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arXiv:submit/0455159 [math.DS] 13 Apr 2012

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Technische Universität Dresden

Herausgeber: Der Rektor

A Hilbert Space Perspective on

Ordinary Diﬀerential Equations

with Memory Term.

Anke Kalauch, Rainer Picard, Stefan Siegmund, Sascha Trostorﬀ & Marcus Waurick

Institut für Analysis

MATH-AN-015-2012

A Hilbert Space Perspective on

Ordinary Differential Equations

with Memory Term

Anke Kalauch, Rainer Picard, Stefan Siegmund,

Sascha Trostorﬀ & Marcus Waurick

Institut für Analysis, Fachrichtung Mathematik

Technische Universität Dresden

Germany

anke.kalauch@tu-dreden.de

rainer.picard@tu-dresden.de

stefan.siegmund@tu-dresden.de

sascha.trostorﬀ@tu-dresden.de

marcus.waurick@tu-dresden.de

April 13, 2012

Abstract. We discuss ordinary diﬀerential equations with delay and memory

terms in Hilbert spaces. By introducing a time derivative as a normal opera-

tor in an appropriate Hilbert space, we develop a new approach to a solution

theory covering integro-diﬀerential equations, neutral diﬀerential equations and

general delay diﬀerential equations within a uniﬁed framework. We show that

reasonable diﬀerential equations lead to causal solution operators.

Keywords and phrases: ordinary diﬀerential equations, causality, memory, delay

Mathematics subject classiﬁcation 2010: 34K05 (Functional diﬀerential equations, general theory);

34K30 (Equations in abstract spaces); 34K40 (Neutral equations); 34A12 (Initial value problems, existence,

uniqueness, continuous dependence and continuation of solutions); 45G15 (Systems of nonlinear integral

equations)

3

1 Introduction

Contents

1 Introduction 4

2 The Time Derivative 7

3 Basic Solution Theory 12

4 Causality and Memory 18

4.1 Causal Solution Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 DelayandMemory ............................... 22

5 Applications 24

5.1 Initial Value Problems for ODE . . . . . . . . . . . . . . . . . . . . . . . . 24

5.2 LocalSolvability ................................ 28

5.3 Classical Delay Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3.1 A Structural Observation . . . . . . . . . . . . . . . . . . . . . . . . 31

5.3.2 Integro-Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . 34

5.3.3 Neutral Diﬀerential Equations . . . . . . . . . . . . . . . . . . . . . 36

1 Introduction

Delay diﬀerential equations (DDE)

9xptq “ fpt, xtq(1)

are important in many areas of engineering and science (see e.g. [2] and the references

therein). Let t0, t1PRwith t0ăt1. If aPRě0denotes the maximal delay and x:

rt0´a, t1s Ñ Rnis a continuous function then for tP rt0, t1sthe function xt:r´a, 0s Ñ Rn

denotes its (restricted) translate, which is deﬁned by xtpθq “ xpt`θqfor θP r´a, 0s. A

continuous function x:rt0´a, t1s Ñ Rnis a solution of (1) which takes at the initial time

t0the initial value φPC:“Cpr´a, 0s,Rnq, if it satisﬁes the following initial condition and

integral equation [10, Lemma 1.1]

xt0“φ, xptq “ φp0q ` żt

t0

fps, xsqds for tP rt0, t1s.(2)

Note that it is assumed that f:DÑRnis deﬁned on an open subset DĎRˆCand

rt0, t1s Q sÞÑ fps, xsq P Rnis well-deﬁned and continuous.

Instead of viewing solutions x:rt0´a, t1s Ñ Rnas functions with values in Rnone can

equivalently consider the induced mapping rt0, t1s Q tÞÑ xtPC(see e.g. [4, 10]). Utilizing

(2) and a contraction argument one can show the following existence theorem.

4

Theorem 1.1 ([10, Theorem 2.3]).Suppose that f:DÑRnis continuous and Lipschitz

continuous w.r.t. the second argument on every compact subset of D. Then for pt0, φq P D

every initial value problem (1) subject to xt0“φ, has a unique solution.

Examples of delay diﬀerential equations (1) include

•ordinary diﬀerential equations (a“0)9xptq “ Fpxptqq

•diﬀerential diﬀerence equations 9xptq “ fpt, xptq, xpt´τ1ptqq,...,xpt´τpptqqq with

0ďτjptq ď a,j“1,...,p

•integro-diﬀerential equations 9xptq “ ş0

´agpt, θ, xpt`θqqdθ

Another important class of delay diﬀerential equations which generalizes (1) are

•neutral delay diﬀerential equations 9xptq “ fpt, xt,9xtq, i.e., 9xptqdepends not only on

the past history of xbut also on the past history of 9x

A modern approach to investigate (1) is based on a reformulation as an abstract integral

equation [4, 3, 5] which has proven to be useful because standard results from the theory

of ordinary diﬀerential equations such as linearized stability or bifurcation results can be

extended to delay equations using the so-called sun-star calculus of adjoint semigroups.

The natural state space is the space L1pr´a, 0s,Rnq, see [3] for an extension to unbounded

delay and a discussion of complications with using this state space (e.g. regularity of the

Nemitzki operator). Already in the 1970s it was observed by Hale [9] that the diﬃculties

of unbounded delay are related to lacking regularity of the solution operator of (2). For an

approach to a general class of partial diﬀerential delay equations, see [19, 20]. Linearized

stability as described in [20] will be addressed with the methods introduced here in another

paper.

Solution theories for DDEs (1) adopt the point of view of semigroup theory and have

as a common feature that they work with the integral equation representation (2). In

this paper we suggest a new approach to delay equations which is based on a uniﬁed

Hilbert space approach for diﬀerential equations by Picard et al. [14, 15, 16] and focusses

on the properties of the time derivative as an operator on taylor-made solution spaces of

exponentially bounded functions on the whole real line. Applying this approach to DDEs,

it turns out that unbounded delay is not more diﬃcult than bounded delay and many

diﬀerent classes of equations with memory, including integro-diﬀerential and neutral delay

equations, can be treated in a uniﬁed way.

In order to illustrate one of the main ideas of this new approach, namely the time derivative

as an operator acting on a function space of potential solutions deﬁned on the whole

real line, consider a simple special case of DDEs: scalar autonomous ordinary diﬀerential

equations and their corresponding initial value problems

9x“fpxptqq for tPRą0, xp0q “ x0PR.

For simplicity assume that f:RÑRis Lipschitz continuous and satisﬁes fp0q “ 0. To

extend this initial value problem on Rě0to R, denote by rx:Rě0ÑRits unique solution.

5

1 Introduction

rxis continuously diﬀerentiable and rxp0`q “ x0. We trivially extend rxby setting

x:RÑR, t ÞÑ "0,for tPRă0,

rxptq,for tPRě0.

The extension xis continuously diﬀerentiable on Rzt0u. Utilizing the Heavy-side function

χRě0, we compute the distributional derivative of x

x1“ pχRě0¨xq1“xp0`qδ`χRě0rx1“x0δ`χRě0rx1

where δis the Dirac delta distribution. Hence, the distributional derivative of xsatisﬁes

(keep in mind that fp0q “ 0)

x1“fpxp¨qq ` x0δ.

With this example in mind, we develop a solution theory for right hand sides being distri-

butions. Moreover, we will cover a comprehensive class of initial value problems within our

perspective (Theorem 5.4). The simplifying assumption fp0q “ 0will be replaced by the

notion of causality (Theorem 4.5). In order to consistently develop an operator-theoretic

point of view, we need a glimpse on extrapolation spaces and view the time derivative as

an operator on embedded Sobolev spaces of exponentially weighted functions on Rwhich

form a Gelfand triple.

The structure of the paper is as follows. In Section 2 we introduce the time-derivative as

a continuously invertible operator in an L2-type space. We deﬁne the time-derivative on

the whole real line in order to be able to use the Fourier transform and have a functional

calculus at hand which allows to consider autonomous linear equations of neutral type very

easily (Theorem 5.15). We keep the paper self-contained but cite the basic concepts from

[16].

Section 3 gives the basic solution theory for diﬀerential equations of the form 9u“Fpuq,

where Fis Lipschitz continuous in some appropriate sense. Here, the term solution theory

means that we deﬁne a suitable Hilbert space in order to establish existence, uniqueness

and continuous dependence on the data. It should be noted that the respective proofs are

based on the contraction mapping theorem and are therefore comparatively elementary.

We emphasize that since our approach is well-suited for the Hilbert space setting, there

are very few technical diﬃculties. In particular, we do not encounter problems due to the

non-reﬂexivity of the underlying space. This may lead to an easier treatment of stability

and bifurcation analysis, since it avoids using the sun-star calculus [4].

Section 4 is devoted to the study of causality which roughly means that a solution up to

time tonly depends on the right hand side up to time t. We also state a notion which

is dual to causality and which may be called amnesic. This concept gives a possible way

to deﬁne delay diﬀerential equations. The notion of causality is of prime importance to

describe physically reasonable equations, it was so far almost not present in the classical

literature on diﬀerential equations and only recently gained some attention [12, 21, 27].

Section 4 also provides a theorem (Theorem 4.5) that states that reasonable right hand

6

sides lead to causal solution operators.

Section 5 illustrates the applicability and versatility of the concepts developed in the Sec-

tions 3 and 4. The ﬁrst part of this section deals with the formulation of initial value

problems within our context. In the second part we present a method to establish local

solvability. We only give an introductory example of how to treat equations of the type

9uptq “ gpt, uptqq, where we assume gto be only locally Lipschitz continuous. Adopting

this strategy to the general case would then also lead to a respective local solution theory.

However, since we aim to cover a broad class of a priori diﬀerent structures, we focus

on global-in-time solutions in order to keep this exposition rather elementary and to avoid

many technicalities. In the third part of this section we study special types of delay diﬀeren-

tial equations including some retarded functional diﬀerential equations, integro-diﬀerential

equations and equations of neutral type.

To ﬁx notation, let Hdenote a Hilbert space over the ﬁeld Kwhere KP tR,Cu.

2 The Time Derivative

To develop our operator-theoretic approach to diﬀerential equations with memory, it is

pivotal to establish the time derivative as a boundedly invertible operator in an adequate

Hilbert space setting. This strategy uses speciﬁc properties of the one-dimensional deriva-

tive on the real line. In order to describe these structural features appropriately, we need

the following operators.

Deﬁnition 2.1. Denoting by ˚

C8pRqthe set of smooth functions with compact support

and by L2pRqthe Hilbert space (of equivalence classes) of square-integrable functions w.r.t.

Lebesgue measure, we deﬁne the time derivative

Bc:˚

C8pRq Ď L2pRq Ñ L2pRq:φÞÑ φ1

and the multiplication operator

mc:˚

C8pRq Ď L2pRq Ñ L2pRq:φÞÑ pxÞÑ xφpxqq.

Moreover, the Fourier transform Fcis deﬁned in the following way

Fc:˚

C8pRq Ď L2pRq Ñ L2pRq

φÞÑ ˆxÞÑ 1

?2πżR

e´ixyφpyqdy˙.

Theorem 2.2. The operators Bc,mc,Fcare densely deﬁned and closable. Moreover, for

B:“ ´B˚

c, m :“mc,F:“Fcthe following properties hold

(a) Fis unitary from L2pRqonto L2pRq,

7

2 The Time Derivative

(b) DpBq “ tfPL2pRq;f1PL2pRqu, where f1denotes the distributional derivative,

(c) Dpmq “ tfPL2pRq;pxÞÑ xfpxqq P L2pRqu,

(d) B,imare skew-selfadjoint,

(e) B “ F˚imF.

Proof. The operators Bc,mc,Fcare clearly densely deﬁned. Closability of Fcis clear,

since Fcis norm-preserving and has dense range [28, Theorem V.2.8 and Lemma V.1.10].

Consequently, (a) follows. The closability of Bcand mcfollows from BcĎ B and mcĎm˚,

respectively. (b) is immediate from the deﬁnition of distributional derivatives and (c) is

easy. The fact that iBis selfadjoint can be found in [11, Example 3.14]. (e) is proved in [1,

Volume 1, p.161-163]. Thus, also (d) follows.

Before we turn to further studies on the operator B, we need to deﬁne weighted L2-type

spaces. This was also done in [14, Section 1.2] and for convenience and to ﬁx some notation

it will be paraphrased shortly.

Deﬁnition 2.3. Let ̺PR. Deﬁne H̺,0pRq:“ tfPL2

locpRq;pxÞÑ expp´̺xqfpxqq P L2pRqu.

We endow H̺,0pRqwith the scalar product

pf, gq ÞÑ xf, gy̺,0:“żR

fpxq˚gpxqexpp´2̺xqdx.

We note that the associated norm, denoted by |¨|̺,0, is an L2-type variant of the Morgen-

stern norm, see also [13]. Moreover, we deﬁne the following unitary operator

expp´̺mq:H̺,0pRq Ñ L2pRq:fÞÑ pxÞÑ expp´̺xqfpxqq.

By the unitarity of expp´̺mqwe have expp´̺mq´1“expp´̺mq˚.

Remark 2.4.We note that for selfadjoint operators Athe spectrum is purely real, i.e.,

σpAq Ď R. Consequently, the spectrum of skew-selfadjoint operators is contained in the

imaginary axis.

Denoting by k¨kLpX,Y qthe operator norm of a bounded linear operator from the Banach

space Xto the Banach space Y, we have the following corollary.

Corollary 2.5. Let ̺PRzt0uand deﬁne B̺:“expp´̺mq´1Bexpp´̺mq. Introducing the

Fourier-Laplace transform L̺:“Fexpp´̺mqand B0,̺ :“ B̺`̺, we have the following

(a) B̺“L˚

̺imL̺,

(b) pim`̺q´1PLpL2pRq, L2pRqq,B´1

0,̺ PLpH̺,0pRq, H̺,0pRqq,

(c) B´1

0,̺ “L˚

̺pim`̺q´1L̺,

(d) kB´1

0,̺kLpH̺,0pRq,H̺,0pRqq “kpim`̺q´1kLpL2pRq,L2pRqq “1

|̺|.

8

Moreover, the following formula holds

`B´1

0,̺ϕ˘pxq “ `pB̺`̺q´1ϕ˘pxq “ #şx

´8 ϕptqdt, ̺ ą0,

´ş´8

xϕptqdt, ̺ ă0,

for all ϕP˚

C8pRqand xPR.

Proof. (a) is immediate from Deﬁnition 2.3 and Theorem 2.2. (b) follows from Remark

2.4. (c) is clear by (a). The remaining formulas are elementary.

We emphasize that B0,̺ only depends on ̺with regards to the domain of deﬁnition. More-

over, ˚

C8pRqis a core for B0,̺ and we have for φP˚

C8pRq, the equality

Bφ“ B0,̺φ“ pB̺`̺qφ.

We will show that our solution theory is also largely independent of ̺(cp. Remark 2.11

and Theorem 4.6).

In our approach to delay diﬀerential equations, we will need a glimpse on extrapolation

spaces. The core of the issues needed here is summarized in the next deﬁnition and the

subsequent theorem.

Deﬁnition 2.6. For ̺PRzt0uwe deﬁne H̺,´1pRq:“H´̺,1pRq˚. The respective norm is

denoted by |¨|̺,´1. Moreover, deﬁne H̺,1pRqas the Hilbert space DpB0,̺qendowed with the

norm |¨|̺,1:φÞÑ |B0,̺φ|̺,0.

Remark 2.7.We note that we may identify H̺,0pRq Ď H̺,´1pRqvia the mapping

H̺,0pRq Q φÞÑ pH´̺,1pRq Q ψÞÑ xexpp´̺mqφ, expp̺mqψy0,0“:xφ, ψy0,0PKq P H̺,´1pRq.

We shall do so henceforth.

Theorem 2.8. Let ̺PRzt0u. Then we have the following chain of continuous embeddings

H´̺,1pRqãÑH´̺,0pRqexpp̺mq

ÑH0,0pRqexpp´̺mq´1

ÑH̺,0pRqãÑH̺,´1pRq,

9

2 The Time Derivative

the triple pH´̺,1pRq, H0,0pRq, H̺,´1pRqq is a Gelfand triple. The following mappings

B0,1Ñ0:H´̺,1pRq Ñ H´̺,0pRq

φÞÑ B0,´̺φ

B0,0Ñ1:H´̺,0pRq Ñ H´̺,1pRq

φÞÑ B´1

0,´̺φ

B0,0Ñ´1:H̺,0pRq Ñ H̺,´1pRq

φÞÑ pH´̺,1pRq Q ψÞÑ xφ, ´B0,´̺ψy0,0q

B0,´1Ñ0:H̺,´1pRq Ñ H̺,0pRq

φÞÑ `H´̺,0pRq Q ψÞÑ φp´B´1

0,´̺ψq˘

are unitary.

Proof. This is part of the more general construction of Sobolev chains and can be found

in [16, 23].

Remark 2.9.

(a) For the sake of simplicity, we will write B0,´̺for B0,1Ñ0,B´1

0,´̺for B0,0Ñ1,B0,̺ for B0,0Ñ´1

and B0,´̺for B0,´1Ñ0. With this notation, we arrive at the following duality. For

φ, ψ P˚

C8pRqand ̺PRzt0uwe have

xB0,̺φ, ψy0,0“ xφ, ´B0,´̺ψy0,0and xB´1

0,̺φ, ψy0,0“ xφ, ´B´1

0,´̺ψy0,0.

(b) We shall mention another possible duality, namely that of the Hilbert space adjoint,

which in this setting may be computed. It is straightforward to show that the adjoint

pB´1

0,̺q˚of B´1

0,̺ :H̺,0pRq Ñ H̺,0pRqis given by

pB´1

0,̺q˚“expp̺mq´1expp´̺mqB´1

0,´̺expp´̺mq´1expp̺mq.

We may also use the continuous extension of pB´1

0,̺q˚as a Banach space isomorphism

from H̺,´1pRqto H̺,0pRq.

(c) Assume ̺ą0. Developing our approach to delay equations, we will deﬁne what it

means for a mapping to have delay or to be amnesic in Section 4.2. Prototypes for the

former are B´1

0,̺ and pB´1

0,´̺q˚, for the latter pB´1

0,̺q˚and B´1

0,´̺.

(d) All the theory developed above can be generalized to function spaces with values in a

Hilbert space. A way for doing so is by means of tensor product constructions, which

can be found in [26, 16]. We use the notation introduced in [16], i.e., for two Hilbert

spaces H1and H2the tensor product will be denoted by H1bH2. For total subsets

D1ĎH1and D2ĎH2, we denote the algebraic tensor product by

D1

a

bD2:“spantφbψ;φPD1, ψ PD2u.

10

Let Hbe a Hilbert space and A:DpAq Ď H1ÑH2be a closable, densely deﬁned

linear operator. The identity in His denoted by 1H. Let DĎHbe dense. For

φPDpAqand ψPD, we may deﬁne

pAa

b1Hqpφbψq:“Aφ bψ.

It can be shown that Aa

b1Hhas a well-deﬁned linear extension as an operator from

DpAqa

bDĎH1bHto H2bH, we re-use the name Aa

b1Hfor that extension. The

closure of Aa

b1His again a linear operator, which will be denoted by Ab1H. An

analogue construction can be done for 1HbA.

(e) We specialize the latter remark. If His a Hilbert space of functions, e.g., if H“

L2pΩ,Σ, µqfor some measure space pΩ,Σ, µqthe Hilbert space tensor product

L2pΩ,Σ, µqbH1is isometrically isomorphic to the space of square-integrable H1-valued

L2-functions L2pΩ,Σ, µ;Hq. In that case the operator 1L2pΩ,Σ,µqbAis the canonical

extension of A, which acts as

p1L2pΩ,Σ,µqbAqptÞÑ χSptqφq “ ptÞÑ χSptqAφq,

where SPΣ, µpSq ă 8, φ PDpAqand χSdenotes the characteristic function of the

set S. We shall use the identiﬁcation of the tensor product and the space of vector-

valued functions in the sequel. We will not distinguish notationally between the norm

in H̺,kpRqand H̺,k pRq b Hfor kP t´1,0,1u. Moreover, we will also use the name A

for the extension of an operator to the tensor product.

The operator B´1

0,̺ is a normal operator in H̺,0pRq. Therefore, it admits a functional calcu-

lus, which may be generalized to operator-valued functions:

Deﬁnition 2.10. Let ̺PRzt0u. Deﬁne L̺:“L̺b1H. Let rą1

2|̺|and M:

BCpsgnp̺qr, rq Ñ LpHqbe bounded and analytic. Deﬁne

M`B´1

0,̺˘:“L˚

̺Mˆ1

im`̺˙L̺,

where

Mˆ1

im`̺˙φptq:“Mˆ1

it`̺˙φptq ptPRq

for φP˚

C8pR;Hq.

Remark 2.11.

(a) It is easy to see that kMpB´1

0,̺qkLpH̺,0pRqbH,H̺,0pRqbHqďsupzPBpsgnp̺qr,rqkMpzqkLpHq.

(b) The deﬁnition of MpB´1

0,̺qis largely independent of the choice of ̺in the sense that the

operators MpB´1

0,̺1qand MpB´1

0,̺2qfor two diﬀerent parameters ̺1, ̺2such that ̺1¨̺2ą0

11

3 Basic Solution Theory

coincide on the intersection of the respective domains. A proof of this statement can

be found in [16, Theorem 6.1.4] or [23, Lemma 1.3.2].

(c) The operator MpB´1

0,̺ qhas a unique continuous extension to H̺,´1pRq b H, which will

be denoted by MpB´1

0,̺qas well. More precisely, we may deﬁne

MpB´1

0,̺ q:H̺,´1pRq b HÑH̺,´1pRq b H

φÞÑ pH´̺,1pRq b HQψÞÑ xMpB´1

0,̺qB´1

0,̺φ, ´B0,´̺ψy0,0q.

This deﬁnition is justiﬁed by observing that MpB´1

0,̺qB´1

0,̺φ“ B´1

0,̺MpB´1

0,̺qφfor all φP

H̺,0pR;Hq.

A prominent example of an analytic and bounded function of B´1

0,̺ is the delay operator,

which itself is a special case of the time translation:

Example 2.12 (Time translation).Let rPRą0,̺PRą1{2r,hPRand uPH̺,0pRq b H.

We deﬁne

τhu:“up¨ ` hq.

The operator τhPLpH̺,0pRqbH, H̺,0pRqbHqis called time translation operator. We note

that τhB´1

0,̺ “ B´1

0,̺τhholds. Therefore, there is a unique continuous extension of τhto the

space H̺,´1pRq b H. We will use the same name for that extension. The operator norm

of τhequals expph̺q. If hă0the operator τhis also called a delay operator. In that case

the mapping

BCpr, rq Q zÞÑ Mpzq:“exppz´1hq1H

is analytic and uniformly bounded. An easy computation shows for uPH̺,0bHthat

expp`B´1

0,̺˘´1hq1Hu“MpB´1

0,̺qu“L˚

̺expppim`̺qhqL̺u“up¨ ` hq.

Analytic, bounded, operator-valued functions of B´1

0,̺ have the following properties, which

can be found in [15, Theorem 2.10] or [23, Lemma 1.3.2] (the link between the causality

notions is established in [23, Lemma 1.2.3]).

Theorem 2.13. Let rPRą0,Ha Hilbert space and M:BCpr, rq Ñ LpHqbe an analytic,

bounded mapping. Then for all ̺PRą1{2rthe operator MpB´1

0,̺qis a causal (in the sense of

the Deﬁnition 4.1), bounded linear operator in H̺,0pRq b H.

We shall discuss other possible analytic functions of B´1

0,̺ in Example 5.13.

3 Basic Solution Theory

Let Hbe a Hilbert space. In the framework prepared in the previous section we consider

equations of the form

B0u“Fpuq.(3)

12

Before we desribe the properties of F, we introduce a particular type of test function space

and a particular type of distributions.

Deﬁnition 3.1. Denoting by supp φthe support of a function φ, we deﬁne

˚

C`

8pR;Hq:“ tφPC8pR;Hq; sup supp φă 8, there is nPNwith φpnqP˚

C8pR;Hqu

and ˚

C`

8pR;Hq1:“ tφ:˚

C`

8pR;Hq Ñ K;φlinearu.

The right-hand side Fin (3) is assumed to be a mapping

F:˚

C8pR;Hq Ñ ˚

C`

8pR;Hq1.

Moreover, assume the existence of ̺0PRą0and sP p0,1qsuch that for all ̺PRą̺0there

is KPRą0such that for all u, w P˚

C8pR;Hqand ψP˚

C`

8pR;Hqwe have

|Fp0qpψq|ďK|ψ|´̺,1and |Fpuqpψq ´ Fpwqpψq|ďs|ψ|´̺,1|u´w|̺,0.(4)

It is straightforward that for all ̺PRą̺0the mapping Fpossesses a unique Lipschitz

continuous extension F̺from H̺,0pRq b Hto H̺,´1pRq b H. The respective Lipschitz

constant may be chosen strictly less than 1. In order to obtain well-posedness of (3),

the task is in ﬁnding ̺such that (3) admits a unique solution u, which continuously

depends on the right hand side Fin some adapted sense. In this setting (3) should hold in

H̺,´1pRq b Hnoting that B0,̺ can be regarded as the unitary operator from H̺,0pRq b H

onto H̺,´1pRq b H(Theorem 2.8). In this situation we arrive at the following result.

Theorem 3.2 (Picard-Lindelöf).Let ̺0PRą0,sP p0,1qand let F:˚

C8pR;Hq Ñ

˚

C`

8pR;Hq1be such that the estimates (4) hold for all ̺PRą̺0. Then for all ̺PRą̺0there

exists a uniquely determined uPH̺,0pRq b Hsuch that

B0,̺u“F̺puqin H̺,´1pRq b H.

Proof. Let ̺PRą̺0. We consider the ﬁxed point problem

u“ B´1

0,̺F̺puq.(5)

According to Theorem 2.8 the operator B´1

0,̺ is unitary from H̺,´1pRq b Hto H̺,0pRq bH.

Since the Lipschitz constant of F̺is strictly less than 1, the contraction mapping theorem

implies the existence of a unique uPH̺,0pRq b Hsatisfying (5). For this ﬁxed point it

follows that

B0,̺u“F̺puqin H̺,´1pRq b H.

The uniqueness of the solution follows immediately, since for any element xPH̺,0pRqbH

satisfying (3) the ﬁxed point equation (5) holds as well. Since this ﬁxed point is unique,

we conclude u“x.

13

3 Basic Solution Theory

If we discuss a particular class of equations of neutral type (see Theorem 5.16), it turns

out that the respective function Fsatisﬁes another estimate than (4). We consider other

possible cases subsequently.

Corollary 3.3. Let ̺0PRą0,sP p0,1qand let F:˚

C8pR;Hq Ñ ˚

C`

8pR;Hq1be such that

for all ̺PRą̺0, there is KPRą0such that for all u, w P˚

C8pR;Hqand ψP˚

C`

8pR;Hq

we have

|Fp0qpψq|ďK|ψ|´̺,0and |Fpuqpψq ´ Fpwqpψq|ďs|ψ|´̺,0|u´w|̺,1.

For ̺PRą̺0denote by F̺:H̺,1pRqbHÑH̺,0pRqb Hthe strictly contracting extension

of F. Then for all ̺PRą̺0there is a unique uPH̺,1pRq b Hsatisfying

B0,̺u“F̺puq

in H̺,0pRq b H.

Proof. Let ̺PRą̺0. We consider the mapping

G:H̺,0pRq b HÑH̺,´1pRq b H

vÞÑ B0,̺F̺pB´1

0,̺vq.

Since the operators B0,̺ and B´1

0,̺ are unitary, it follows that G̺is strictly contracting. Thus,

by Theorem 3.2 there exists a unique vPH̺,0pRq b Hwith

B0,̺v“Gpvq “ B0,̺ F̺pB´1

0,̺vq

which is equivalent to

v“F̺pB´1

0,̺vq.

By setting u:“ B´1

0,̺vPH̺,1pRq b H, we obtain the desired solution of our diﬀerential

equation. The uniqueness is clear since any solution xPH̺,1pRq b Hsatisﬁes

B0,̺x“F̺pxq “ F̺pB´1

0,̺B0,̺xq.

Hence, by the uniqueness of vwe obtain B0,̺x“vand thus x“u.

If we impose a modiﬁed Lipschitz-type condition on F, it turns out that we gain better

regularity of the solution. It should be noted that in contrary to (4) this estimate does not

impose a strict contractivity condition on F.

Corollary 3.4. Let kP t0,1u,̺0, C PRą0and let F:˚

C8pR;Hq Ñ ˚

C`

8pR;Hq1be

such that for all ̺PRą̺0, there exists KPRą0such that for all u, w P˚

C8pR;Hqand

ψP˚

C`

8pR;Hqwe have

|Fp0qpψq|ďK|ψ|´̺,k and |Fpuqpψq ´ Fpwqpψq|ďC|ψ|´̺,k|u´w|̺,k .

14

For ̺PRą̺0we denote by F̺:H̺,kpRqbHÑH̺,k pRqbHthe unique continuous extension

of F. Then for all ̺PRąmaxtC,̺0uthere is a unique uPH̺,k`1pRq b Hwith

B0,̺u“F̺puqin H̺,k pRq b H.

Proof. Let ̺PRąmaxtC,̺0u. For ψP˚

C`

8pR;Hq, we observe that by Theorem 2.8

|ψ|´̺,k “|B´1

0,´̺B0,´̺ψ|´̺,k ď1

̺|B0,´̺ψ|´̺,k “1

̺|ψ|´̺,k`1.

Consequently, F̺considered as a mapping with values in H̺,k´1pRqbHis strictly contract-

ing. Thus, we are in the situation of Theorem 3.2 if k“0or in the situation of Corollary

3.3 in the case k“1. Hence, we ﬁnd a unique uPH̺,kpRq b Hwith

B0,̺u“F̺puq.

Since F̺puq P H̺,kpRqbH, it follows that indeed uPH̺,k`1pRqbHand the equation even

holds in H̺,kpRq b H.

Remark 3.5.It should be noted that the solution of (3) seems to depend on the choice of

the parameter ̺PRą̺0. This however is not the case and will be shown in Theorem 4.6.

Now we show the continuous dependence of our solution uof (3) on the function Fwith

respect to a suitable topology. For a Lipschitz continuous mapping F:H̺,0pRq b HÑ

H̺,´1pRq b Hwe denote the best Lipschitz constant of Fby |F|Lip.

Theorem 3.6. Let ̺PRą0and

F, G :H̺,0pRq b HÑH̺,´1pRq b H

be two Lipschitz-continuous mappings with

|F|Lip ` |G|Lip

2ă1.

Furthermore let u, v PH̺,0pRq b Hwith

B0,̺u“Fpuqand B0,̺ v“Gpvq.

Then

|u´v|̺,0ď1

1´|F|Lip`|G|Lip

2

sup

xPH̺,0pRqbH|Fpxq ´ Gpxq|̺,´1.

15

3 Basic Solution Theory

Proof. For uand vwe compute in H̺,0pRq b H

u´v“B´1

0,̺Fpuq ´ B´1

0,̺Gpvq

“1

2B´1

0,̺pFpuq ´ Fpvqq ´ 1

2B´1

0,̺pGpvq ´ Fpvqq` 1

2B´1

0,̺Fpuq ´ 1

2B´1

0,̺Gpvq

“1

2B´1

0,̺pFpuq ´ Fpvqq ` 1

2B´1

0,̺pGpuq ´ Gpvqq´ 1

2B´1

0,̺pGpvq ´ Fpvqq ` 1

2B´1

0,̺pFpuq ´ Gpuqq.

The latter yields

|u´v|̺,0ď1

2p|F|Lip ` |G|Lipq|u´v|̺,0`sup

xPH̺,0pRqbH|Fpxq ´ Gpxq|̺,´1

and thus

|u´v|̺,0ď1

1´|F|Lip`|G|Lip

2

sup

xPH̺,0pRqbH|Fpxq ´ Gpxq|̺,´1.

Remark 3.7.

(a) In the case of

F, G :H̺,1pRq b HÑH̺,0pRq b H

with |F|Lip ` |G|Lip

2ă1

we consider the mappings r

F , r

Ggiven by r

Fpvq “ B0,̺FpB´1

0,̺vqand r

Gpvq “ B0,̺GpB´1

0,̺vq

for vPH̺,0pRqbH. These mappings are again Lipschitz continuous from H̺,0pRqbH

to H̺,´1pRq b Hwith

|r

F|Lip “ |F|Lip and |r

G|Lip “ |G|Lip.

Thus we have

|ru´rv|̺,0ď1

1´|r

F|Lip`| r

G|Lip

2

sup

xPH̺,0pRqbH|r

Fpxq ´ r

Gpxq|̺,´1

for B0,̺ru“r

Fpruqand B0,̺rv“r

Gprvqby Theorem 3.6. Since u:“ B´1

0,̺ ruand v:“ B´1

0,̺rv

satisfy B0,̺u“Fpuqand B0,̺v“Gpvqwe conclude that, by using the unitarity of B´1

0,̺

|u´v|̺,1ď1

1´|F|Lip`|G|Lip

2

sup

xPH̺,1pRqbH|Fpxq ´ Gpxq|̺,0.

(b) If

F, G :H̺,k pRq b HÑH̺,k pRq b H

16

with |F|Lip ` |G|Lip

2ă̺

for kP t0,1uwe consider the functions

r

F , r

G:H̺,kpRq b HÑH̺,k ´1pRq b H

with r

Fpxq “ Fpxqand r

Gpxq “ Gpxqfor xPH̺,kpRq b H. Then we can estimate

the Lipschitz-constants of r

Fand r

Gby ̺´1|F|Lip and ̺´1|G|Lip, respectively. Thus we

conclude

|r

F|Lip ` | r

G|Lip

2ă1

and hence, we can apply Theorem 3.6 in the case k“0and Remark 3.7(a) in the case

k“1. Thus for u, v PH̺,k pRq b Hwith B0,̺u“Fpuqand B0,̺v“Gpvqwe get

|u´v|̺,k ď1

1´|r

F|Lip`| r

G|Lip

2

sup

xPH̺,kpRqbH|r

Fpxq ´ r

Gpxq|̺,k´1

ď1

̺´|F|Lip`|G|Lip

2

sup

xPH̺,kpRqbH|Fpxq ´ Gpxq|̺,k .

Another common case is that Fhas two arguments, i.e.,

F:pH̺,0pRq b Hq ‘ pH̺,´1pRq b Hq Ñ H̺,´1pRq b H

pu, f q ÞÑ Fpu, fq,

where the second argument fshould be interpreted as a certain source term, cf. Theorem

5.4. Then we arrive at the problem of ﬁnding uPH̺,0pRq b Hsuch that

B0,̺u“Fpu, f q(6)

for a ﬁxed fPH̺,´1pRq b H. We now deﬁne what it means for (6) to be autonomous.

Deﬁnition 3.8. An ordinary diﬀerential equation of the form (6) is called autonomous if

Fcommutes with time translation, i.e., for all uPH̺,0pRq b H, f PH̺,´1pRq b Hand

hPRwe have

Fpτhu, τhfq “ τhFpu, f q.

Remark 3.9.In the situation of an autonomous equation (6) it follows that for a solution

uPH̺,0pRq b Hand for each hPR

B0,̺τhu“τhB0,̺u“τhFpu, f q “ Fpτhu, τhfq.

This means that the translated solution solves the equation for the translated source term

f.

17

4 Causality and Memory

4 Causality and Memory

4.1 Causal Solution Operators

If a solution of an evolutionary problem up to time aonly depends on the equation up to

time a, then the solution operator is called causal or nonanticipative. Already Volterra im-

plicitely used nonanticipative operators in his work on integral equations. Later Tychonoﬀ

made contributions in developing the theory of functional equations involving causal op-

erators, cp. also [22, 12]. We consider equations of the form (3) with time on the whole

real line. In this setting causality is a natural property of the solution operator. At ﬁrst

we give a deﬁnition of causality in our framework.

Deﬁnition 4.1. Let X, Y be Hilbert spaces, ̺PR. A mapping

W:DpWq Ď H̺,0pRq b XÑH̺,0pRq b Y

is called causal if for all aPR, x, y PDpWq

pχRăapm0qpx´yq “ 0ùñ χRăapm0q pWpxq ´ Wpyqq “ 0q.

Remark 4.2.

(a) An equivalent formulation of causality is the following, cf. also [21, 27]. A mapping

W:DpWq Ď H̺,0pRq b XÑH̺,0pRq b Y,

is causal, if for all aPR

χRăapm0qW“χRăapm0qW χRăapm0q,

where, for a Hilbert space Hand a bounded and measurable function φ:RÑR, we

denote

pφpm0qfqptq:“φptqfptq ptPR, f PH̺,0pRq b Hq

(b) If ̺‰0, then it is immediate from the formulas in Theorem 2.8 that B´1

0,̺ is causal if

and only if ̺ą0.

It is remarkable that causality is actually implied by the uniform Lipschitz continuity we

required in Theorem 3.2. In order to prove this fact, i.e., Theorem 4.5, we need a deﬁnition.

Deﬁnition 4.3. Let wP˚

C`

8pR;Hq1. Then deﬁne

ż¨

´8

w:˚

C`

8pR;Hq Ñ R:ψÞÑ wpż8

¨

ψq.

18

4.1 Causal Solution Operators

Remark 4.4.Let ̺PRą0. We choose to identify wP˚

C`

8pR;Hq1, for which there exists

CPRą0such that for all ψP˚

C`

8pR;Hqit holds |wpψq|ďC|ψ|´̺,1, with an element of

H̺,´1pRq b Hby appropriate continuous extension. Then we have the equality

ż¨

´8

w“ B´1

0,̺wPH´̺,0pRq˚bH–H̺,0pRq b H.

Indeed, let ψP˚

C`

8pR;Hq. Then we have

ż¨

´8

wpψq “ wpż8

¨

ψq “ wp´B´1

0,´̺ψq “ B´1

0,̺wpψq.

Theorem 4.5. With the assumptions and the notation from Theorem 3.2, we have that for

all ̺PRą̺0the mapping B´1

0,̺F̺is causal as a mapping from H̺,0pRqbHto H̺,0pRq b H.

Proof. Let aPR,̺PRą̺0and let φPC8pRqbe bounded. Now, let vP˚

C8pR;Hqand

ψP˚

C8pR;Hqbe such that sup supp ψďa. For ηPRě̺we compute

|B´1

0,̺F̺pvqpψq ´ B´1

0,̺F̺pφpm0qvqpψq|“|ż¨

´8

Fpvqpψq ´ ż¨

´8

Fpφpm0qvqpψq|

“|B´1

0,ηFηpvqpψq ´ B´1

0,η Fηpφpm0qvqpψq|

“|Fηpvqp´B´1

0,´ηψq ´ Fηpφpm0qvqp´B´1

0,´ηψq|

ď|´B´1

0,´ηψ|´η,1|v´φpm0qv|η,0.

“|ψ|´η,0|v´φpm0qv|η,0

ď|ψ|0,0eηa|v´φpm0qv|η,0.

Summarizing, we get for ηPRě̺

|B´1

0,̺F̺pvqpψq ´ B´1

0,̺F̺pφpm0qvqpψq|ď|ψ|0,0eηa|v´φpm0qv|η,0.

By continuity, we deduce for ηPRě̺

|B´1

0,̺F̺pvqpψq ´ B´1

0,̺F̺pχRăapm0qvqpψq|

ď|ψ|0,0eηa|v´χRăapm0qv|η,0

“|ψ|0,0eηa|χRąapm0qv|η,0“|ψ|0,0eηa ˆż8

a

|vptq|2e´2ηtdt˙1

2

“|ψ|0,0ˆż8

0

|vpt`aq|2e´2ηtdt˙1

2

.

Letting ηÑ 8 in the above inequality, we conclude that

|B´1

0,̺F̺pvqpψq ´ B´1

0,̺F̺pχRăapm0qvqpψq|“0.

19

4 Causality and Memory

Hence, by the choice of ψ, the function B´1

0,̺F̺pvq ´ B´1

0,̺F̺pχp´8,aqpm0qvqis not supported

on the set Răa. This yields the claim.

Of course, as it was already noted in Remark 3.5, it is tempting to believe that the solution

of (3) provided by the above Picard-Lindelöf-type theorems (cf. Theorem 3.2 and the

Corollaries 3.3 and 3.4) depends on the particular choice of ̺. This is not the case as our

next result conﬁrms.

Theorem 4.6. With the assumptions and the notation from Theorem 3.2, the following

holds. The respective solutions w̺kPH̺k,0pRq b H,kP t1,2u, of (3) for ̺1, ̺2ě̺0,

coincide, i.e.,

w̺1“w̺2PH̺1,0pRq b HXH̺2,0pRq b H

provided that

B0,̺1w̺1“F̺1pw̺1qand B0,̺2w̺2“F̺2pw̺2q

holds in H̺1,´1pRq b Hand H̺2,´1pRq b H, respectively.

Proof. Let aPR,̺PRě̺0. Denoting by w̺the solution of

B0,̺w̺“F̺pw̺q P H̺,´1pRq b H,

we recall w̺PH̺,0pRq b H. Moreover, we have due to causality, i.e., Theorem 4.5

χRăapm0qw̺“χRăapm0qB´1

0,̺Fpw̺q

“χRăapm0qB´1

0,̺FpχRăapm0qw̺q.

Let ̺PRě̺0be such that mint̺1, ̺2u ě ̺. Then, as B´1

0,̺F̺leaves H̺,0pRq b Hinvariant,

we have for kP t1,2u, since χRăapm0qw̺kPH̺,0pRq b H, the following equality

χRăapm0qB´1

0,̺kF̺kpχRăapm0qw̺kq “ χRăapm0qB´1

0,̺F̺pχRăapm0qw̺kq.

Hence,

@w̺1´w̺2ˇˇχRăapm0qpw̺1´w̺2qD̺,0“

“@w̺1´w̺2ˇˇχRăapm0q`B´1

0,̺F̺pχRăapm0qw̺2q ´ B´1

0,̺F̺pχRăapm0qw̺2q˘D̺,0.

Using the Cauchy-Schwarz inequality, we get

|χRăapm0qpw̺1´w̺2q|2

̺,0

ď |FpχRăapm0qw̺1q ´ FpχRăapm0qw̺2q|̺,´1|χRăapm0qpw̺1´w̺2q|̺,0

ďs|χRăapm0qpw̺1´w̺2q|2

̺,0.

Since să1, we deduce |χRăapm0qpw̺1´w̺2q|̺,0“0. Since aPRwas arbitrary, the

desired result follows.

20

4.1 Causal Solution Operators

In the spirit of the continuous dependence result Theorem 3.6, we now show causality of

the solution operator in a suitably adapted sense. Thereby, we strengthen the causality

result by showing that the solution is independent of any future of F. Before, however,

stating the theorem, we deﬁne the set of possible right hand sides in (3) for some ̺PRą0.

Deﬁnition 4.7. Let Hbe a Hilbert space. For ̺PRą0we deﬁne ConevpH̺,0pRq b

H;H̺,´1pRqbHqthe set of all eventually contracting mappings, i.e., FPConevpH̺,0pRqb

H;H̺,´1pRqbHqif and only if F:H̺,0pRqbHÑH̺,´1pRqbHand there exists sP p0,1q

such that for all ηPRě̺, there is KPRą0such that for all u, w P˚

C8pR;Hqand

ψP˚

C`

8pR;Hqit holds

|Fp0qpψq|ďK|ψ|´η,1and |Fpuqpψq ´ Fpwqpψq|ďs|ψ|´η,1|u´w|η,0.

We may summarize the solution operators of (3) in the following way. Deﬁne

S̺:ConevpH̺,0pRq b H;H̺,´1pRq b Hq Ñ H̺,0pRq b H

FÞÑP ` xPH̺,0pRq b H|x“ B´1

0,̺Fpxq(˘,

where for singletons tau, we deﬁne P ptauq :“a. Theorem 3.2 ensures that for FP

ConevpH̺,0pRq b H;H̺,´1pRq b Hqthe set

xPH̺,0pRq b H|x“ B´1

0,̺Fpxq(

is indeed a singleton. Therefore, S̺, which maps a right-hand side to the corresponding

solution of (3), is well-deﬁned.

We may formulate the strengthened causality result.

Theorem 4.8. Let ̺PRą0,aPRand let Hbe a Hilbert space. Let F, G PConevpH̺,0pRqb

H;H̺,´1pRq b Hq. Assume χRăapm0qB´1

0,̺F“χRăapm0qB´1

0,̺G. Then χRăapm0qS̺pFq “

χRăapm0qS̺pGq.

Proof. For x, y PH̺,0pRq b Hwe need to show χRăapm0qx“χRăapm0qy, if

x“ B´1

0,̺Fpxq, y “ B´1

0,̺Gpyq.

Due to the causality of B´1

0,̺Fand B´1

0,̺G, cf. Theorem 4.5, we have

χRăapm0qx“χRăapm0qB´1

0,̺FpχRăapm0qxq

χRăapm0qy“χRăapm0qB´1

0,̺GpχRăapm0qyq

“χRăapm0qB´1

0,̺FpχRăapm0qyq

We see that χRăapm0qxand χRăapm0qyare both solutions of a ﬁxed point problem for

the same contractive mapping

χRăapm0qB´1

0,̺F,

21

4 Causality and Memory

which implies

χRăapm0qx“χRăapm0qy.

4.2 Delay and Memory

In this section we provide a new deﬁnition for operators having delay. In order to do so,

we deﬁne the opposite, i.e., we introduce the concept of an operator being memoryless or

amnesic. It is remarkable that this notion is dual to the concept of causality.

Deﬁnition 4.9. Let ̺PRand X, Y be Hilbert spaces. A mapping

W:DpWq Ď H̺,0pRq b XÑH̺,0pRq b Y,

is called amnesic or said to have no delay if for all aPRand x, y PDpWq

`χRąapm0qpx´yq “ 0ùñ χRąapm0q pWpxq ´ Wpyqq “ 0˘.

If a mapping Wis not amnesic, we also say Whas memory or has delay.

We observe that, by the very deﬁnition, a ﬁrst example for an amnesic operator is B´1

0,̺ for

̺ă0or pB´1

0,̺q˚for ̺ą0. We may also give other examples of amnesic operators, namely

operators of Nemitzki type.

Example 4.10 (Nemitzki operators).Let Hbe a Hilbert space, f:RˆHÑH,̺PRą0.

We assume that fis uniformly Lipschitz continuous with respect to the ﬁrst variable, i.e.,

there exists Lą0such that for all tPRand x, y PHwe have

|fpt, xq ´ fpt, yq|HďL|x´y|H.

Moreover, assume fpt, 0q “ 0for all tPR. We may deﬁne the following mapping

F̺:H̺,0pRq b HÑH̺,0pRq b H:uÞÑ ptÞÑ fpt, uptqqq.

The uniform Lipschitz continuity of ftogether with fpt, 0q “ 0for all tPRensures that

F̺is well-deﬁned. We claim that F̺is amnesic. Indeed, let aPRand u, v PH̺,0pRq b H

be such that χRąapm0qpu´vq “ 0. Then for a.e. tPRąawe have

F̺pt, uptqq “ F̺pt, χRąaptquptqq “ F̺pt, χRąaptqvptqq “ F̺pt, vptqq.

Thus, the claim follows.

An example of an operator, which has memory is B´1

0,̺ for ̺ą0. Now, we want to deﬁne,

when a diﬀerential equation of the form (3) is a delay diﬀerential equation, which, to the

best of our knowledge, has not been done yet in a mathematically rigorous way. For a

22

4.2 Delay and Memory

deﬁnition of delay diﬀerential equations of the form (3), one should take into account that

this should only depend on the right hand side F. However, this right hand side is in

general not of the form to be described as amnesic or to have delay, since Fmaps in

general into a space with negative index. Taking into account that the operator pB´1

0,̺q˚

is amnesic for ̺ą0and the application of pB´1

0,̺q˚transforms Finto a mapping within

H̺,0pRq b H(cf. Remark 2.9(b)), we arrive at the following possible deﬁnition.

Deﬁnition 4.11. Let Hbe a Hilbert space, ̺ą0and FPConev pH̺,0pRqbH;H̺,´1pRqb

Hq. A diﬀerential equation of the form (3), i.e.,

B0,̺u“Fpuq

is called a delay diﬀerential equation if `B´1

0,̺˘˚Fhas delay.

We illustrate this deﬁnition by means of the following examples.

Example 4.12. Let Hbe a Hilbert space and let f:HÑHsatisfy analogue conditions

as in Example 4.10, i.e., fis Lipschitz continuous with Lipschitz constant Lą0and

fp0q “ 0. Let ̺PRąL,gPH̺,´1pRq b H. We consider a diﬀerential equation of the

following form1, with F̺being analogously deﬁned as in Example 4.10,

B0,̺u“F̺puq ` g. (7)

The latter equation admits a unique solution uPH̺,0pRq b Hby our choice of ̺, cf.

Corollary 3.4. Moreover, this diﬀerential equation is not a delay diﬀerential equation, since,

as a composition of amnesic mappings, the mapping uÞÑ pB´1

0,̺q˚pF̺puq`gqis amnesic itself.

We may change equation (7) a little by introducing a time translation. Let hPRą0and

τ´hbe the time translation of Example 2.12. Consider the diﬀerential equation

B0,̺u“F̺pτ´huq ` g. (8)

The equation (8) is a delay diﬀerential equation, if fis not constant. Indeed, if fis not

constant, then there is x1, x2PHsuch that fpx1q ‰ fpx2q. Deﬁne u:“χr´h,0sp¨qx1, v :“

χr´h,0sp¨qx2. Then χRą0pm0qpu´vq “ 0. Using the fact that pB´1

0,̺q˚is amnesic, we get

χRą0pm0qppB´1

0,̺q˚pF̺pτ´huq ` gq ´ pB´1

0,̺q˚pF̺pτ´hvq ` gqq

“χRą0pm0qppB´1

0,̺q˚pF̺pτ´huqq ´ pB´1

0,̺q˚pF̺pτ´hvqqq

“χRą0pm0qppB´1

0,̺q˚χRą0pm0qpF̺pτ´huqq ´ pB´1

0,̺q˚χRą0pm0qpF̺pτ´hvqqq

“χRą0pm0qpB´1

0,̺q˚χRą0pm0qpF̺pτ´huq ´ F̺pτ´hvqq.

Since for a.e. tP r0, hswe have pF̺pτ´huqptq´F̺pτ´hvqptqq “ fpx1q´fpx2q ‰ 0, we deduce

χRą0pm0qpF̺pτ´huq ´ F̺pτ´hvqq ‰ 0.

1Such type of problems will be discussed later, when we come to initial value problems.

23

5 Applications

Hence, as pB´1

0,̺q˚is one-to-one, we have

χRą0pm0qpB´1

0,̺q˚χRą0pm0qpF̺pτ´huq ´ F̺pτ´hvqq ‰ 0.

This yields that (8) is a delay diﬀerential equation. This justiﬁes a posteriori the name

delay operator for τ´hsince, by setting fto be the identity on H, the equation (8) is indeed

a delay diﬀerential equation.

5 Applications

In this section we illustrate the versatility of the concepts developed above. In particular

we give several examples of delay problems in order to show that many delay problems

ﬁt into the uniﬁed framework which we developed. We start with a discussion on how to

investigate initial value problems within our context.

5.1 Initial Value Problems for ODE

We discuss the initial value problem

B0u“Fpuq, up0q “ u0.(9)

This problem seems not to be covered by our previous reasoning, however, as we will show

now, our abstract solution theory also applies to this initial value problem. Since our

approach basically builds on L2-space based arguments, we need a theorem, which justiﬁes

point-wise evaluation of functions. In order to do so, we deﬁne a weighted Hölder-type

space.

Deﬁnition 5.1. Let ̺PRą0,HHilbert space. Then we deﬁne for a continuous function

φPCpR;Hq

|φ|̺,8,1{2:“sup |exp p´̺tqφptq|HˇˇtPR(

`sup #|exp p´̺tqφptq ´ exp p´̺sqφpsq|H

|t´s|1{2ˇˇˇˇˇt, s PR^t“ s+.

Moreover, deﬁne C̺,8,1{2pR;Hq:“ tφPCpR;Hq;|φ|̺,8,1{2ă 8u. The vector space

C̺,8,1{2pR;Hqbecomes a Banach space under the norm |¨|̺,8,1{2.

We have the following form of a Sobolev embedding result.

Lemma 5.2 ([16, Lemma 3.1.59]).Let ̺PRzt0u. Then the mapping

˚

C8pR;Hq Ď H̺,1pRq b HÑC̺,8,1{2pR;Hq

uÞÑ pRQtÞÑ uptq P Hq

24

5.1 Initial Value Problems for ODE

has a continuous extension Γto all of H̺,1pRqbH(sometimes called the “trace operator”

or the “operator of point-wise evaluation in time”). Moreover, for all uPH̺,1pRq b H

sup |exp p´̺tqpΓuqptq|HˇˇtPR(ď1

a2|̺||u|H̺,1pRqbH

and

sup #|exp p´̺tqpΓuqptq ´ exp p´̺sq pΓuqpsq|H

|t´s|1{2ˇˇˇˇˇt, s PR^t“ s+ď |u|H̺,1pRqbH.

Furthermore, the mapping Γis injective.

Proof. Let φP˚

C8pR;Hq. We have for s, t PRwith sďtinvoking Hölder’s inequality and

the mean value theorem

|φptq ´ φpsq|H“ˇˇˇˇżt

sB0,̺φpuqduˇˇˇˇH

ďdˇˇˇˇżt

s

exp p2̺uqduˇˇˇˇdˇˇˇˇżt

s|B0,̺φpuq|2

Hexp p´2̺uqduˇˇˇˇ

ďd|exp p2̺tq ´ exp p2̺sq|

2|̺||φ|̺,1

ďa|t´s|d|exp p2̺tq ´ exp p2̺sq|

2|̺||t´s||φ|̺,1

ďa|t´s|max exp p̺xqˇˇxP ts, tu(|φ|̺,1

from which we can read oﬀ the desired Hölder continuity. With sÑ ´8 we also see from

the second inequality that

|φptq| ď exp p̺tqd1

2|̺||φ|̺,1.

Moreover, using the relations B0,̺ “ B̺`̺, cf. Corollary 2.5, and |φ|2

̺,1“|̺φ|2

̺,0`|B̺φ|2

̺,0,

we calculate

|exp p´̺tqφptq ´ exp p´̺sqφpsq|H“ˇˇˇˇżt

spB0,̺ pexp p´̺m0qφqqpuqduˇˇˇˇH

ďa|t´s|dˇˇˇˇżt

s|pB0,̺ pexp p´̺m0qφqqpuq|2

Hduˇˇˇˇ

25

5 Applications

ďa|t´s|dˇˇˇˇżt

s|pB0,̺φ´̺φqpuq|2

Hexp p´2̺uqduˇˇˇˇ

ďa|t´s|dˇˇˇˇżt

s|ppB0,̺ ´̺qφqpuq|2

Hexp p´2̺uqduˇˇˇˇ

ďa|t´s| |pB̺qφ|̺,0,

ďa|t´s|b|pB̺qφ|2

̺,0` |̺φ|2

̺,0

“a|t´s| |φ|̺,1,

which shows that the mapping under consideration is a well-deﬁned continuous linear

mapping. This mapping can now be extended by the obvious uniform continuity to all of

H̺,1pRq b Hdue to the density of ˚

C8pR;Hqin H̺,1pRq b H.

Finally, to see that Γ : H̺,1pRq b HÑC̺,8,1{2pR;Hqis injective, assume pφkqkis a

sequence in ˚

C8pR;Hqwith the property that |φk|̺,8,1{2

kÑ8

Ñ0is a Cauchy sequence in

H̺,1pRqbHwith limit fPH̺,1pRqbH. We need to show that f“0. Letting kÑ 8 in

the equality

@B0,̺φkˇˇψD0,0“@φkˇˇ´ B0,´̺ψD0,0,

where ψP˚

C8pR;Hqis arbitrary, we obtain

@B0,̺fˇˇψD0,0“0

for every ψP˚

C8pR;Hqfrom which we conclude that

B0,̺f“0

and hence

f“0in H̺,1pRq b H

follows.

Lemma 5.2 gives a criterion when it may be reasonable to impose initial conditions. The

solution of the respective diﬀerential equation has to lie in some sense in the space H̺,1pRqb

H. We need the following deﬁnition.

Deﬁnition 5.3 (Dirac delta distribution).Let ̺PRą0. Then χRą0PH̺,0pRq. We deﬁne

the Dirac delta distribution δin the point zero as the derivative of the Heavyside function:

δ:“ B0,̺χRą0.

26

5.1 Initial Value Problems for ODE

Clearly, δPH̺,´1pRq. Moreover, it is easy to see that

δ:H´̺,1pRq Ñ K:φÞÑ φp0q.

For a Hilbert space Hand wPHwe denote by δbwPH̺,´1pRq b Hthe derivative of

tÞÑ χRą0ptqw.

Now our perspective on initial value problems is as follows.

Theorem 5.4. Let ̺0PRą0,Ha Hilbert space, Cą0. Let F:˚

C8pR;Hq Ñ ˚

C`

8pR;Hq1

be such that for all ̺PRą̺0there exists KPRą0such that for all u, w P˚

C8pR;Hqand

ψP˚

C`

8pR;Hqthe estimates

|Fp0qpψq|ďK|ψ|´̺,0and |Fpuqpψq ´ Fpwqpψq|ďC|ψ|´̺,0|u´w|̺,0

hold. Let ̺PRą̺0,u0PHand denote by F̺:H̺,0pRq b HÑH̺,0pRq b Hthe unique

Lipschitz continuous extension of F. Then the equation

B0,̺u“F̺puq ` δbu0

admits a unique solution uPH̺,0pRq b Hsuch that u´χRą0pm0quPH̺,1pRq b Hand

up0`q “ u0.

Proof. The unique existence of uPH̺,0pRqbHfollows from Theorem 3.2. Moreover, from

the equation that is satisﬁed by uwe get

u“ B´1

0,̺pF̺puq ` δbu0q.

Therefore,

u´χRą0u“u´ B´1

0,̺δbu0“ B´1

0,̺F̺puq.

Hence, we read oﬀ that

tÞÑ uptq ´ χRě0ptqu0

lies in H̺,1pRq b Hand thus in C̺,8,1{2pR;Hq, by Lemma 5.2. As a consequence,

up0`q ´ u0“ pu´χRě0bu0qp0`q

“ pu´χRě0bu0qp0´q

“up0´q

.

Theorem 4.5 yields the causality of B´1

0,̺F̺. It follows that up0´q “ 0and therefore the

initial condition

up0`q “ u0

is satisﬁed.

27

5 Applications

The fact that the solution depends continuously on the initial data is formulated in our

setting as follows.

Theorem 5.5. Let Hbe a Hilbert space, C, D PRą0. Let F, G :˚

C8pR;Hq Ñ ˚

C`

8pR;Hq1

be such that for all ̺PRąmaxtC,Du, there exists KPRą0such that for all u, w P˚

C8pR;Hq

and ψP˚

C`

8pR;Hqwe have

|Fpuqpψq ´ Fpwqpψq|ďC|ψ|´̺,0|u´w|̺,0and |Gpuqpψq ´ Gpwqpψq|ďD|ψ|´̺,0|u´w|̺,0.

and

|Fp0qpψq|ďK|ψ|´̺,0and |Gp0qpψq|ďK|ψ|´̺,0

Let u0, w0PHand denote by F̺, G̺the respective extensions of Fand Gas continuous

mappings within H̺,0pRqbH. Moreover, let u, w PH̺,0pRqbHbe the respective solutions

of the diﬀerential equations

B0,̺u“F̺puq ` δbu0and B0,̺w“G̺pwq ` δbw0.

Then the continuous dependence estimate

|u´w|̺,0ď1

p2̺´ pC`Dqq ˜a2̺|u0´w0|H`2 sup

xPH̺,0pRqbH

|F̺pxq ´ G̺pxq|̺,0¸

holds.

Proof. Deﬁne r

F̺:H̺,0pRqbHÑH̺,´1pRqbH:uÞÑ F̺puq` δbu0and analogously r

G̺.

Applying Theorem 3.6 and using Corollary 2.5, we get

|u´w|̺,0ď1

1´|r

F̺|Lip`|r

G̺|Lip

2

sup

xPH̺,0pRqbH

|r

F̺pxq ´ r

G̺pxq|̺,´1

ď1

1´|F̺|Lip`|G̺|Lip

2̺

sup

xPH̺,0pRqbH

|B´1

0,̺ pF̺pxq ` δbu0´G̺pxq ` δbu0q|̺,0

ď2̺

2̺´ pC`Dq˜|χRą0b pu0´w0q|̺,0`sup

xPH̺,0pRqbH

|B´1

0,̺ pF̺pxq ´ G̺pxqq|̺,0¸

ď2̺

2̺´ pC`Dq˜1

?2̺|u0´w0|H`1

̺sup

xPH̺,0pRqbH

|F̺pxq ´ G̺pxq|̺,0¸.

5.2 Local Solvability

It appears that the above solution theory only deals with global solutions. This is, however,

not the case. To illustrate how to get local existence results we consider an initial value

28

5.2 Local Solvability

problem in a Hilbert space H

B0uptq “ gpt, uptqq,(10)

up0q “ u0,

where g:r0, T s ˆ HÑHis a measurable function. Moreover, for every xPHit holds

gp¨, xq P L8pr0, T s, H qand there exists a radius ηPRą0and a constant LPRą0such that

for all y, z PBHpx, ηq “ twPH;|w´x|Hďηuwe have

|gp¨, yq ´ gp¨, zq|L8pr0,T s;HqďL|y´z|H.(11)

In this situation we derive the following Lemma.

Lemma 5.6. Let u0PHand ηPRą0such that (11) is satisﬁed for each y, z PBHpu0, ηq.

We denote the projection on the closed, convex set BHpu0, ηqby P. Then for each ̺PRą0

the operator F̺deﬁned by

F̺:Cpr0, T s;Hq X pH̺,0pRq b Hq Ď H̺,0pRq b HÑH̺,0pRq b H

uÞÑ ptÞÑ χr0,T sptqgpt, P puptqqqq

is a Lipschitz-continuous mapping with lim sup̺Ñ8 |F̺|Lip ă 8. Moreover, the continuous

extension of F̺, for which we will use the same name is causal.

Proof. For u, v PCpr0, T s;Hq X H̺,0pRq b Hwe estimate

żR|F̺puqptq ´ F̺pvqptq|2

He´2̺tdt “żT

0|gpt, P puptqqq ´ gpt, P pvptqqq|2

He´2̺tdt

ďL2żT

0|Ppuptqq ´ Ppvptqq|2

He´2̺tdt

ďL2żT

0|uptq ´ vptq|2

He´2̺tdt

ďL2|u´v|2

̺,0.

This would prove the Lipschitz continuity, if we ensure that F̺is well-deﬁned. This,

however, follows by using the above estimate to obtain

|F̺puq|̺,0ď |F̺puq ´ F̺pχRě0bu0q|̺,0` |F̺pχRě0bu0q|̺,0

ďL2|u´χRě0bu0|̺,0` |gp¨, u0q|L8pr0,T s;Hqd1´e´2̺T

2̺ă 8.

The causality of F̺is straightforward.

This Lemma shows that F̺satisﬁes the conditions of our solution theorem concerning

29

5 Applications

initial value problems Theorem 5.4. Thus, there is ̺0PRą0such that for all ̺PRą̺0we

ﬁnd a unique solution vPH̺,0pRq b Hof

B0,̺v“F̺pvq ` δbu0.(12)

The next theorem asserts that a solution to (12) satisﬁes Equation (10) at least for some

non-vanishing time interval.

Theorem 5.7. Let ̺PRą̺0and let vPH̺,0pRq b Hbe the solution of (12). Then there

exists t˚Ps0, T ssuch that vsatisﬁes Equation (10) on the interval r0, t˚s.

Proof. For each tP r0, T swe obtain due to causality of F̺

B0,̺pχRďtpm0qv`χRątbvptqq “ χRďtpm0qB0,̺v

“χRďtpm0qF̺pvq ` δbu0

“χRďtpm0qF̺pχRďtpm0qvq ` δbu0

and thus we estimate using the ﬁrst inequality in Lemma 5.2

a2̺supt|vpsq ´ u0|He´̺s |sP r0, tsu

ď |B0,̺pχRďtpm0qv`χRątbvptq ´ χRě0bu0q|̺,0

“ |χRďtpm0qF̺pχRďtpm0qvq|̺,0

ď |χRďtpm0qF̺pχRďtpm0qvq ´ χRďtpm0qF̺pχr0,tsbu0q|̺,0` |χRďtpm0qF̺pχr0,tsbu0q|̺,0

ďL|χRďtpm0qv´χr0,tsbu0|̺,0` |gp¨, u0q|L8pr0,ts;Hqd1´e´2̺t

2̺

ďLsupt|vpsq ´ u0|He´̺s |sP r0, tsu?t`|gp¨, u0q|L8pr0,ts;Hqd1´e´2̺t

2̺.

If we choose tă2̺

L2we can conclude that

supt|vpsq ´ u0|He´̺s |sP r0, tsu ď b1´e´2̺t

2̺

?2̺´L?t|gp¨, u0q|L8pr0,ts;Hq.

Hence,

supt|vpsq ´ u0|H|sP r0, tsu ď e̺t b1´e´2̺t

2̺

?2̺´L?t|gp¨, u0q|L8pr0,ts;Hq.

Therefore, we can ﬁnd t˚P r0, T ssuch that

supt|vpsq ´ u0|H|sP r0, t˚su ď η,

30

5.3 Classical Delay Equations

or in other words vrr0, t˚ss Ď BHpu0, ηqwhich implies

B0,̺vptq “ F̺pvqptq “ gpt, vptqq

for each tP p0, t˚q. Since the initial condition vp0q “ u0is satisﬁed by deﬁnition of vand

Theorem 5.4, the assertion follows.

5.3 Classical Delay Equations

In this section we apply the existence theory to classical delay equations, in particular

integro-diﬀerential and neutral equations. As a ﬁrst step we observe that delay equations

typically show a special structure which can be utilized to formulate natural assumptions

for existence of solutions.

5.3.1 A Structural Observation

In many applications it turns out, that the function Fof Theorem 3.2 factorizes as F“

Φ˝Θ, where for Hilbert spaces Hand Vwe have

Φ : č

ηPRą0

Hη,0pRq b VÑ˚

C`

8pR;Hq1

and

Θ : ˚

C8pR;Hq Ñ č

ηPRą0

Hη,0pRq b V

with appropriate Hilbert spaces Hand Vand suitable conditions on Φand Θ, which we

specify later on. Thus, we arrive at a specialized form of the general problem (3) given by

B0u“ΦpΘuq.

We will prove well-posedness results for two particular cases. The ﬁrst one, Theorem 5.8,

describes discrete delay, the second one, Theorem 5.10, deals with the whole past of u.

Theorem 5.8. Let NPN,HHilbert space, pθ0,...θN´1q P pRď0qN,sP p0,1q,̺0PRą0,

Φ : ˚

C8pR;HNq Ñ ˚

C`

8pR;Hq1. Assume that for all ̺PRą̺0, there is KPRą0such that

for all u, w P˚

C8pR;HNqand ψP˚

C`

8pR;Hqwe have

|Φp0qpψq|ďK|ψ|´̺,1and |Φpuqpψq ´ Φpwqpψq|ďs|ψ|´̺,1|u´w|̺,0.

Denote by Φ̺the continuous extension of Φas a mapping from H̺,0pRqbHNto H̺,´1pRqb

H. For ̺PRą̺0let Θ̺:H̺,0pRq b HÑH̺,0pRq b HNbe given by

Θ̺x“`τθ0x,...,τθN´1x˘PH̺,0