will provide a
generalisation of (1) towards (non-autonomous) functional differential
equations on $H^{1}(-h,T;\mathbb{R}^{n})$. Similar to [4, 18], we have to
suitably restrict to Lipschitz continuous functions for the initial data.

There are no other restrictions however and we obtain global in time solutions
for (1) right away. The theorem containing the solution theory spelled out for
functional differential equations reads as follows (in Theorem 5.1 below we
also state and prove a blow-up result); in which we derive a local solution
theory.

######  Theorem 1.1.

Let $h,T>0$, $G\colon[0,T]\times H^{1}(-h,0;\mathbb{R}^{n})\to\mathbb{R}^{n}$
continuous. Assume that for all $\beta>0$ there exists $L\geq 0$ such that for
all $\phi,\psi\in H^{1}(-h,0;\mathbb{R}^{n})$ with
$\|\phi^{\prime}\|_{\infty},\|\psi^{\prime}\|_{\infty}\leq\beta$ and
$t\in[0,T]$

$\|G(t,\phi)-G(t,\psi)\|\leq L\|\phi-\psi\|_{H^{1}}.$

Then for all Lipschitz continuous $\phi\colon(-h,0)\to\mathbb{R}^{n}$ there
exists $0<T_{1}\leq T$ and a unique solution $x\in
H^{1}(-h,T_{1};\mathbb{R}^{n})$ of

$\begin{cases}x^{\prime}(t)=G(t,x_{(t)}),&(0<t<T_{1}),\\\
x_{(0)}=\phi.\end{cases}$

It follows that $x$ is Lipschitz continuous.

Already for (1), by means of a counterexample we show that the space
$H^{1}(-h,T;\mathbb{R}^{n})$ for the initial data to lie in is too big
warranting well-posedness. Our approach here is based on exponentially
weighted spaces and is inspired by Morgenstern’s proof for the classical
Picard–Lindelöf theorem using an exponentially weighted norm for the space of
continuous functions. Exponentially weighted $L_{2}$-type spaces are natural
in the context of partial differential equations (see [23, 25]) and can
successfully be applied to ordinary (delay) differential equations (see [16,
24, 25]). Note that in these references the time-derivative is established as
a continuously invertible linear operator; in fact this is also some part of
the reason why the present approach works. However, we shall suppress this
viewpoint in the following in the interest of readability. In contrast to the
approach (iii) ([14]) from above, the Lipschitz continuity of the solution is
not built in in the underlying space the contraction mapping principle is
applied to, but rather a consequence of the regularity properties of the
equation. Moreover, the exponential weights permit the consideration of the
whole final time horizon right away and no time-stepping is needed. We note in
passing that the mixed norm space in [14] is no Hilbert space.

Using the Hilbert space $H^{1}(-h,T;\mathbb{R}^{n})$ instead, we are in the
position to make use of the projection theorem. This particularly proves
useful when we employ retraction techniques to extend (Lipschitz) continuous
functions on $H^{1}$ as every closed convex subset $C\subseteq
H^{1}(-h,T;\mathbb{R}^{n})$ is a retract of $H^{1}(-h,T;\mathbb{R}^{n})$; the
retraction is given by the metric projection onto $C$, $P_{C}$, see in
particular Proposition 3.3 below. Hence, Theorem 1.1 also contains a _local_
solution theory. Indeed, some particulars of which are presented in
Proposition 6.1. In this proposition, we show that the functional differential
equations guaranteeing unique local existence as considered in [18, 4] provide
a particular case of the solution theory spelled out here: Using $P_{C}$, we
can always extend a suitably Lipschitz continuous functional to the whole of
$H^{1}$ and then apply Theorem 1.1. We emphasise that Kirszbraun’s theorem,
see [15], even asserts that given a Lipschitz continuous
$f\colon\operatorname{dom}(f)\to H_{2}$, $H_{1},H_{2}$ Hilbert spaces, defined
on _any_ subset $\operatorname{dom}(f)\subseteq H_{1}$ can be Lipschitz
continuously extended to the whole of $H_{1}$; hinting at the flexibility
allowed for in the present Hilbert space approach.

Concerning our proof strategy of Theorem 1.1, Proposition 3.3 is in fact
applied to $V_{\beta}\subseteq H^{1}(-h,T;\mathbb{R}^{n})$ the set of weakly
differentiable functions with derivative bounded by $\beta$. This will be used
to identify the evaluation functional defined on
$H^{1}(-h,0;\mathbb{R}^{n})\times[-h,0]$ suitably composed with the delay
functional $r$ as a Lipschitz continuous mapping on $V_{\beta}$ (see Theorem
3.2). The existence of solutions up to the final time $T$ for the more
particular equations of the type (1) is then guaranteed with a contradiction
argument and a blow-up technique.

Another technical advantage of $H^{1}(-h,T;\mathbb{R}^{n})$ (or rather
$L_{2}$) is that the pre-history mapping assigning to each function $x$ the
pre-histories $s\mapsto x_{(s)}$ can be realised as a Lipschitz continuous
mapping with arbitrarily small norm given the considered norm is suitably
(exponentially) weighted, see Theorem 2.1 or [16]. This observation already
proved useful in the context of neutral (delay) differential equations, see
[16, Section 5.3.3]. Note that an analogue property for the space of
continuous functions is not true, see [24, Remark 4.15]. Since $H^{1}$ embeds
continuously into the space of continuous functions, the presented approach
lies somewhat inbetween the retraction approach in [4] and the
$C^{1}$-technique in [26]. Note that, however, similar to the results
presented in [4], the developed assumptions are easily applicable and more
often than not boil down to routine arguments.

We remark that the solution theory developed here is independent of previous
solution results in the literature and is based on the contraction mapping
principle on $H^{1}$ combined with a retraction technique valid for Hilbert
spaces. We emphasise that the present approach addresses the non-autonomous
case right away and the assumptions are formulated in such a way that the
well-known (non-autonomous) Picard–Lindelöf theorem for ODEs is a simple
corollary of the theory developed here. Indeed, classical Picard–Lindelöf
needs to be given an initial value only rather than an initial pre-history.
Hence, the restriction to Lipschitz continuous pre-histories imposed in the
well-posedness theorem here offers no constriction for the classical case.

Next, we briefly summarise the outline of the paper. In Section 2, we gather
some facts about weighted $L_{2}$\- and $H^{1}$-spaces, particularly
concerning the pre-history mapping. Section 3 is concerned with the key
estimate showing that the evaluation mapping (and thus the delay functional)
can be suitably established as a Lipschitz continuous mapping. The solution
theory for (2) on $H^{1}$ is provided in Section 4. The sharpness of the
result is illustrated by an example similar to the one in [28]. We also
provide a permanence principle that eventually yields local solution theories
for when the delay functional $r$ is a priori not globally Lipschitz
continuous on $H^{1}$. In Section 5 one finds a suitable generalisation to
functional differential equations giving more details and a proof for Theorem
1.1. By means of applications in Section 6 we illustrate our theory with a
classical example from [26] and an equation from mathematical biology taken
from [9, 4]. In this section, we will particularly show how the assumptions on
unique existence of solutions in [4] can be used to obtain a local solution
theory also within the present situation. We close with a conclusion in
Section 7.

##  2 Functional analytic prerequisites and the pre-history map

We use standard terminology for the function spaces involved and briefly
recall the most important ones here, see, also, e.g. [25]. For this, let
$I\subseteq\mathbb{R}$ be a non-empty interval. For $\rho\in\mathbb{R}$,
$L_{2,\rho}(I)$ is the space of equivalence classes of square-integrable
functions on an interval $I$ with respect to the measure with Lebesgue density
$t\mapsto\exp(-2\rho t)$; and $H_{\rho}^{1}(I)$ is the space of once weakly
differentiable functions with derivative in $L_{2,\rho}(I)$. In case of
$\rho=0$, we also use $H^{1}(I)$ and $L_{2}(I)$, respectively. For
$\rho\in\mathbb{R}$, we define

$\langle x,y\rangle_{0,\rho}\coloneqq\int_{I}x(t)^{*}y(t)\exp(-2\rho
t)dt\text{ and }\langle x,y\rangle_{1,\rho}\coloneqq\langle
x,y\rangle_{0,\rho}+\langle x^{\prime},y^{\prime}\rangle_{0,\rho}$

and, analogously, the norm, $\|x\|_{L_{2,\rho}}$ and $\|x\|_{H^{1}_{\rho}}$,
respectively. Corresponding Hilbert space $X$-valued functions spaces will be
denoted by $L_{2,\rho}(I;X)$, etc.

Throughout this section, we let $h>0$ be our fixed pre-history horizon and
$T\in(0,\infty]$ be the final time considered in this section. We define for
$\rho\in\mathbb{R}$ the pre-history map

$\Theta\colon L_{2,\rho}(-h,T)\to L_{2,\rho}(0,T,L_{2}(-h,0))$

via

$\Theta f(t)\coloneqq(s\mapsto f(t+s))=f_{(t)}.$

We compute the operator-norm of $\Theta$ first.

######  Theorem 2.1.

Let $\rho>0$. Then

$\|\Theta\|\leq\frac{1}{\sqrt{2\rho}}$

###### Proof.

Let $f\in L_{2,\rho}(-h,T)$. Then we compute

$\displaystyle\|\Theta f\|^{2}_{L_{2,\rho}(0,T;L_{2}(-h,0))}$
$\displaystyle=\int_{0}^{T}\int_{-h}^{0}|f(t+s)|^{2}\,\mathrm{d}s\exp(-2\rho
t)\,\mathrm{d}t=\int_{-h}^{0}\int_{0}^{T}|f(t+s)|^{2}\exp(-2\rho
t)\,\mathrm{d}t\,\mathrm{d}s$
$\displaystyle=\int_{-h}^{0}\int_{0}^{T}|f(t+s)|^{2}\exp(-2\rho(t+s))\,\mathrm{d}t\exp(2\rho
s)\,\mathrm{d}s\leq\int_{-h}^{0}\exp(2\rho
s)\,\mathrm{d}s\|f\|_{L_{2,\rho}(-h,T)}^{2}$
$\displaystyle\leq\frac{1}{2\rho}\|f\|_{L_{2,\rho}(-h,T)}^{2}.\qed$

######  Remark 2.2.

Note that $\Theta$ is linear and, thus, commutes with the weak derivative. As
a consequence, we have for

$\hat{\Theta}\colon H_{\rho}^{1}(-h,T)\to L_{2,\rho}(0,T,H^{1}(-h,0))$

given by $f\mapsto\Theta f$ the same estimate, that is,

$\|\hat{\Theta}\|\leq\frac{1}{\sqrt{2\rho}}$

We introduce the time-shift, $f\mapsto\tau_{t}f\coloneqq f(\cdot+t)$. Then a
different representation of $\Theta$ is

$\Theta f=(t\mapsto\tau_{t}f).$

This perspective helps us to prove the next observation concerning the pre-
history map.

######  Proposition 2.3.

Let $\rho\in\mathbb{R}$. Then for all $f\in L_{2,\rho}(-h,T)$, we have

$\Theta f\in C([0,T);L_{2}(-h,0)).$

If, in addition, $T<\infty$, then $\Theta f\in C([0,T];L_{2}(-h,0))$ and

$\Theta\colon L_{2,\rho}(-h,T)\to C_{\rho}([0,T];L_{2}(-h,0))$

is continuous with operator norm bounded by $1$, where the latter space is
endowed with the norm given by
$\|f\|_{\rho}\coloneqq\sup_{t\in[0,T]}\mathrm{e}^{-\rho
t}\|f(t)\|_{L_{2}(-h,0)}$.

###### Proof.

The proof rests on a standard approximation technique and similar results can
be found in (almost) any basic text on $C_{0}$-semi-group theory. We provide
the short arguments here. First of all, since for all $t\in(0,\infty)$, we
have $L_{2,\rho}(-h,\infty)\hookrightarrow L_{2,\rho}(-h,t),f\mapsto
f|_{(0,t)}$ continuously it suffices to treat the case $T<\infty$.

Let $f\in C[-h,T]$. Then $\tau_{t}f\to\tau_{s}f$ in $L_{2,\rho}(-h,0)$ as
$t\to s$ in $[0,T]$ by continuity of $f$ and Lebesgue’s dominated convergence
theorem.

Next, it is elementary to see that for each $t\in[0,T]$, we have

$\|\tau_{t}f\|_{L_{2}(-h,0)}\leq\mathrm{e}^{\rho t}\|f\|_{L_{2,\rho}(-h,T)}.$

Thus,

$C[-h,T]\subseteq L_{2,\rho}(-h,T)\to
C_{\rho}([0,T];L_{2}(-h,0)),f\mapsto\Theta f$

is continuous with operator norm bounded by $1$. Continuous extension by
density of $C[-h,T]$ in $L_{2,\rho}(-h,T)$ shows the remaining assertions. ∎

######  Remark 2.4.

Similar to Theorem 2.1, we can also spell out the corresponding continuity
statement for $H^{1}$ instead of $L_{2}$. More precisely, assuming $T<\infty$,
it follows from Proposition 2.3 that for $f\in H^{1}_{\rho}(-h,T)$, we have

$\hat{\Theta}f\in C_{\rho}([0,T];H^{1}(-h,0)).$

Note that $\|\hat{\Theta}\|\leq 1$ as well.

In order for the objects in the next section to be well-defined, we quickly
recall the Sobolev embedding theorem.

######  Theorem 2.5 (Sobolev-embedding theorem).

Let $I\subseteq\mathbb{R}$ be a non-empty interval. Then every $f\in
H^{1}_{\textnormal{loc}}(I)$ admits a continuous representative. If $I=(a,b)$
for some $-\infty<a<b<\infty$, then

$\|f\|_{C[a,b]}\leq((b-a)^{1/2}+(b-a)^{-1/2})\|f\|_{H^{1}(a,b)}.$

###### Proof.

The proof can be found in [2, Theorem 4.9]; see also the standard reference
[1, Theorem 4.12, Case A]. ∎

##  3 A key estimate for the solution theory

In order to provide a solution theory for state-dependent delay equations, we
need to derive a key estimate. This estimate will render the anticipated
functional differential equation as ordinary differential equation with
Lipschitz continuous right-hand side. For this, we introduce the Lipschitz
semi-norm, $\|g\|_{\textnormal{Lip}}$ for a mapping $g\colon X\to Y$ between
metric spaces $(X,d)$ and $(Y,e)$ given by

$\|g\|_{\textnormal{Lip}}\coloneqq\sup_{x_{1},x_{2}\in X,x_{1}\neq
x_{2}}e(g(x_{1}),g(x_{2}))/d(x_{1},x_{2})\in[0,\infty].$

Evidently, $g$ is Lipschitz continuous if and only if
$\|g\|_{\textnormal{Lip}}<\infty$.

For $\beta>0$, we introduce

$V_{\beta}\coloneqq\\{\psi\in
H^{1}(-h,0;\mathbb{R}^{n});\|\psi^{\prime}\|_{\infty}\leq\beta\\},$

where $\|\cdot\|_{\infty}$ denotes the norm in
$L_{\infty}(-h,0;\mathbb{R}^{n})$ where $h>0$.

######  Remark 3.1.

$V_{\beta}\subseteq H^{1}(-h,0;\mathbb{R}^{n})$ is closed and convex. Indeed,
convexity being clear; we briefly address closedness. Given a sequence
$(\psi_{k})_{k}$ in $V_{\beta}$ converging to some $\psi\in
H^{1}(-h,0;\mathbb{R}^{n})$. Then, by the Fischer–Riesz theorem, we may assume
without loss of generality that $(\psi_{k}^{\prime})_{k}$ converges almost
everywhere to $\psi^{\prime}$; hence the $L_{\infty}$-norm estimate is
preserved.

In order to present the key ingredient warranting well-posedness of state-
dependent delay equations, we reformulate said equation as a functional
differential equation. For this, similar to as in [26], we introduce the
evaluation mapping $\operatorname{ev}\colon
H^{1}(-h,0;\mathbb{R}^{n})\times[-h,0]\to\mathbb{R}^{n}$ given by

$\operatorname{ev}(\phi,t)\coloneqq\phi(t),$

which is well-defined by Theorem 2.5.

######  Theorem 3.2.

Let $h,\beta>0$. Let both $g\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ and
$r\colon H^{1}(-h,0;\mathbb{R}^{n})\to[-h,0]$ be Lipschitz continuous.

Then $f\colon V_{\beta}\to\mathbb{R}^{n}$ with

$f=g\circ\operatorname{ev}\circ(\operatorname{id}\times r)$

satisfies

$\|f\|_{\textnormal{Lip}}\leq\|g\|_{\textnormal{Lip}}(2h^{1/2}+\beta\|r\|_{\textnormal{Lip}}+h^{-1/2}).$

###### Proof.

We compute for $\phi,\psi\in V_{\beta}$

$\displaystyle\|f(\phi)-f(\psi)\|_{\mathbb{R}^{n}}$
$\displaystyle=\|g\circ\operatorname{ev}\circ(\operatorname{id}\times
r)(\phi)-g\circ\operatorname{ev}\circ(\operatorname{id}\times
r)(\psi)\|_{\mathbb{R}^{n}}$
$\displaystyle\leq\|g\|_{\textnormal{Lip}}\|\phi(r(\phi))-\psi(r(\psi))\|_{\mathbb{R}^{n}}.$

Focussing on the right-hand factor of the right-hand side, we further estimate

$\displaystyle\|\phi(r(\phi))-\psi(r(\psi))\|_{\mathbb{R}^{n}}$
$\displaystyle\leq\|\phi(0)-\phi(r(\phi))+\psi(0)-\psi(r(\psi))\|_{\mathbb{R}^{n}}+\|\psi(0)-\phi(0)\|_{\mathbb{R}^{n}}$
$\displaystyle\leq\Big{\|}\int_{r(\phi)}^{0}\phi^{\prime}(s)\,\mathrm{d}s-\int_{r(\psi)}^{0}\psi^{\prime}(s)\,\mathrm{d}s\Big{\|}_{\mathbb{R}^{n}}+(h^{1/2}+h^{-1/2})\|\psi-\phi\|_{H^{1}},$

using Theorem 2.5. Assuming without restriction that $r(\phi)\leq r(\psi)$ and
using the Cauchy–Schwarz inequality, we treat the integral terms next:

$\displaystyle\Big{\|}\int_{r(\phi)}^{0}\phi^{\prime}(s)\,\mathrm{d}s-\int_{r(\psi)}^{0}\psi^{\prime}(s)\,\mathrm{d}s\Big{\|}$
$\displaystyle\leq\Big{\|}\int_{r(\psi)}^{0}\phi^{\prime}(s)\,\mathrm{d}s-\int_{r(\psi)}^{0}\psi^{\prime}(s)\,\mathrm{d}s\Big{\|}_{\mathbb{R}^{n}}+\Big{\|}\int_{r(\phi)}^{r(\psi)}\phi^{\prime}(s)\,\mathrm{d}s\Big{\|}_{\mathbb{R}^{n}}$
$\displaystyle\leq\|1\|_{L_{2}(-h,0)}\|\phi^{\prime}-\psi^{\prime}\|_{L_{2}}+|r(\psi)-r(\phi)|\beta$
$\displaystyle\leq\sqrt{h}\|\phi^{\prime}-\psi^{\prime}\|_{L_{2}}+\beta\|r\|_{\textnormal{Lip}}\|\psi-\phi\|_{H^{1}}.$

Putting the above inequalities together, we conclude the proof. ∎

In applications, the delay functional $r$ assigning the time at which the
unknown is to be evaluated on the right-hand side might not be Lipschitz
continuous on the whole of $H^{1}(-h,0;\mathbb{R}^{n})$. Moreover, $f$ as
given in Theorem 3.2 is only Lipschitz continuous on $V_{\beta}$ for all
$\beta>0$. However, in all the cases we consider in Section 6, $r$ admits a
Lipschitz continuous _extension_ to $H^{1}(-h,0;\mathbb{R}^{n})$ and so does
$f$ from Theorem 3.2. The reason for this is the Hilbert space structure of
$H^{1}(-h,0;\mathbb{R}^{n})$, which provides the possibility of defining
projections onto closed convex subsets. Further to the comments in the
introduction, we mention that in general also the metric projection in general
Banach spaces is not as well-behaved as in the Hilbert space case. Indeed, any
closed convex admitting a best-approximation as in Proposition 3.3 requires
the space considered to be reflexive and strictly convex excluding the space
of continuous functions or continuously differentiable functions, see Theorem
of Day–James in [20, Chapter 5, p 436]. Moreover, if one was to extend $r$
assumed to be a $C^{1}$-function defined on a subset of
$H^{1}(-h,0;\mathbb{R}^{n})$ to a $C^{1}$-function on the whole space, one is
confronted with a challenging matter on its own, see, e.g., [3] and the
references therein. For extending $r$ to a Lipschitz map on
$H^{1}(-h,0;\mathbb{R}^{n})$, we briefly recall the following abstract result
for (minimal) projections in Hilbert spaces.

######  Proposition 3.3 (see, e.g., [6, p 142]).

Let $H$ be a Hilbert space, $C\subseteq H$ closed and convex. Then for all
$x\in H$ there exists a unique $P_{C}x\in C$ such that

$\|x-P_{C}x\|=\inf_{y\in C}\|x-y\|.$

Moreover, $P_{C}$ is a Lipschitz continuous projection with Lipschitz constant
bounded by $1$.

As a consequence of the latter result, we can now extend any Lipschitz
continuous function on the full Hilbert space:

######  Corollary 3.4 (Lipschitz continuous extension).

Let $H$ be a Hilbert space, $\emptyset\neq C\subseteq H$ closed and convex,
$Y$ a metric space. If $r\colon C\to Y$ is Lipschitz continuous, then
$r_{H}\coloneqq r\circ P_{C}$ extends $r$ on $H$ with

$\|r_{H}\|_{\textnormal{Lip}}=\|r\|_{\textnormal{Lip}}$

###### Proof.

Since by Proposition 3.3, $P_{C}^{2}=P_{C}$ it follows that

$r_{H}|_{C}=r,$

which leads to $\|r_{H}\|_{\textnormal{Lip}}\geq\|r\|_{\textnormal{Lip}}$.
Proposition 3.3 yields that $r_{H}$ is Lipschitz continuous and
$\|r_{H}\|_{\textnormal{Lip}}\leq\|r\|_{\textnormal{Lip}}$. ∎

##  4 Solution theory for state-dependent delay equations

Throughout, we let $h,T>0$. We emphasise that $T<\infty$. We aim to provide a
solution theory for any Lipschitz continuous given pre-history
$\phi:[-h,0]\to\mathbb{R}^{n}$, of the following initial value problem on
$(0,T)$,

$\begin{cases}x^{\prime}(t)=g(t,x(t),x(t+r(x_{(t)}))),&t\in(0,T);\\\
x_{(0)}=\phi.\end{cases}$ (2)

The main assumptions for the equation just introduced are the following.

  1. (H1)

$g\colon[0,T]\times\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}^{n}$ is
continuous and uniformly Lipschitz continuous with respect to the spatial
variables, i.e., there exists $L\geq 0$ such that for all $t\in[0,T]$ and
$x,y,u,v\in\mathbb{R}^{n}$ we have

$\|g(t,x,u)-g(t,y,v)\|\leq L(\|x-y\|+\|u-v\|).$

  2. (H2)

$r\colon H^{1}(-h,0;\mathbb{R}^{n})\to[-h,0]$ is Lipschitz continuous.

We aim to show the following theorem.

######  Theorem 4.1 (Picard–Lindelöf for SDDEs).

Let $g$ and $r$ be as in (H1) and (H2). Then for all $\phi\in
H^{1}(-h,0;\mathbb{R}^{n})$ with bounded derivative, there exists a unique
$x\in H^{1}(-h,T;\mathbb{R}^{n})$ satisfying (2).

######  Remark 4.2.

(a) By the Sobolev embedding theorem, any solution of (2) is continuous. This,
however, does not require the derivative of $x$ to be continuous—even if $x$
solves the equation. This has the following effect: the continuity of
$t\mapsto\tau_{t}x$ as an $L_{2}$ (or $H^{1}$)-valued function (see
Proposition 2.3) forces the right-hand side of (2) to be continuous in $t$.
Thus, as $t\to 0+$, the right-hand side converges to
$g(0,\phi(0),\phi(r(\phi))$; the left-hand side does also converge; the limit,
however, is $x^{\prime}(0+)$ and is _a priori unrelated_ to $\phi^{\prime}$.
Therefore, $\phi$ does not ‘see’ the solution manifold. As a consequence, in
Theorem 4.1 there is no restriction on the initial value other than its
(bounded) weak differentiability. In particular, the solution manifold is not
needed for the formulation of the theorem.

(b) It follows from the equation, that any solution $x$ of (2) satisfies $x\in
H^{1}(-h,T;\mathbb{R}^{n})\subseteq C([-h,T],\mathbb{R}^{n})$, $x|_{[0,T]}\in
C^{1}([0,T];\mathbb{R}^{n})$. Hence, if $T>h$ and $x\in
H^{1}(-h,T;\mathbb{R}^{n})$ solves (2) for some $\phi\in
H^{1}(-h,0;\mathbb{R}^{n})$ with bounded derivative; then $x(h+\cdot)$ belongs
to the solution manifold as introduced in [26].

(c) A special case of Theorem 4.1 is the classical form of the theorem of
Picard–Lindelöf, see e.g. [25, Theorem 4.2.6 (and proof)].

(d) Let $-\infty<a<b<\infty$ and $\phi\colon(a,b)\to\mathbb{R}^{n}$. Then
$\phi\in H^{1}(a,b;\mathbb{R}^{n})$ with bounded derivative if and only if
$\phi$ is Lipschitz continuous. Indeed, the ‘only if’-part is a
straightforward consequence of integration by parts and a standard estimate
for the integral. For the ‘if’-part note that Lipschitz continuity of $\phi$
implies boundedness of $\phi$ on $(a,b)$. Hence, $\phi\in
L_{2}(a,b;\mathbb{R}^{n})$. By Rademacher’s theorem (see, e.g., [8, Section
3.1.2]), the differential quotient of $\phi$ exists at almost every point; by
Lipschitz continuity, the differential quotient is bounded by
$\|\phi\|_{\textnormal{Lip}}$. Hence, Lebesgue’s dominated convergence theorem
and Fubini’s theorem imply that $\phi$ is weakly differentiable with
derivative given by the almost everywhere limit of the difference quotients.
As these are uniformly bounded, the weak derivative $\phi^{\prime}$ is bounded
and hence in $L_{2}(a,b;\mathbb{R}^{n})$, eventually yielding $\phi\in
H^{1}(a,b;\mathbb{R}^{n})$.

(e) A combination of the arguments in (a) and (d) shows that any solution of
(2) is Lipschitz continuous. Indeed, for this it suffices to observe that
$[0,T]\ni t\mapsto g(t,x(t),x(t+r(x_{(t)})))$ is continuous and that the pre-
history $\phi$ is Lipschitz continuous by assumption.

The next (non-)example of Theorem 4.1 is a variant of the one provided in
[28]. This example shows that the assumptions in Theorem 4.1 are sharp. We
emphasise, however, that the considered equation does satisfy (H1) and (H2)
and, thus, Lipschitz continuous pre-histories imply long-term unique existence
of solutions by Theorem 4.1.

######  Example 4.3.

Consider the state-dependent delay equation given by

$x^{\prime}(t)=-x(t-\min\\{|x(t)|,2\\})$

with $h=2$ and $T>0$. Then $g(t,x,u)=-u$ and $r\colon H^{1}(-h,0)\to[-h,0]$
given by $r(\phi)=-\min\\{|\phi(0)|,2\\}$ satisfy the assumptions of Theorem
4.1. Indeed, (H1) is clear. For (H2), we estimate for $\phi,\psi\in
H^{1}(-h,0)$

$|r(\phi)-r(\psi)|=|\min\\{|\phi(0)|,2\\}-\min\\{|\psi(0)|,2\\}|\leq|\phi(0)-\psi(0)|\leq\tfrac{3\sqrt{2}}{2}\|\phi-\psi\|_{H^{1}},$

by Theorem 2.5. For $T>0$ sufficiently small both $x_{1}(t)=1+t$ and
$x_{2}(t)=1+t-t^{3}$ satisfy the differential equation with the pre-history

$\phi(t)\coloneqq\begin{cases}-1,&-2\leq t<-1\\\ 3(t+1)^{2/3}-1,&-1\leq
t\leq-\tfrac{\sqrt{27}-1}{\sqrt{27}},\\\
\tfrac{\sqrt{27}}{\sqrt{27}-1}t+1,&-\tfrac{\sqrt{27}-1}{\sqrt{27}}<t\leq 0,\\\
\end{cases}$

As $\phi\in H^{1}(-h,0)$ and uniqueness fails for this pre-history, a solution
theory on $H^{1}(-h,0)$ alone cannot be expected even if $r$ is supposed to be
globally Lipschitz a priori. Note that Theorem 4.1 does not apply as $\phi$ is
not Lipschitz continuous for its derivative blows up at $t=-1$.

The proof of Theorem 4.1 requires several preliminary results. To begin with,
as in the classical setting of ordinary differential equations, we shall
reformulate the initial value problem (2) into an integral equation. Similar
to [14], we define

$\hat{\phi}(t)\coloneqq\begin{cases}\phi(t),&t\in(-h,0]\\\
\phi(0),&t\in(0,T)\end{cases}$

and aim to derive a fixed point problem for functions vanishing on $[-h,0]$.
Note that $\hat{\phi}\in H^{1}(-h,T;\mathbb{R}^{n})$. Indeed, this is a direct
consequence of Fubini’s theorem.

######  Lemma 4.4.

Let $x\in H^{1}(-h,T;\mathbb{R}^{n})$ and $p\colon
H^{1}(-h,0;\mathbb{R}^{n})\to H^{1}(-h,0;\mathbb{R}^{n})$ be a continuous
mapping. Then the following conditions are equivalent:

  1. (i)

$x$ satisfies

$x^{\prime}=g(\cdot,x(\cdot),\operatorname{ev}(\operatorname{id}\times
r)(p(x_{(\cdot)}))))\text{ on }(0,T)$

and $x_{0}=\phi$.

  2. (ii)

$x=y+\hat{\phi}$, where $y(t)=0$ on $(-h,0]$ and for $t\in(0,T)$

$y(t)=\int_{0}^{t}g(s,y(s),\operatorname{ev}(\operatorname{id}\times
r)(p((y+\hat{\phi})_{(s)})))\,\mathrm{d}s$

and $y\in H^{1}(-h,T;\mathbb{R}^{n})$.

###### Proof.

We note that the right-hand side of the differential equation (in (i) and the
integrand in (ii)) is continuous by composition of continuous maps and the
continuity of the shift taking values in $H^{1}$, see Proposition 2.3.

(i)$\Rightarrow$(ii) Integration over $(0,t)$ for some $t<T$ of both left- and
right-hand sides yield

$x(t)-x(0)=\int_{0}^{t}g(s,x(s),x(s+r(p(x_{(s)}))))\,\mathrm{d}s.$

Defining $y(t)\coloneqq x(t)-\hat{\phi}(t)$, we obtain the desired integral
equation; where we also used that $x(0)=\phi(0)$ due to continuity of both $x$
and $\phi$.

(ii)$\Rightarrow$(i) The claim follows using the fundamental theorem of
calculus from the integral equation after having differentiated with respect
to $t$. ∎

######  Remark 4.5.

We apply Lemma 4.4 in two different cases, one for $p=\operatorname{id}$ and
two for $p=\pi_{\beta}$, where for $\beta>0$, we denote $\pi_{\beta}\coloneqq
P_{V_{\beta}}\colon H^{1}(-h,0)\to H^{1}(-h,0)$, the orthogonal projection
onto $V_{\beta}$ as defined in Proposition 3.3.

Next, we prove exponential bounds for any solution of (2).

######  Lemma 4.6.

Let $\phi\in H^{1}(-h,0;\mathbb{R}^{n})$ with bounded derivative and let $x\in
H^{1}(-h,T;\mathbb{R}^{n})$ be a solution of (2). Then for all $t\in[0,T]$, we
have

$\|x(t)\|\leq(\|\phi\|_{\infty}+\int_{0}^{t}\|g(s,0,0)\|\,\mathrm{d}s)\mathrm{e}^{2Lt}.$

###### Proof.

We start off by recalling that $x\in H^{1}(-h,T;\mathbb{R}^{n})$ is continuous
by Theorem 2.5. For $t\in[0,T]$, we introduce

$\kappa(t)\coloneqq\sup_{s\in[-h,t]}\|x(s)\|=\max_{s\in[-h,t]}\|x(s)\|.$

Since $x$ is continuous, so is $\kappa$. Hence, for $t\in[0,T]$, we estimate
using Lemma 4.4,

$\displaystyle\|x(t)\|$
$\displaystyle\leq\|\phi(0)\|+\big{\|}\int_{0}^{t}g(s,x(s),x(s+r(x_{(s)})))\,\mathrm{d}s\big{\|}$
$\displaystyle\leq\|\phi(0)\|+\int_{0}^{t}\|g(s,x(s),x(s+r(x_{(s)})))-g(s,0,0)\|\,\mathrm{d}s+\int_{0}^{t}\|g(s,0,0)\|\,\mathrm{d}s$
$\displaystyle\leq\|\phi(0)\|+\int_{0}^{t}L(\|x(s)\|+\|x(s+r(x_{(s)}))\|)\,\mathrm{d}s+\int_{0}^{t}\|g(s,0,0)\|\,\mathrm{d}s$
$\displaystyle\leq\|\phi(0)\|+\int_{0}^{t}\|g(s,0,0)\|\,\mathrm{d}s+\int_{0}^{t}2L\max_{\tau\in[0,s]}\|x(\tau)\|\,\mathrm{d}s.$

Hence, with
$\alpha(t)\coloneqq\|\phi(0)\|+\int_{0}^{t}\|g(s,0,0)\|\,\mathrm{d}s$,

$\kappa(t)\leq\alpha(t)+2L\int_{0}^{t}\kappa(s)\,\mathrm{d}s.$

Thus, Grönwall’s inequality implies

$\|x(t)\|\leq\kappa(t)\leq\alpha(t)\mathrm{e}^{2Lt}.\qed$

The following auxiliary statement is a variant of [25, p 44 (see also Lemma
3.2.1)] (see also [23, p 1772]) on finite time intervals.

######  Proposition 4.7.

For all $\rho>0$, the operator

$I_{\rho}\colon L_{2,\rho}(0,T)\to L_{2,\rho}(0,T)$

given by

$I_{\rho}f(t)\coloneqq\int_{0}^{t}f(s)\,\mathrm{d}s$

is continuous with $\|I_{\rho}\|\leq 1/\rho$.

###### Proof.

Let $f\in L_{2,\rho}(0,T)$. Then we compute

$\displaystyle\|I_{\rho}f\|^{2}$
$\displaystyle=\int_{0}^{T}\big{\|}\int_{0}^{t}f(s)\,\mathrm{d}s\big{\|}^{2}\mathrm{e}^{-2\rho
t}\,\mathrm{d}t$
$\displaystyle\leq\int_{0}^{T}\big{(}\int_{0}^{T}\|f(s)\|\mathds{1}_{(0,T)}(t-s)\,\mathrm{d}s\big{)}^{2}\mathrm{e}^{-2\rho
t}\,\mathrm{d}t$
$\displaystyle=\int_{0}^{T}\big{(}\int_{0}^{T}\|f(s)\|\mathrm{e}^{-\rho
s}\mathds{1}_{(0,T)}(t-s)\mathrm{e}^{-\rho(t-s)}\,\mathrm{d}s\big{)}^{2}\,\mathrm{d}t$
$\displaystyle\leq\int_{0}^{T}\int_{0}^{T}\|f(s)\|^{2}\mathrm{e}^{-2\rho
s}\mathds{1}_{(0,T)}(t-s)\mathrm{e}^{-\rho(t-s)}\,\mathrm{d}s\big{(}\int_{0}^{T}\mathds{1}_{(0,T)}(t-s)\mathrm{e}^{-\rho(t-s)}\,\mathrm{d}s\big{)}\,\mathrm{d}t$
$\displaystyle\leq\frac{1}{\rho}\int_{0}^{T}\int_{0}^{T}\|f(s)\|^{2}\mathrm{e}^{-2\rho
s}\mathds{1}_{(0,T)}(t-s)\mathrm{e}^{-\rho(t-s)}\,\mathrm{d}t\,\mathrm{d}s\leq\frac{1}{\rho^{2}}\|f\|_{L_{2,\rho}}^{2}.\qed$

We introduce, for $\rho>0$,

$H_{0,\rho}^{1}(0,T;\mathbb{R}^{n})\coloneqq\\{y\in
H_{\rho}^{1}(-h,T;\mathbb{R}^{n});y=0\text{ on }(-h,0]\\}.$

As a consequence of the previous proposition, we get the following estimate.

######  Lemma 4.8.

Let $\rho>0$, $f\in H_{0,\rho}^{1}(0,T;\mathbb{R}^{n})$. Then
$\|f\|_{L_{2,\rho}}\leq\frac{1}{\rho}\|f^{\prime}\|_{L_{2,\rho}}$.

###### Proof.

By the fundamental theorem, we obtain $I_{\rho}f^{\prime}=f$ due to $f$
vanishing at $0$; thus, the claim follows from Proposition 4.7. ∎

We recall that $\pi_{\beta}\coloneqq P_{V_{\beta}}\colon
H^{1}(-h,0;\mathbb{R}^{n})\to H^{1}(-h,0;\mathbb{R}^{n})$ denotes the
orthogonal projection onto $V_{\beta}$ as defined in Proposition 3.3.

######  Theorem 4.9.

Let $\beta>0$ and $\phi\in H^{1}(-h,0;\mathbb{R}^{n})$. Then there is a unique
$y\in H^{1}(-h,T;\mathbb{R}^{n})$ satisfying

$y(t)=\int_{0}^{t}g(s,y(s)+\phi(0),\operatorname{ev}(\operatorname{id}\times
r)(\pi_{\beta}((y+\hat{\phi})_{(s)})))\,\mathrm{d}s\quad(0<t<T)$

and $y=0$ on $(-h,0)$.

###### Proof.

Consider the mapping $F\colon H_{0,\rho}^{1}(0,T)\to H_{0,\rho}^{1}(0,T)$
given by

$Fy(t)\coloneqq\int_{0}^{t}g(s,y(s),\operatorname{ev}(\operatorname{id}\times
r)(\pi_{\beta}(y+\hat{\phi})_{(s)}))\,\mathrm{d}s.$

We shall prove that $F$ is a strict contraction for $\rho$ sufficiently large.
For this, recall that $\|\pi_{\beta}\|_{\textnormal{Lip}}\leq 1$ by
Proposition 3.3. For $u,v\in H_{0,\rho}^{1}(0,T)$ we compute using Proposition
4.7 (in the second line), Theorem 3.2 (in the fourth line), and Theorem 2.1
(and subsequent remark; in the seventh line)

$\displaystyle\|Fu-Fv\|_{L_{2,\rho}}$
$\displaystyle\leq\frac{1}{\rho}\big{\|}g(\cdot,u(\cdot),\operatorname{ev}(\operatorname{id}\times
r)(\pi_{\beta}(u+\hat{\phi})_{(\cdot)}))-g(\cdot,v(\cdot),\operatorname{ev}(\operatorname{id}\times
r)(\pi_{\beta}(v+\hat{\phi})_{(\cdot)}))\big{\|}_{L_{2,\rho}}$
$\displaystyle\leq\frac{1}{\rho}L(\|u-v\|_{L_{2,\rho}}+\|\operatorname{ev}(\operatorname{id}\times
r)(\pi_{\beta}(u+\hat{\phi})_{(\cdot)})-\operatorname{ev}(\operatorname{id}\times
r)(\pi_{\beta}(v+\hat{\phi})_{(\cdot)})\|_{L_{2,\rho}})$
$\displaystyle\leq\frac{1}{\rho}L(\|u-v\|_{L_{2,\rho}}+(2h^{1/2}+\beta\|r\|_{\textnormal{Lip}}+h^{-1/2})\|\|\pi_{\beta}((u+\hat{\phi})_{(\cdot)})-\pi_{\beta}(v+\hat{\phi})_{(\cdot)})\|_{H^{1}}\|_{L_{2,\rho}})$
$\displaystyle\leq\frac{1}{\rho}L(\|u-v\|_{L_{2,\rho}}+(2h^{1/2}+\beta\|r\|_{\textnormal{Lip}}+h^{-1/2})\|\|u_{(\cdot)}-v_{(\cdot)}\|_{H^{1}}\|_{L_{2,\rho}})$
$\displaystyle=\frac{1}{\rho}L(\|u-v\|_{L_{2,\rho}}+(2h^{1/2}+\beta\|r\|_{\textnormal{Lip}}+h^{-1/2})\|\hat{\Theta}u-\hat{\Theta}v\|_{L_{2,\rho}(0,T;H^{1}(-h,0;\mathbb{R}^{n}))})$
$\displaystyle\leq\frac{1}{\rho}L(\|u-v\|_{L_{2,\rho}}+\frac{1}{\sqrt{2\rho}}(2h^{1/2}+\beta\|r\|_{\textnormal{Lip}}+h^{-1/2})\|u-v\|_{H^{1}_{\rho}})$

Similarly, we get

$\displaystyle\|(Fu)^{\prime}-(Fv)^{\prime}\|_{L_{2,\rho}}$ $\displaystyle\leq
L\|u-v\|_{L_{2,\rho}}+L\frac{1}{\sqrt{2\rho}}(2h^{1/2}+\beta\|r\|_{\textnormal{Lip}}+h^{-1/2})\|u-v\|_{H^{1}_{\rho}}$
$\displaystyle\leq
L\big{(}\frac{1}{\rho}+\frac{1}{\sqrt{2\rho}}(2h^{1/2}+\beta\|r\|_{\textnormal{Lip}}+h^{-1/2})\big{)}\|u-v\|_{H^{1}_{\rho}}$

Hence, choosing $\rho$ large enough, we identify $F$ as a strict contraction.
Thus, by the contraction mapping principle, unique existence of a fixed point
is guaranteed. ∎

Next, we provide a criterion for when we can drop the projection $\pi_{\beta}$
onto $V_{\beta}$.

######  Theorem 4.10.

Let $\phi\in H^{1}(-h,0;\mathbb{R}^{n})$, $\beta>0$. If $\phi\in V_{\beta}$
and

$\|g(0,\phi(0),\phi(r(\phi)))\|<\beta,$

then there exists $0<T_{\beta}\leq T$ such that $y+\phi$ is a solution of (2)
on $(-h,T_{\beta})$.

###### Proof.

Note that $y(0)=0$. Moreover,

$\displaystyle\|(y+\hat{\phi})_{s}-\phi\|_{H^{1}(-h,0;\mathbb{R}^{n})}^{2}=\|\tau_{s}(y+\hat{\phi})-\phi\|_{H^{1}(-h,0;\mathbb{R}^{n})}^{2}\to
0$

as $s\to 0$, by Proposition 2.3 (and Remark 2.4). Hence,

$y^{\prime}(t)=g(t,y(t)+\phi(0),\operatorname{ev}(\operatorname{id}\times
r)(\pi_{\beta}(y+\phi)_{(t)}))\to g(0,\phi(0),\phi(r(\phi)))$

as $t\to 0+$. As $\|g(0,\phi(0),\phi(r(\phi))\|<\beta$, we find
$0<T_{\beta}\leq T$ such that

$\sup_{t\in(0,T_{\beta})}\|y^{\prime}(t)\|<\beta$

Hence, for all $t\in(0,T_{\beta})$:
$\pi_{\beta}((y+\phi)_{(t)})=(y+\phi)_{(t)}$ and the theorem is proven. ∎

Before we turn to a proof of Theorem 4.1, we establish uniqueness.

######  Lemma 4.11.

Let $\phi\in H^{1}(-h,0;\mathbb{R}^{n})$ with bounded derivative. Then there
exists at most one solution $x\in H^{1}(-h,T;\mathbb{R}^{n})$ of (2).

###### Proof.

Let $x^{(1)},x^{(2)}\in H^{1}(-h,T;\mathbb{R}^{n})$ satisfy (2) and consider
$w(t)\coloneqq x^{(1)}(t)-x^{(2)}(t)$. By Lemma 4.4, for all $t\in[-h,0]$,
$w(t)=0$. Moreover, by Remark 4.2 (e), both $x^{(1)},x^{(2)}$ are Lipschitz
continuous. Let $\beta>0$ be a common Lipschitz constant. For $t>0$, we obtain

$\displaystyle\sup_{\tau\in[-h,t]}\|w(\tau)\|$
$\displaystyle\leq\int_{0}^{t}\|g(s,x^{(1)}(s),x^{(1)}(s+r(x^{(1)}_{s})))-g(s,x^{(2)}(s),x^{(2)}(s+r(x^{(2)}_{s})))\|\,\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}L(1+\beta)\sup_{\tau\in[-h,s]}\|w(\tau)\|\,\mathrm{d}s.$

Grönwall’s inequality yields $w=0$; that is, $x^{(1)}=x^{(2)}$. ∎

######  Remark 4.12.

Using the same technique as in the proof of this lemma; we can show a
continuous dependence result on the data. In fact, one obtains

$\sup_{t\in[0,T]}\|x_{\phi}(t)-x_{\psi}(t)\|\leq C\|\phi-\psi\|_{\infty},$

for some $C$ depending on $T$, where $x_{\phi}$ and $x_{\psi}$ are solutions
of (2) with initial conditions $(x_{\phi})_{0}=\phi$ and
$(x_{\psi})_{0}=\psi$, respectively.

The preparations are complete for the main result of the present paper.

###### Proof of Theorem 4.1.

Uniqueness has been established in Lemma 4.11. By assumption, we find
$\beta_{0}>0$ such that $\phi\in V_{\beta_{0}}$ and
$\|g(0,\phi(0),\phi(r(\phi)))\|\leq\beta_{0}$. Let $\beta>\beta_{0}$. Then, by
Theorems 4.9 and 4.10, there exists $0<T_{\beta}\leq T$ and a unique solution
$x$ on $(0,T_{\beta})$. Without loss of generality, we may choose the maximal
$T_{\beta}$; that is, such that either $T_{\beta}=T$ or for all $t<T_{\beta}$,
$\|x^{\prime}(t)\|<\beta$ and $\|x^{\prime}(T_{\beta})\|=\beta$, which is
possible as $t\mapsto x^{\prime}(t)=g(t,x(t),x(t+r(x_{(t)})))$ is continuous.
We define $T_{2\beta}$ along the same lines. It follows that either
$T_{2\beta}=T$ or $T_{2\beta}<T$ and $T_{\beta}<T_{2\beta}$. The latter
inequality follows by uniqueness of the solution of (2) on $(-h,T_{\beta})$ by
Lemma 4.11 and continuity of $t\mapsto
x^{\prime}(t)=g(t,x(t),x(t+r(x_{(t)})))$. We obtain a sequence
$(T_{k\beta})_{k\in\mathbb{N}}$ of strictly positive numbers, which is
increasing and either eventually constant $T$; or converges to some
$T_{\infty}<T$. The latter case implies that

$\|g(T_{k\beta},x(T_{k\beta}),x(T_{k\beta}+r(x_{(T_{k\beta})})))\|=\|x^{\prime}(T_{k\beta})\|=k\beta\to\infty\quad(k\to\infty).$

It follows that $\sup_{t\in[-h,T_{\infty}]}\|x(t)\|=\infty$ for otherwise $g$
is bounded on bounded sets by continuity of $g$. Lemma 4.6, however, implies
$\sup_{t\in[-h,T_{\infty}]}\|x(t)\|<\infty$, which is a contradiction. Hence,
$(T_{k\beta})_{k\in\mathbb{N}}$ is eventually constant equal to $T$ and the
claim follows. ∎

We conclude this section with a permanence principle: Given a solution $x$ of
(2), for every $\varepsilon>0$ there exists a time $0<T_{\varepsilon}\leq T$
such that both the pre-history of $x$ and $x^{\prime}$ does not deviate from
$\phi$ more than $\varepsilon$ in the _sup-norm_. This permanence principle is
important, if $r$ is given as a Lipschitz continuous extension of the type
considered in Corollary 3.4 and one wants to find a time-horizon, where the
projection is the identity.

######  Theorem 4.13 (Permanence principle).

Let $\phi\in H^{1}(-h,0;\mathbb{R}^{n})$ with bounded derivative and let $x\in
H^{1}(-h,T;\mathbb{R}^{n})$ be a solution of (2). Let $\varepsilon>0$. Then
there exists $T_{\beta}>0$ such that for all $0\leq t\leq T_{\beta}$

$\sup_{s\in[-h,0]}(\|x(s+t)-\phi(s)\|_{\mathbb{R}^{n}})+\|x^{\prime}(t)-g(0,\phi(0),\phi(r(\phi)))\|_{\mathbb{R}^{n}})\leq\varepsilon.$

###### Proof.

The solution $x\in H^{1}(-h,T;\mathbb{R}^{n})$ agrees with $\phi$ on $[-h,0]$.
Since $x$ is continuous by Theorem 2.5, it is uniformly continuous and bounded
on the compact set $[-h,T]$. In particular, $g$ is uniformly continuous on
$[0,T]\times K\times K$ with $K=x[[-h,T]]$. We find $\delta_{0}>0$ such that
for all $t,s\in[0,T],u,v,w,z\in K$ with $|t-s|+\|u-v\|+\|w-z\|\leq\delta_{0}$,
we infer

$|g(t,u,w)-g(s,v,z)|\leq\varepsilon/2.$

By uniform continuity of $x$, we find $\delta_{1}>0$ such that
$\|x(s+t)-\phi(s)\|\leq\min\\{\varepsilon/2,\delta_{0}/3\\}$ for all
$s\in[-h,0]$ as long as $0\leq t\leq\delta_{1}$. By continuity of the shift
(see Proposition 2.3 and Remark 2.4) and continuity of $r$, we find
$\delta_{2}>0$ such that $\|r(x(t+\cdot))-r(\phi)\|\leq\delta_{1}/2$ for all
$0\leq t\leq\delta_{2}$.

Thus, for
$T_{\varepsilon}\coloneqq\min\\{\delta_{0}/3,\delta_{2},\delta_{1}/2\\}$ and
$0<t\leq T_{\varepsilon}$ we estimate

$\|x(s+t)-\phi(s)\|\leq\varepsilon/2\quad(s\in[-h,0])$

and

$\displaystyle\|x^{\prime}(t)-g(0,\phi(0),\phi(r(\phi)))\|$
$\displaystyle=\|g(t,x(t),x(t+r(x_{(t)})))-g(0,\phi(0),\phi(r(\phi)))\|\leq\varepsilon/2,$

as

$|t|+\|x(t)-\phi(0)\|+\|x(t+r(x_{(t)}))-\phi(r(\phi))\|\\\
\leq\tfrac{\delta_{0}}{3}+\tfrac{\delta_{0}}{3}+\|x(t+r(x_{(t)})-r(\phi)+r(\phi))-\phi(r(\phi))\|\leq\delta_{0},$

where we used that $r(\phi)\in[-h,0]$ and
$|t+r(x_{(t)})-r(\phi)|\leq\delta_{1}$. ∎

In applications, the permanence principle will serve as a means to provide a
local solution theory. Before we come to some applications we shall consider
the extension of the techniques developed here to functional differential
equations.

##  5 Local solution theory for functional differential equations

An easy adaptation of the techniques carried out in the proof of Theorem 4.1
will show the following more general theorem for functional differential
equations (FDEs). The right-hand side of the differential equation being more
general as opposed to (2), we deduce a local unique existence result only.
Again, let $h,T>0$. We say that $G\colon[0,T]\times
H^{1}(-h,0;\mathbb{R}^{n})\to\mathbb{R}^{m}$ is almost uniformly Lipschitz
continuous, if it is continuous and if for all $\beta>0$, there exists $L\geq
0$ such that for all $u,v\in V_{\beta}$ and $t\in[0,T]$

$\|G(t,u)-G(t,v)\|_{\mathbb{R}^{n}}\leq L\|u-v\|_{H^{1}}$

We shall revisit Theorem 1.1 from the introduction. Note that a certain
permanence principle is also true. We refrain from stating and proving this
for the functional differential equation at hand as the arguments are very
similar to case for (2).

######  Theorem 5.1 (Picard–Lindelöf for FDEs).

Let $G\colon[0,T]\times H^{1}(-h,0;\mathbb{R}^{n})\to\mathbb{R}^{n}$ be almost
uniformly Lipschitz continuous. Then for all $\phi\in
H^{1}(-h,0;\mathbb{R}^{n})$ with bounded derivative, there exists $0<T_{0}\leq
T$ and $x\colon(-h,T_{0})\to\mathbb{R}^{n}$ such that for all
$T_{1}\in(0,T_{0})$, $x|_{(-h,T_{1})}\in H^{1}(-h,T_{1};\mathbb{R}^{n})$ is
the unique solution of

$\displaystyle x^{\prime}(t)=G(t,x_{(t)})\quad(t\in(0,T_{1})),\quad
x_{0}=\phi.$ (FDE)

If $T_{0}$ is chosen maximally, then either $\|x\|_{\textnormal{Lip}}=\infty$
or both $T_{0}=T$ and $x\in H^{1}(-h,T;\mathbb{R}^{n})$.

###### Proof.

At first we show that any solution $x\in H^{1}(-h,T;\mathbb{R}^{n})$ of (FDE)
is Lipschitz continuous. Since $\phi$ is Lipschitz, this follows if we show
that any solution $x$ of (FDE) has a bounded derivative on $(0,T)$, see Remark
4.2(d). Note that by Remark 2.4, the mapping $[0,T]\ni t\mapsto x_{(t)}\in
H^{1}(-h,0;\mathbb{R}^{n})$ is continuous. Hence, the image, $X$, of said
mapping is a compact subset of $H^{1}(-h,0;\mathbb{R}^{n})$. Since $G$ is
continuous, $G[[0,T]\times X]\subseteq\mathbb{R}^{n}$ is compact and hence
bounded. Thus, $\|x^{\prime}(t)\|\leq\max G[[0,T]\times X]<\infty$ for all
$t\in[0,T]$.

For the existence part, let $\beta>0$. Then it is not difficult to see that

$\Phi_{\beta}\colon H_{0,\rho}^{1}(0,T;\mathbb{R}^{n})\to
H_{0,\rho}^{1}(0,T;\mathbb{R}^{n})$

defined by

$(\Phi_{\beta}y)(t)=\int_{0}^{\max\\{t,0\\}}G(s,\pi_{\beta}(y+\hat{\phi})_{(s)})\,\mathrm{d}s$

is Lipschitz continuous and a strict contraction if $\rho>0$ is large enough,
where again $\pi_{\beta}$ is the continuous projection onto $V_{\beta}$. As in
Theorem 4.10 one shows that if
$\beta>\max\\{\|\phi\|_{\textnormal{Lip}},G(0,\phi)\\}$, then there exists
$T_{\beta}>0$ such that $x_{\beta}\coloneqq y_{\beta}+\hat{\phi}$, where
$y_{\beta}$ is the fixed point of $\Phi_{\beta}$, solves (FDE) on
$(0,T_{\beta})$. This shows local existence of a solution. Let
$\widetilde{x}\in H^{1}(-h,T_{\beta})$ be a solution of (FDE) on
$(0,T_{\beta})$. Then, by our preliminary observation, $\widetilde{x}$ is
Lipschitz continuous. Hence, we find $\widetilde{\beta}>\beta$ such that
$\widetilde{x}\in V_{\widetilde{\beta}}$. Next,
$\widetilde{y}\coloneqq\widetilde{x}-\hat{\phi}$ is a fixed point of
$\Psi_{\beta}$, the restriction of $\Phi_{\widetilde{\beta}}$ to
$H_{0,\rho}^{1}(0,T_{\beta};\mathbb{R}^{n})$ taking values in
$H_{0,\rho}^{1}(0,T_{\beta};\mathbb{R}^{n})$. By inspection, it follows that
$y_{\beta}|_{(-h,T_{\beta})}$ is also a fixed point of $\Psi_{\beta}$.
Choosing $\rho>0$ large enough, one realises that $\Psi_{\beta}$ is a strict
contraction and, thus, has one fixed point only. Hence,
$\widetilde{y}=y_{\beta}|_{(-h,T_{\beta})}$, and thus,
$x_{\beta}=\widetilde{x}$ on $(-h,T_{\beta})$ establishing uniqueness of the
(local) solution of (FDE).

Finally, without loss of generality, either $T_{\beta}=T$ (in which case we
are done) or $T_{\beta}<T$ and
$\|x^{\prime}(T_{\beta})\|=\beta=\|G(T_{\beta},x_{(T_{\beta})})\|$. In the
latter case, we consider a sequence $(T_{k\beta})_{k\in\mathbb{N}}$ of final
times corresponding to solutions $x_{k\beta}=y_{k\beta}+\hat{\phi}$, where
$y_{k\beta}$ is the fixed point of $\Phi_{k\beta}$. Again, without loss of
generality, either the sequence of final times is eventually constant equal to
$T$ or $T_{k\beta}\to T_{0}\leq T$ as $k\to\infty$ with
$\|x^{\prime}(T_{k\beta})\|=k\beta\to\infty$. In the former case, $x\in
H^{1}(-h,T;\mathbb{R}^{n})$ and in the latter
$\|x\|_{\textnormal{Lip}}=\infty$. ∎

##  6 Applications

In this section, we consider two different state-dependent delay equations.
One example is an elementary problem from Newtonian mechanics and the other
one is an application from mathematical biology. In either cases, we shall not
go through the model assumptions in full detail as this section is merely
meant to illustrate the versatility of the solution theory developed in the
previous section. More so, we rather focus on the delay functional $r$ in each
application in turn. We briefly sketch how the model assumptions can be
relaxed to obtain a more general solution theory for the individual equation
at hand.

###  6.1 On a problem from mathematical biology

In this section, we revisit the example from cell population biology, which
has been analysed in [9, 4]. Here, we will use the slightly more general
theorem on the solution theory for a class of functional differential
equations, see Theorem 5.1. Again, we emphasise that we merely require
appropriate Lipschitz continuity; the assumptions are then similar to the ones
developed in [4]. Note that, however, the main abstract assumption in [4]
establishing both uniqueness and existence is that of so-called ‘almost
locally Lipschitz’ continuity for the right-hand side of the functional
differential equation involved. More precisely, let $h>0$. Then
$f\colon\operatorname{dom}(f)\subseteq
C([-h,0];\mathbb{R}^{n})\to\mathbb{R}^{m}$ is called almost locally Lipschitz,
if $f$ is continuous and for all $\phi_{0}\in\operatorname{dom}(f)$ and
$\beta>0$ there exists $\delta>0$ and $L\geq 0$ such that

$\|f(u)-f(v)\|_{\mathbb{R}^{m}}\leq L\|u-v\|_{C[-h,0]},$

whenever $u,v\in B_{C[-h,0]}[\phi_{0},\delta]$, the closed $\delta$-ball
around $\phi_{0}$ and
$\|u\|_{\textnormal{Lip}},\|v\|_{\textnormal{Lip}}\leq\beta$. Any almost
locally Lipschitz continuous mapping can be suitably extended to an almost
uniformly Lipschitz continuous mapping in the sense of Theorem 5.1. In fact,
one can even extend it to a Lipschitz continuous mapping as the following
result shows. We consider the autonomous case here only, the non-autonomous
situation as treated in [12] can be addressed similarly.

######  Proposition 6.1.

Let $f\colon\operatorname{dom}(f)\subseteq
C([-h,0];\mathbb{R}^{n})\to\mathbb{R}^{m}$ almost locally Lipschitz with
$\operatorname{dom}(f)\subseteq C([-h,0];\mathbb{R}^{n})$ open. Then for all
Lipschitz continuous $\phi_{0}\in\operatorname{dom}(f)$ there exists $F\colon
H^{1}(-h,0;\mathbb{R}^{n})\to\mathbb{R}^{m}$ Lipschitz continuous and
$\delta>0$ such that for all $\beta>0$

$F|_{B_{C[-h,0]}[\phi_{0},\delta]\cap
V_{\beta}}=f|_{B_{C[-h,0]}[\phi_{0},\delta]\cap V_{\beta}}$

###### Proof.

At first, we show for $\beta>\|\phi_{0}\|_{\textnormal{Lip}}$, $V_{\beta}\cap
B_{C[-h,0]}[\phi_{0},\delta]\subseteq H^{1}(-h,0;\mathbb{R}^{n})$ is convex
and closed and non-empty. Since $D\coloneqq V_{\beta}\cap
B_{C[-h,0]}[\phi_{0},\delta]$ is a subset of Lipschitz continuous functions by
Remark 4.2 (d), it is a subset of $H^{1}$. Convexity being evident as an
intersection of convex sets, it remains to show closedness. For this, let
$(\psi_{k})_{k}$ in $D$ converging $H^{1}(-h,0;\mathbb{R}^{n})$ to some
$\psi\in H^{1}(-h,0;\mathbb{R}^{n})$. By Theorem 2.5, it follows that $\psi\in
B_{C[-h,0]}[\phi_{0},\delta]$. Remark 3.1 confirms that $\psi\in V_{\beta}$.

For the construction of the extension, let $\phi_{0}\in\operatorname{dom}(f)$
be Lipschitz continuous. As $\operatorname{dom}(f)$ is open, we find
$\delta_{0}>0$ such that
$B[\phi_{0},\delta_{0}]\subseteq\operatorname{dom}(f)$; let
$\beta>\|\phi\|_{\textnormal{Lip}}$. As $f$ is almost locally Lipschitz, we
find $0<\delta<\delta_{0}$ such that $f|_{D}\colon D\subseteq
C([-h,0];\mathbb{R}^{n})\to\mathbb{R}^{m}$ is Lipschitz continuous, where
$D\coloneqq V_{\beta}\cap B[\phi_{0},\delta]$. Thus, using Theorem 2.5, we
estimate for all $\phi,\psi\in D$

$\|f|_{D}(\phi)-f|_{D}(\psi)\|\leq\|f|_{D}\|_{\textnormal{Lip}}\|\phi-\psi\|_{C[-h,0]}\leq(h^{1/2}+h^{-1/2})\|f|_{D}\|_{\textnormal{Lip}}\|\phi-\psi\|_{H^{1}}.$

Thus, $f|_{D}$ is Lipschitz continuous as mapping in $H^{1}$. Since, by our
preliminary observation, $D\subseteq H^{1}$ is closed and convex, $f|_{D}$
admits a Lipschitz continuous extension, $F$, to the whole of
$H^{1}(-h,0;\mathbb{R}^{n})$, by Corollary 3.4. ∎

The model in [9] reads as follows

$\begin{pmatrix}w(t)\\\
v(t)\end{pmatrix}^{\prime}=\begin{pmatrix}q(v(t))w(t)\\\ -\mu
v(t)+\frac{\gamma(v(t-r(v_{(t)})))g(x_{2},v(t))w(t-r(v_{(t)}))}{g(x_{1},v(t-r(v_{t})))}\mathrm{e}^{\int_{0}^{r(v_{(t)})}d(y(s,v_{(t)}),v(t-s))\,\mathrm{d}s}\end{pmatrix},$

where $g,q,\gamma,d$ are suitable functions and $x_{1}<x_{2}$, $\mu$ are given
real parameters. Assuming merely Lipschitz continuity and suitable boundedness
requirements, Theorem 5.1 applies here and provides long-time existence. We
shall discuss the mapping $r$ in some more detail and we will show that the
assumptions given in [9, 4] yield a Lipschitz continuous mapping on the whole
of $H^{1}(-h,0)$ already.

We assume that $g\colon\mathbb{R}^{2}\to\mathbb{R}$ is Lipschitz continuous
satisfying

$\varepsilon\leq g(y,p)\leq K\quad(y,p\in\mathbb{R})$

for some $0<\varepsilon<K$. Then consider for $\psi\in H^{1}(-h,0)$ the
differential equation

$y^{\prime}(s)=-g(y(s),\psi(-s)),\quad y(0)=x_{2}.$

The time $r(\psi)\in[-h,0]$ is then given by $y(r(\psi))=x_{1}$. Recall that
the classical Picard–Lindelöf theorem yields unique existence of $y$. Also,
the assumptions on $g$ yields existence and uniqueness of $r(\psi)\in[-h,0]$
if $h$ is large enough, see again, e.g., [9].

The rest of this section is devoted to establishing the following result,
which confirms even global Lipschitz continuity of the delay functional $r$.

######  Theorem 6.2.

The mapping

$r\colon H^{1}(-h,0)\to[-h,0]$

is Lipschitz continuous.

###### Proof.

Let $\phi,\psi\in H^{1}(-h,0)$. We denote the corresponding solutions of the
above ODEs determining $r(\phi)$ and $r(\psi)$ by $y_{\phi}$ and $y_{\psi}$,
respectively. Then by integrating the ODEs in question, we obtain

$x_{1}=x_{2}-\int_{0}^{r(\psi)}g(y_{\psi}(t),\psi(-t))\,\mathrm{d}t=x_{2}-\int_{0}^{r(\phi)}g(y_{\phi}(t),\phi(-t))\,\mathrm{d}t.$

Hence, from $r(\psi)<r(\phi)$ (without restriction) we deduce from the
assumption on $g$,

$\displaystyle\varepsilon|r(\phi)-r(\psi)|$
$\displaystyle\leq\int_{r(\phi)}^{r(\psi)}g(y_{\phi}(t),\phi(-t))\,\mathrm{d}t$
$\displaystyle=\int_{0}^{r(\psi)}(g(y_{\psi}(t),\psi(-t))-g(y_{\phi}(t),\phi(-t)))\,\mathrm{d}t$
$\displaystyle\leq\int_{0}^{h}\|(g(y_{\psi}(t),\psi(-t))-g(y_{\phi}(t),\phi(-t)))\|\,\mathrm{d}t$
$\displaystyle\leq\int_{0}^{h}L\|y_{\psi}(t)-y_{\phi}(t)\|\,\mathrm{d}t+\int_{0}^{h}L\|\psi(t)-\phi(t)\|\,\mathrm{d}t,$

where $L$ is the Lipschitz constant of $g$. Using the differential equations
for $y_{\phi}$ and $y_{\psi}$ and integrating, we get

$|y_{\psi}(t)-y_{\phi}(t)|\leq\int_{0}^{t}L|y_{\psi}(s)-y_{\phi}(s)|\,\mathrm{d}s+\int_{0}^{t}L|\psi(s)-\phi(s)|\,\mathrm{d}s.$

Grönwall’s inequality thus implies

$|y_{\psi}(t)-y_{\phi}(t)|\leq\int_{0}^{t}L|\psi(s)-\phi(s)|\,\mathrm{d}s\mathrm{e}^{Lt}\leq
L\sqrt{t}\mathrm{e}^{Lt}\|\phi-\psi\|_{L_{2}}$

Thus, combining the estimates, we get

$\varepsilon|r(\phi)-r(\psi)|\leq C\|\phi-\psi\|_{L_{2}}\leq
C\|\phi-\psi\|_{H^{1}}$

as required. ∎

######  Remark 6.3.

Note that the estimate derived in the proof of Theorem 6.2 is better than
anticipated in the claim of the theorem. However, the unique existence for
$r(\phi)$ requires $\phi\in H^{1}\subseteq C$.

###  6.2 A positioning problem

The second illustration of our results is taken from [26] and describes a
device positioned at the point $x(t)$ on the real axis determining its
position by emitting signals to a given reference point $-w<0$, where the
signal with constant speed $c$ is reflected. After some time, $s(t)$,
depending on the position of the device, the signal is received again by the
device. The time $s(t)$ yields a position $\hat{x}$ according to

$\hat{x}=\tfrac{c}{2}s(t)-w,$

this derived position being the actual position if $x(t)=0$. For the right-
hand side this computed position is used for the acceleration $a(\hat{x})$. As
in [26, Section 3], we consider the given model only if $x$ is in between $-w$
and some a priori given $w_{+}>0$. For some Lipschitz continuous
$a\colon\mathbb{R}\to\mathbb{R}$, the set of equations we consider here is

$\begin{pmatrix}x\\\ v\end{pmatrix}^{\prime}=\begin{pmatrix}v\\\ -\mu
v+a(s\tfrac{c}{2}-w)\end{pmatrix}\text{ with }cs(t)=x(t-s(t))+x(t)+2w.$

for some fixed real numbers $c,\mu$.

As in [26], we let $h\coloneqq(2w+2w_{+})/c$ and obtain $0<s\leq h$ assuming
$x(t)<w_{+}$. Thus, $h$ is our final time horizon (in negative direction).
Next, we apply our main theorem (Theorem 4.1) and the permanence principle
(Theorem 4.13) for obtaining a local solution theory for all initial values
$\phi\in H^{1}(-h,0;\mathbb{R}^{2})$ with bounded derivative. For this, (H1)
is evident by the Lipschitz continuity of $a$; note that in [26] following up
on a solution theory in $C^{1}$, $a$ needed to be continuously differentiable.
For the satisfaction of (H2), we need to find a Lipschitz estimate for $s(t)$
in terms of $x$ on $V_{\beta}$. This is provided in the next result, which is
a Sobolev regular variant of the similar result [26, Proposition 8]. We
emphasise that we do not need a differentiability statement here. We define
for $0<\alpha<\min\\{c,w,w_{+}\\}$

$W_{\alpha}\coloneqq\\{\phi\in H^{1}(-h,0);-w+\alpha\leq\phi\leq
w_{+}-\alpha,\sup_{s\in[-h,0]}\|\phi^{\prime}(s)\|\leq c-\alpha\\}.$

It is not difficult to see that $W_{\alpha}$ is a closed and convex subset of
$H^{1}(-h,0)$.

######  Theorem 6.4.

Let $\phi\in W_{\alpha}$. Then there exists a unique $s_{\phi}\in[0,h]$ such
that

$cs_{\phi}=\phi(-s_{\phi})+\phi(0)+2w.$

Moreover,

$W_{\alpha}\ni\phi\mapsto s_{\phi}\in[0,h]$

is Lipschitz continuous.

###### Proof.

Let $s,t\in[-h,0]$, $s<t$. Then by the fundamental theorem of calculus for
$H^{1}$, we obtain for $\phi\in W_{\alpha}$

$|\phi(s)-\phi(t)|\leq\int_{s}^{t}|\phi^{\prime}(\tau)|\,\mathrm{d}\tau\leq(c-\alpha)|t-s|.$

Thus, $s\mapsto\tfrac{1}{c}(\phi(-s)+\phi(0)+2w)$ is a self-mapping strict
contraction and thus admits a unique fixed point. Similarly to the proof of
the same property in [26], we estimate for $\phi,\psi\in W_{\alpha}$

$\displaystyle c|s_{\phi}-s_{\psi}|$
$\displaystyle=|\phi(s_{\phi})-\psi(s_{\psi})+\phi(0)-\psi(0)|$
$\displaystyle\leq(c-\alpha)|s_{\phi}-s_{\psi}|+2\sup_{s\in[-h,0]}|\phi(s)-\psi(s)|.$

Hence, by Theorem 2.5,

$|s_{\phi}-s_{\psi}|\leq\frac{2}{\alpha}(h^{1/2}+h^{-1/2})\|\phi-\psi\|_{H^{1}(-h,0)}.\qed$

Next, with the projection $\pi_{\alpha}\colon H^{1}(-h,0)\to H^{1}(-h,0)$ on
$W_{\alpha}$ and $r_{\alpha}\colon H^{1}(-h,0)\to[-h,0]$ given by
$r_{\alpha}(\phi)=s_{\pi_{\alpha}(\phi)}$, the assumptions of Theorem 4.1 are
satisfied, for the right-hand side

$g(t,(x(t),v(t)),x(t+r_{\alpha}(x_{(t)})))=\begin{pmatrix}v(t)\\\ -\mu
v(t)+a(\frac{x(t+r_{\alpha}(x_{(t)}))+x(t)}{2})\end{pmatrix}.$

A local solution theory with initial pre-history $\phi\in W_{\alpha}$ for some
$\alpha>0$ can thus be obtained by the permanence principle Theorem 4.13.

######  Remark 6.5.

For simplicity, we treat $n=1$ in the following.

(a) In this context let us also point out the subtle difference of the
functional differential equation in [4, 18] and the one in Theorem 1.1. In
[18], the existence of solutions was shown for continuous $G\colon
C[-h,0]\to\mathbb{R}$; uniqueness requires almost local Lipschitz continuity
for $G$. The delay functional, $r$, considered in Theorem 6.4 is defined on
Lipschitz continuous functions satisfying a certain bound for the functions
itself and for the Lipschitz norm. In order to apply the retraction approach
also in this situation, one is required to analyse, whether there exists a
continuous extension of $f\colon W_{\alpha}\subseteq C[-h,0]\to\mathbb{R}$ to
the whole of $C[-h,0]$ given $f$ is continuous. Since $W_{\alpha}$ is a closed
and convex subset of $C[-h,0]$ it is possible to apply [5, Proposition
1.19(i)] to obtain the retraction $p\colon C[-h,0]\to W_{\alpha}$ (i.e., a
continuous map such that $\rho|_{W_{\alpha}}=\operatorname{id}_{W_{\alpha}}$).
Thus, $F\coloneqq f\circ p$ extends $f$ continuously to the whole of
$C[-h,0]$. Whether this extension retains to be almost locally Lipschitz if
$f$ has the said property is a priori unclear. Note that for this, by [4,
Lemma 2.7], it suffices to show that $p$ can be chosen to be (locally)
Lipschitz. The quest is to show whether or not $p$ can be chosen to be
Lipschitz continuous.

In the following we describe possible arguments that are relevant to this
problem.

(b) An argument in [18] shows that $C([-h,0];[A,B])$ for some
$-\infty<A<B<\infty$ is a Lipschitz retract. Next, McShane’s Lemma shows
$W_{1,0}\coloneqq\\{f\in C[-h,0];\|f\|_{\text{Lip}}\leq 1,f(0)=0\\}$ is a
Lipschitz retract of $C[-h,0]$, see [19]. Indeed, for this, consider the
mapping

$E\colon C[-h,0]\to C[-h,0]\cap
V_{1},f\mapsto\big{(}s\mapsto\min_{t\in[-h,0]}(f(t)-|s-t|)\big{)},$

which is a Lipschitz retract on the space of Lipschitz continuous functions
with Lipschitz semi-norm bounded by $1$. The mapping $H\colon C[-h,0]\cap
V_{1}\to W_{1,0},f\mapsto f-f(0)$ is, in turn, Lipschitz continuous. Then
$E\circ H$ is the desired retraction.

(c) Whether or not $W_{\alpha}$ from above is a Lipschitz retract of
$C([-h,0];\mathbb{R}^{n})$ does not follow from (b) and needs to be addressed
separately. More generally, let $W\subseteq C[-h,0]$ be closed and convex.
Note that by the Sobolev embedding theorem, Theorem 2.5, any closed and convex
subset $W\subseteq C[-h,0]$ yields a closed and convex subset $W\cap
H^{1}(-h,0)\subseteq H^{1}(-h,0)$. Hence, $W\cap H^{1}(-h,0)$ is a Lipschitz
retract by Proposition 3.3 and any Lipschitz continuous mapping on $G\colon
W\cap H^{1}(-h,0)\subseteq C[-h,0]\to\mathbb{R}$ can be extended to a
Lipschitz continuous mapping on the whole of $H^{1}(-h,0).$

(d) By [5, Theorem 1.6], $C[-h,0]$ is an absolute Lipschitz retract, that is,
for every metric space $M$ containing $C[-h,0]$, there exists a Lipschitz
continuous retraction from $M$ onto $C[-h,0]$. Thus, whether or not $W$ from
(c) is a Lipschitz retract of $C[-h,0]$ is equivalent to $W$ being an absolute
Lipschitz retract, see [5, Proposition 1.2 or subsequent remark (ii)]. In
fact, this puts one into the situation of [11], where it was shown that if $W$
is additionally a compact subset of $C[-h,0]$ such sets are so-called absolute
uniform retracts, still a weaker property than being an absolute Lipschitz
retract. Whether or not (a closed, compact) $W$ is an absolute Lipschitz
retract apparently needs to be addressed on an individual basis, see in
particular the counterexample in [11, Section 5].

##  7 Conclusion

We have developed a solution theory for state-dependent delay equations (and
in passing) for a certain class of functional differential equation that
require Lipschitz continuity assumptions only. The advantage of the developed
theory is that it does work for the classical case as well as for the state-
dependent delay case _simultaneously_. The assumptions are easily verified in
practice and do not involve potentially complicated computations and analysis
of certain derivatives. The underlying Hilbert space structure permits
localisation techniques using the projection theorem so that many equations
can be first dealt with on the whole domain. The local (or global) solution
theory is then consequence of a permanence principle using the equation in
question. The concept of a solution manifold is not needed to consider the
problem in the first place. Being an element of the solution manifold for the
initial value is a mere requirement for regular solutions. Solutions starting
off of the solution manifold will eventually end up on the solution manifold
if the solution exists for a long enough time. Other approaches not using the
solution manifold are based on compactness results, Schauder’s fixed point
theorem and a retraction technique. The present approach works for a larger
class of equations and initial pre-histories, provides uniqueness and
existence results at the same time and its application is technically not as
demanding and more widely possible as previously known rationales.

## Competing Interests Statement

There are no competing interests to declare.

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