Retarded functional differential equations with general delay structure

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Title

Retarded functional differential equations with general delay

structure( Dissertation_全文 )

Author(s)

Nishiguchi, Junya

Citation

京都大学

Issue Date

2017-03-23

URL

https://doi.org/10.14989/doctor.k20156

Right

許諾条件により本文は2018-03-22に公開

Type

Thesis or Dissertation

Textversion

ETD

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general delay structure

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Abstract

A differential equation of a dependent variable x with one independent vari-able t ∈ R is called a delay differential equation (abbrev. DDE), if the derivative of an unknown function x = x(·) at t depends on the past before t. DDEs appear in the fields of mathematical physics, population dynamics, engineering, and so on, and the way the equation depends on the past largely varies in such problems in application. In this thesis, we consider DDEs from dynamics viewpoint and propose a new proper setting as dynamical systems for DDEs.

Unlike ordinary differential equations without delay, dynamical systems generated by DDEs cannot be formulated on the space of dependent variable x but on the space of histories of the unknown function x(·). We call this feature the delay structure of the dynamics of DDEs.

The dynamics viewpoint of DDEs was brought by Hale (1963), who introduced the retarded functional differential equations (RFDEs) with finite delay. For a given DDE with finite delay, one can obtain an RFDE with finite delay by choosing the space of continuous initial histories with uniform topology and by defining the associated right-hand side F . However, there is no reason to restrict the space of histories to this space for other types of delays.

As our new setting for DDEs, we shall treat DDEs with various types of delays in a unified way and clarify the delay structure in those dynamics. For this purpose, we introduce the RFDEs with general delay structure. Key features of our approach are that natural and crucial topological properties of the space of initial histories are identified, and that the Lipschitz condition of the right-hand side F is formulated independently from the metric structure of the space of initial histories.

One of the main theorems of this thesis is a necessary and sufficient condition on the space of initial histories for which the corresponding ini-tial value problem is well-posed for any “admissible” right-hand side of an RFDE with general delay structure. This necessary and sufficient condi-tion for the well-posedness is sharper than similar condicondi-tions in the earlier result obtained by Hale & Kato (1978) and clarifies their meaning of their conditions.

This result can be applied to the space of initial histories whose topology i

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is not given by a seminorm. An example of such topology is the compact-open topology which is frequently used for DDEs with unbounded time- and state-dependent delays.

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Acknowledgement

I am deeply grateful to my supervisor, Professor Hiroshi Kokubu, for his encouragement, support, and advise. I started to study delay differential equations by his suggestion when I was a master course student. During my master course, I could take part in the First International Conference on Dynamics of Differential Equations held in Georgia Tech by his financial support. This was a valuable chance to meet the researches on infinite-dimensional dynamical systems and delay differential equations.

I am grateful for the support of the Top Global University project for Kyoto University (abbrev. KTGU project). My co-supervisor by the KTGU project, Professor Hans-Otto Walther at Justus Liebig University Giessen (abbrev. JLU Giessen), kindly supported me for my visit by KTGU project to Giessen three times. I would like to thank Hans-Otto for his hospitality. I would also like to thank Professor Bernhard Lani-Wayda at JLU Giessen. It was fortunate that I was accepted as a research fellowship of Japan Society for the Promotion of Science (JSPS).

With the support of the KTGU project, I had a chance to visit Professor Roger Nussbaum at Rutgers University and Professor John Mallet-Paret at Brown University. This was a great opportunity to discuss my research and to consider my future works. I would appreciate their hospitality during my visit.

I am grateful to Professor Hisashi Inaba at University of Tokyo for the financial support by the JSPS Bilateral Joint Research Project (Open Part-nership). I would like to thank Dr. Yukihiko Nakata at Shimane University for the interesting discussion. I would also like to thank Dr. Gergely R¨ost at University of Szeged for inviting me to the school about the computational tools of delay differential equations at Szeged.

I extend my gratitude to the administrative staff at the Department of Mathematics, Kyoto University. My research life would not have been done without their kind help.

Finally, I owe a debt of gratitude to my family for their encouragement. Without their mental support, I could not continue my research in Ph.D. course.

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Abstract i

Acknowledgement iii

1 Introduction 1

1.1 Delay differential equations . . . 1

1.1.1 Types of time-delay . . . 1

1.1.2 Delay differential equations from applications . . . 3

1.2 Dynamics of delay differential equations with finite delay . . . 4

1.2.1 Functional differential equations . . . 5

1.2.2 Initial value problem . . . 6

1.2.3 Continuous semi-flow . . . 6

1.2.4 Linear stability analysis . . . 7

1.3 State-dependent delay . . . 8

1.4 Estimation of parameter range . . . 10

1.4.1 Transcendental equation with complex coefficients . . 10

1.4.2 Lambert W function . . . 11

1.5 General delay structure and main theorems . . . 11

1.6 Organization . . . 13

2 General delay structure 14 2.1 Continuations and continuability . . . 15

2.1.1 Continuations . . . 15

2.1.2 Continuability . . . 16

2.1.3 Examples . . . 18

2.2 Trivial retarded functional differential equation . . . 19

2.3 Comments on preceding results about continuability . . . 20

2.3.1 Axiom by Hale & Kato . . . 20

2.3.2 Hypotheses by Schumacher . . . 21

3 Uniqueness and maximal solutions 25 3.1 Maximality . . . 25

3.2 Solution process . . . 27

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CONTENTS v

4 Local semi-flows 29

4.1 Local semi-flows on topological spaces . . . 30

4.1.1 Local semi-flows and local processes . . . 30

4.1.2 Continuity of local semi-flows and local processes . . . 32

4.2 Semi-groups of linear operators . . . 37

5 Existence and uniqueness theorems 38 5.1 Picard–Lindel¨of theorem . . . 40

5.1.1 Lipschitzian about continuations . . . 40

5.1.2 Lipschitzian about memories . . . 42

5.2 Cauchy–Peano existence theorem . . . 43

5.3 The restricted initial value problem . . . 45

6 Necessary and sufficient condition 48 6.1 Well-posedness of initial value problem . . . 48

6.2 Examples . . . 52

7 Applications 54 7.1 Proof of well-posedness result under Hale–Kato axiom . . . . 54

7.2 Well-posedness for differential equations with time and state-dependent delays . . . 56

8 Concluding remarks 61 A Calculus for vector-valued functions 63 B Uniform contraction theorem 66 C Uniform spaces and uniform continuity 68 C.1 Topological spaces . . . 68 C.2 Nets . . . 68 C.3 Uniform spaces . . . 70 C.3.1 Uniformity . . . 70 C.3.2 Uniform continuity . . . 72 Bibliography 74

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Introduction

1.1

Delay differential equations

A differential equation of a dependent variable x ∈ E with one independent variable t ∈ R is called a delay differential equation (abbrev. DDE), if the derivative of an unknown function x = x(·) at t depends on the past before t:

{ x(s) ∈ E : s < t }

Here E = (E, k · k) is a Banach space. There are two major types of delays for DDEs; finite delay and infinite delay. A DDE is said to be of finite delay if there is r > 0 such that for all t, the derivative ˙x(t) only depends on

{ x(s) ∈ E : t − r ≤ s ≤ t }. Otherwise, the equation is said to be of infinite delay.

The dependent variable x of a DDE can be scalar-valued (x ∈ R) or vector-valued (x ∈ Rn) for some n ≥ 2. Sometimes, the dependent variables

may belong to an infinite dimensional space. A DDE with a dependent variable x = (xi)i∈Z being in a Banach space `∞(Z, R) appears as a lattice differential equation (LDE) with delay (see Caraballo et al. [5]).

We refer the reader to Walther [76] for a general survey of the dynamics of DDEs.

1.1.1 Types of time-delay

Let us see examples of the types of time-delay in DDEs. A differential equation with a constant delay

˙

x(t) = f (t, x(t), x(t − r)), r > 0 (1.1)

falls in a class of DDEs which appears to be the simplest. Here f : R×E×E → E is a map. If the delay r depends on the present value of the unknown

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CHAPTER 1. INTRODUCTION 2

function, we obtain a DDE of the form ˙

x(t) = f t, x(t), x(t − ρ(x(t)), (1.2)

Such a DDE is called a differential equation with a state-dependent de-lay, and ρ : E → R+ := [0, +∞) is called the delay function. When ρ is

unbounded, DDE (1.2) is of infinite delay. Another example of a state-dependent delay can be found in the following DDE:

˙

x(t) = f t, x(t), x(t − ρ(x(t − δ)), δ > 0. (1.3)

For this DDE, the delay depends not on the present value but on the past value of the unknown function.

We can also see a DDE where the delay is time-dependent. A differential equation with a proportional delay

˙

x(t) = f (t, x(t), x(λt)), t ≥ 0, (1.4)

where 0 < λ < 1, is an example of a DDE with time-dependent delay, because the argument λt is decomposed as

λt = t − (1 − λ)t.

DDE (1.4) is also of infinite delay since (1 − λ)t diverges to +∞ as t → +∞. An example of the proportional delay appears as the pantograph equation

˙

x(t) = ax(t) + bx(λt), (t ≥ 0, 0 < λ < 1),

where a, b ∈ R (e.g., see Kato & McLeod [37] and Ockendon & Taylor [53]). In the above examples of DDEs, the derivative ˙x(t) depends on the spe-cific past value of the unknown function x(·). We can consider a DDE where the derivative ˙x(t) depends on the mean value of the unknown function. For a map g : [−r, 0] × R × E2 → E, ˙ x(t) = Z 0 −r g(θ, t, x(t), x(t + θ)) dθ (1.5)

is an example of a differential equation with a distributed delay. DDE (1.5) is of finite delay. However, if g is defined on (−∞, 0]×R×E2 → E, and if the

domain of integration is changed to (−∞, 0], then the corresponding DDE is of infinite delay. We can also consider another DDE with a distributed delay as follows: ˙ x(t) = f  t, x(t),1 ε Z −r+ε −r x(t + θ) dθ  , r > 0, ε > 0. Since 1 ε Z −r+ε −r x(t + θ) dθ → x(t − r) as ε → 0

under the assumption that x(·) is continuous, DDE (1.1) can be considered as an approximation of the above DDE.

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1.1.2 DDEs from applications

DDEs appear in the variety of fields in application, such as mathematical physics, population dynamics, engineering, and so on.

Finite speed of propagation

Driver [14] considered the relativistic equations of motion for the two-body problem of classical electrodynamics and obtained a system of DDEs with state-dependent delay, in which the time-delays

τji(t) = 1 c · xi(t) − xj(t − τji(t)) , (i, j) = (1, 2), (2, 1) (1.6) are brought by the finite speed c (the speed of light) of propagation of the electric fields. Here x1(t), x2(t) ∈ R denote the positions of particles moving

on the straight line. A DDE with state-dependent delays is also obtained by the equations of motion with gravitational forces (Chicone [7]). See also Chicone [8, 9] for the investigation of the dynamics of DDEs with a small delay motivated by these equations of motions. Walther [70] considered the position control by echo, and studied a DDE with a state-dependent delay whose time-delay is similar to (1.6). See also Walther [74] for a model of the automatic soft landing.

Nuclear reactor

Levin [46] considered a differential equation of the form ˙

x(t) = − Z t

0

a(t − s)g(x(s)) ds, t ≥ 0, x(t) ∈ R

for studying the dynamics of nuclear reactor without using the framework of RFDEs.

Instantaneous destruction and delayed production

Mackey & Glass [49] modeled the dynamics of the concentration of circu-lating blood cells by using the following scalar DDE with a single constant delay

˙

x(t) = β x(t − r)

1 + x(t − r)n − γx, x(t) ∈ R, r > 0, (1.7)

where β, γ, n are positive numbers. This equation is now called the Mackey– Glass equation. See also Glass & Mackey [19]. The introduction of the time-delay to the equation is important in the sense that an oscillation phenomenon appears in spite of the first order scalar differential equation. In this model, the constant delay is used, but one can consider a different time-delay.

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CHAPTER 1. INTRODUCTION 4

The Mackey–Glass equation describes the “instantaneous destruction and delayed production.” By a reparametrization of time, this type of DDE can be reformulated as

˙

x(t) = −µx(t) + g(x(t − 1)), x(t) ∈ R, (1.8)

where µ ≥ 0 and g : R → R is a smooth function. DDE (1.8) is called of positive feedback if ξ · g(ξ) > 0 for all ξ 6= 0. If ξ · g(ξ) < 0 for all ξ 6= 0, DDE (1.8) is called of negative feedback. It is known that the monotonic-ity of g influences the asymptotic dynamics (e.g., the global attractor) of DDE (1.8). See Walther [69], Krisztin & Walther [45], R¨ost & Wu [60], and Liz & R¨ost [47] for example. We refer the reader to Diekmann et al. [13, Chapter XV] and Krisztin [43] for further references on the dynamics of DDE (1.8).

Delayed feedback control

Pyragas [56] considered the delayed feedback control of an unstable periodic orbit Γ = {p(t)}t∈[0,T ] of an ODE ˙x = f (x) on the Euclidean space Rn by adding a delayed feedback term to the original ODE as follows:

˙

x(t) = f (x(t)) + K[x(t − r) − x(t)]. (1.9)

Here K is an n × n real matrix, and r is an integer multiple mT of the minimal period T of that unstable periodic orbit Γ which we would like to stabilize. p(t) is a periodic solution of DDE (1.9), and our purpose is to find m ∈ Z and K ∈ Mn(R) such that p(t) is an orbitally stable periodic

solution of DDE (1.9). This feedback control is motivated by the control of chaos. Pyragas [56] showed the efficiency of the delayed feedback control numerically, but the mathematical analysis still remains incomplete. We refer the reader to Pyragas [57] for a survey article of the delayed feedback control.

1.2

Dynamics viewpoint of delay differential

equa-tions with finite delay

The objective of the thesis is to consider DDEs from dynamics viewpoint and to find a proper and unified setting as dynamical systems for DDEs. Unlike ordinary differential equations (ODEs) without delay, dynamical systems generated by DDEs cannot be formulated on the space of dependent variable x but on the space of histories of the unknown function x(·). We call this feature the delay structure of the dynamics of DDEs.

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1.2.1 Functional differential equations

This dynamics viewpoint was brought by Hale [21, 22], who introduced the retarded functional differential equations (RFDEs) with finite delay. Let C(S, E) be the linear space of all continuous maps from a subset S of R to E. When S is a closed and bounded interval, let C(S, E)udenote the Banach

space C(S, E) with the supremum norm: kxk∞= sup

t∈S

kx(t)k, x ∈ C(S, E).

We consider Rna Banach space with a suitable norm k · k.

Definition 1.1 (ref. Hale & Verduyn Lunel [30]). Let r > 0 be fixed. For real numbers a < b and for a continuous map x : [a − r, b] → Rn, define xt ∈ C([−r, 0], Rn)u as xt: [−r, 0] 3 θ 7→ x(t + θ) for every t ∈ [a, b]. For a

map F : D ⊂ R × C([−r, 0], Rn)u→ Rn, a differential equation

˙

x+(t) = F (t, xt)

is called a retarded functional differential equation with finite delay (RFDE with finite delay). Here ˙x+(t) denotes the right derivative of x at t. For

a < b, a map x : [a − r, b) → Rnis said to be a solution of this RFDE, if (i) x is continuous on [a−r, a], (ii) x is right differentiable on [a, b), (iii) (t, xt) ∈ D

holds for all t ∈ [a, b), and (iv) x satisfies the equation for t ∈ [a, b).

We call the above xt the history of x : [a − r, b] → Rn at t ∈ [a, b] with

length r. The uniform continuity of x shows that the map [a, b] 3 t 7→ xt∈ C([−r, 0], Rn)u

is continuous.

The above RFDE represents how the right derivative ˙x+(t) of an

un-known function x(·) depends on its past information before t. In this sense, we call F the history functional. Usually, F is assumed to be continuous. Therefore, a solution x : [a − r, b) → Rn has a continuous right derivative

on [a, b), which implies that x is continuously differentiable on [a, b) (see Knight [42, Corollary]).

In this thesis, we consider the usual derivative ˙x(t), and hence an RFDE with finite delay becomes

˙

x(t) = F (t, xt) (1.10)

for a map F : D ⊂ R × C([−r, 0], Rn)u → Rn. We say that a map x : [a −

r, b] → Rn (a < b) is a solution of RFDE (1.10), if (i) x is continuous on [a − r, a], (ii) x is differentiable on [a, b], (iii) (t, xt) ∈ D holds for all

t ∈ [a, b], and (iv) x satisfies the equation for t ∈ [a, b]. ˙x(a) denotes the right derivative at a.

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CHAPTER 1. INTRODUCTION 6

1.2.2 Initial value problem

To formulate a dynamical system generated by RFDE (1.10), we consider the initial value problem. Since the derivative ˙x(t) depends on the history xt, it is natural to set an initial condition of the form xt0 = φ0 for some

(t0, φ0) ∈ D ⊂ R × C([−r, 0], Rn)u.

Definition 1.2 (ref. [30]). We consider RFDE (1.10) for some r > 0. The system ( ˙ x(t) = F (t, xt), t ≥ t0, xt0 = φ0, (t0, φ0) ∈ D (1.11) is called the initial value problem (IVP) of RFDE (1.10).

The following theorem are fundamental for the IVPs of RFDEs.

Theorem 1.3 (ref. [30]). If D is open and F is continuous, then IVP (1.11) has a solution for each (t0, φ0) ∈ D.

Theorem 1.4 (ref. [30]). Suppose that D is open and F is continuous. If F (t, φ) is Lipschitzian in φ in each compact set in D, then for each (t0, φ0) ∈

D, there is a unique solution of (1.11).

Theorem 1.5 (ref. [30]). Suppose that D is open, F is continuous, and x0 is a solution of IVP (1.11) for a given (t0, φ0) ∈ D which exists and is unique

on [t0− r, b]. If a sequence ((σk, ψk))∞k=1 in D satisfies (σk, ψk) → (t0, φ0)

as k → ∞, then there is k0 ≥ 1 such that each solution xk of RFDE (1.10)

with xkσk = ψkexists on [σk−r, b] and xk→ x0 uniformly on [t0−r, b]. Since

all xk may not be defined on [t0− r, b], by xk→ x0 uniformly on [t0− r, b],

we mean that for any ε > 0, there is a k1(ε) such that xk(t), k ≥ k1(ε), is

defined on [t0− r + ε, b], and xk→ x0 uniformly on [t0− r + ε, b].

In Theorem 1.4, F (t, φ) is said to be Lipschitzian in φ in a compact set K ⊂ D, if there exists L > 0 such that for all (t, φ1), (t, φ2) ∈ K,

kF (t, φ1) − F (t, φ2)k ≤ L · kφ1− φ2k∞

holds.

1.2.3 Continuous semi-flow

We concentrate our discussion here on autonomous RFDEs of the form ˙

x(t) = F (xt) (1.12)

for F : D ⊂ C([−r, 0], Rn)u→ Rn. The corresponding IVP is

( ˙

x(t) = F (xt), t ≥ 0,

x0 = φ0, φ0 ∈ D.

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Under the assumptions that (i) D is open, (ii) F is continuous, and (iii) F satisfies the Lipschitz condition stated above, Theorems 1.3, 1.4, and 1.5 show that IVP (1.13) has the unique maximal solution

x(·, φ0) : [−r, b(φ0)) → Rn, 0 < b(φ0) ≤ +∞

for each φ0∈ D. In this way, we obtain the solution semi-flow ΦF = (ΦtF)t≥0

defined by ΦF: [ φ∈D [0, b(φ)) × {φ} → C([−r, 0], Rn)u, ΦtF(φ) := ΦF(t, φ) = x(·, φ)t.

From Theorem 1.5, the function b(·) : D → (0, +∞] is lower semi-continuous. Moreover, ΦF is continuous, because

F(t, φ) − ΦF(t0, φ0)k∞

≤ kx(·, φ)t− x(·, φ0)tk∞+ kx(·, φ0)t− x(·, φ0)t0k∞

≤ k[x(·, φ) − x(·, φ0)][−r,b]k∞+ kx(·, φ0)t− x(·, φ0)t0k∞

holds for 0 < b < b(φ0). Therefore, we obtain the following theorem.

Theorem 1.6. ΦF given as above defines a continuous local semi-flow.

Theorems 1.3, 1.4, and 1.5 are the fundamental in order for an au-tonomous RFDE (1.12) generates a topological local semi-dynamical system (D, ΦF). The terminology of local dynamical systems was used in Sell [62].

This dynamical system is infinite-dimensional because the phase space is a subset of C([−r, 0], Rn). We refer the reader to Hale et al. [27], Temam [68], and Sell & You [63] for general references of the infinite-dimensional dynam-ical systems.

For a given DDE of finite delay, one can obtain an RFDE with finite delay by defining the associated right-hand side F . However, there is no reason to restrict the space of histories to the space of continuous maps with uniform topology (e.g., Breda [2], Mallet-Paret [50, Example 3.1]).

1.2.4 Linear stability analysis

We consider an autonomous RFDE (1.12) with E = Rn. We suppose that

the map F is continuously differentiable in the Fr´echet sense. A constant map φ ∈ C([−r, 0], Rn)uis called an equilibrium of RFDE (1.12) if F (φ) = 0.

Then ΦtF(φ) ≡ φ for all t ≥ 0, where ΦF is the semi-flow generated by this

RFDE.

Let φ∗(θ) ≡ p be an equilibrium of RFDE (1.12). Then x∗(t) ≡ p (t ∈ R) is an equilibrium of this RFDE. The (first) variational equation along x∗(t) becomes

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CHAPTER 1. INTRODUCTION 8

where DF (φ∗) : C([−r, 0], Rn)u→ Rn is the Fr´echet derivative of F at φ∗.

In general, we consider the following linear RFDE ˙

x(t) = Lxt, (1.14)

where L : C([−r, 0], Rn)

u→ Rnis a bounded linear operator. Let (SL(t))t≥0

denote the solution semi-group generated by (1.14). This is a (C0)

semi-group on C([−r, 0], Rn)u. For the infinitesimal generator A of this solution

semi-group, the following theorem is fundamental.

Theorem 1.7 (ref. Hale & Verduyn Lunel [30]). Let σ(A) and σp(A) denote

the spectrum of A and the point spectrum of A, respectively. Then we have σ(A) = σp(A) =  λ ∈ C : det  λI − Z 0 −r eλθdη(θ)  = 0  .

Here η : [−r, 0] → Mn(R) is a function of bounded variation satisfying

Lφ = Z 0

−r

dη(θ) φ(θ), φ ∈ C([−r, 0], Rn)u,

and I ∈ Mn(R) is the identity matrix.

The above integral is the Riemann–Stieltjes integral with respect to a matrix-valued function η. The equation det(λI −R−r0 eλθdη(θ)) = 0 is called the characteristic equation of linear RFDE (1.14), and a solution of this equation is called a characteristic root.

The following theorem holds.

Theorem 1.8 (e.g., Diekmann et al. [13]). For an equilibrium of RFDE (1.12) x∗(t) and its linearized equation, the following statements hold:

(i) x∗(t) is exponentially stable if <(λ) < 0 for all characteristic roots λ, (ii) x∗(t) is unstable if <(λ) > 0 for some characteristic root λ.

Here a characteristic root means that of the characteristic equation of the linearized equation.

1.3

State-dependent delay

DDEs with a constant delay often appear as mathematical models since the delay structure in these equations is simple. Various applications of DDEs with a constant delay can be found in Erneux [16]. However, DDEs with a state-dependent delay sometimes appear as more realistic mathematical models, and it is interesting how the dynamics of DDEs with a constant

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delay and those with a state-dependent delay are different. We refer the reader to Hartung et al. [32] for a survey which studies examples of DDEs with a state-dependent delay and their dynamics.

Here we consider DDEs with a state-dependent delay which are of finite delay, namely, DDEs with a bounded state-dependent delay. Suppose

ρ(E) ⊂ [−r, 0]

for some r > 0 in DDEs (1.2) and (1.3), where δ < r is assumed. The difficulty for these DDEs is that when we consider these equations as RFDEs with finite delay, the associated history functionals Fi: R × C([−r, 0], E)u→

E (i = 1, 2) defined as

F1(t, φ) = f t, φ(0), φ(−ρ(φ(0))),

F2(t, φ) = f t, φ(0), φ(−ρ(φ(−δ))), δ > 0

are not smooth in general, even if f : R × E → E is a smooth function (e.g., see Krisztin & Arino [44], Louihi et al. [48], Mallet-Paret at al. [51], Walther [71]). The reason is that the arguments −ρ(φ(0)) and −ρ(φ(−δ)) vary in φ, therefore, the smoothness of φ is needed to ensure the smoothness of F1 and F2. In the same way, F1(t, ·) and F2(t, ·) are not locally Lipschitz

continuous uniformly in t in general. This means that the classical results stated in Section 1.2 cannot be applied to DDEs with a state-dependent delay, and hence, the notion of dynamical systems generated by these equa-tions is unclear.

This feature can also be understood as follows: One consider DDE (1.2) with an initial condition xt0 = φ0 for a time-interval [t0, t0+ β], where β is

some positive number. For t ∈ [t0, t0+ β], x t − ρ(x(t)) is determined by

the initial history φ0 if and only if

t − ρ(x(t)) ≤ t0. (1.15)

Suppose that the inequality (1.15) holds. Then DDE (1.2) with initial con-dition xt0 = φ0 becomes the IVP of a non-autonomous ODE

( ˙

x = f t, x, φ0(t − ρ(x) − t0), t0 ≤ t ≤ t0+ β,

x(t0) = φ0(0).

(1.16)

This argument is called the method of steps (ref. Smith [65]). We notice that a solution of IVP (1.16) is not necessarily unique, even if f (t, ·, ·) is locally Lipschitz continuous uniformly in t. We need the Lipschitz continuity of a map φ0: [−r, 0] → E in general.

The method of steps is also valid for DDE (1.3). If t − ρ(x(t − δ))) ≤ t0, β ≤ δ,

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CHAPTER 1. INTRODUCTION 10

then we have the IVP of a non-autonomous ODE (

˙

x = f t, x, φ0(t − ρ(φ0(−δ)) − t0), t0 ≤ t ≤ t0+ β,

x(t0) = φ0(0).

Unlike IVP (1.16), this IVP has a unique solution with continuous initial history φ0.

The difference between these IVPs is brought by the property of delay functional Ri: C([−r, 0], E)u → [0, r] (i = 1, 2) defined by

R1(φ) = ρ(φ(0)), R2(φ) = ρ(φ(−δ)).

See Section 7.2 for the detail. To clarify the difference of these delay func-tionals, we define the notion of Lipschitzian about continuations in Subsec-tion 5.1.1.

1.4

Estimation of parameter range

1.4.1 Transcendental equation with complex coefficients

To apply Theorem 1.8, it is important to find the parameter range for which all the roots of

det  λI − Z 0 −r eλθdη(θ)  = 0

belong to the left half plane of C. Here r > 0 is an upper bound of the time-delay, and η : [−r, 0] → Mn(R) is a function of bounded variation. For

the equation with a single constant delay ˙

x(t) = Ax(t) + Bx(t − r), A, B ∈ Mn(R),

the characteristic equation becomes

det(λI − [A + e−λrB]) = 0.

It is difficult to find a necessary and sufficient condition on A, B, and r > 0 for which all the roots of this equation belong to the left half plane (see Hale & Verduyn Lunel [30, pp. 136]). See Pontrjagin [54] and St´ep´an [67] for related general results.

The case n = 1 looks simple. The characteristic equation is

λ + α − βe−λr = 0, (1.17)

where α and β are real numbers (we have changed the sign of α). Hayes [33] found the exact region on α, β ∈ R for which all the roots of (1.17) belong to the left half plane with r = 1. If we consider the case n ≥ 2, then the transcendental equation (1.17) with α, β ∈ C may appear. Breda [3] considered this case and obtained some results.

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1.4.2 Lambert W function

The set of roots of (1.17) with α, β ∈ C can be expressed by using the Lambert W function as follows:

1

rW (rβe

αr) − α.

Here W is the multi-valued inverse of a complex function zez, namely W (ζ) = { z ∈ C : zez = ζ }.

We refer the reader to Corless et al. [11] for a survey of the Lambert W function and its applications. Hayes [33] studied the equation zez = ζ for ζ ∈ R, and Wright [79] studied this equation for ζ ∈ C.

An attempt to analyze the transcendental equation (1.17) by using the Lambert W function is not new (e.g., see Asl & Ulsoy [1] and Shinozaki & Mori [64]). Finding the expressions of each complex branches of the W function, we obtain the following theorem.

Theorem 1.9 (Nishiguchi [52]). Let α ∈ C, β ∈ C \ {0}, and r > 0. Then all the roots of (1.17) belong to the left half plane of C if and only if α, β and r satisfy the following conditions (a) or (b):

(a) <(α) > |β|,

(b) −|β| < <(α) ≤ |β| and

Z(=(α)r + Arg(β)) > Arccos(<(α)/|β|) + r ·p|β|2− <(α)2.

Here Z is a real function defined as Z(x) = Arccos(cos(x)).

This theorem has been applied to the stabilization of an unstable equi-librium solution by the delayed feedback control and to obtain a necessary and sufficient condition for which a synchronous state in some oscillator net-works with a delayed coupling is linearly stable (see [52]). For the oscillator networks with a delayed coupling, see Earl & Strogatz [15] and Punetha et al. [55].

1.5

General delay structure and main theorems

In this thesis, we shall treat DDEs with various types of delays in a unified way and clarify the delay structure in the associated dynamics. For this pur-pose, we introduce the RFDEs with general delay structure which consists of a history interval I, an initial value space X, and a history functional F . The interval I may be a bounded interval [−r, 0] for some r > 0 or the unbounded interval R− := (−∞, 0]. We consider the linear space Map(I, E)

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CHAPTER 1. INTRODUCTION 12

of all not necessarily continuous maps from I to a Banach space E with linear operations:

(φ + ψ)(θ) := φ(θ) + ψ(θ), (α · φ)(θ) := αφ(θ).

The zero element 0 of Map(I, E) is the constant map whose value is identi-cally equal to 0 ∈ E. Let X denote a linear subspace of Map(I, E). For a map F : D ⊂ R × X → E, we consider a differential equation

˙

x(t) = F (t, xt) (1.18)

and its initial value problem (IVP) (

˙

x(t) = F (t, xt), t ≥ t0,

xt0 = φ0, (t0, φ0) ∈ D.

(1.19) See Chapter 2 for details.

The following are the main theorems of this thesis, where X is assumed to be a topological linear space. Terminologies are explained in later chapters. Theorem A. Suppose that X is continuable. Let (t0, φ0) ∈ D be arbitrarily

given. If F is continuous and is Lipschitzian about continuations of (t0, φ0),

and if D is a neighborhood of (t0, φ0), then IVP (1.19) has a locally unique

solution.

Theorem B. For a continuable X, the following conditions are equivalent: (a) For any map F which is continuous on an open set D and is uni-formly Lipschitzian about continuations, the solution process UF is a

continuous local process. In other words, IVP (1.19) is well-posed. (b) The solution semi-group (S0(t))t≥0on X generated by the trivial RFDE

˙

x = 0 is continuous. Namely, the semi-flow

[0, +∞) × X 3 (t, φ) 7→ S0(t)φ ∈ X

is continuous.

Notice that we do not need the metric structure of a space X for these theorems, because the notion of Lipschitzian about continuations is weaker than the usual Lipschitz condition. For “Lipschitzian about continuations,” we only compare the difference of the values of F on “continuations.” The definitions of “continuations and continuability,” and “Lipschitzian about continuations” will be given in Chapters 2 and 5, respectively. The definition of solution process and the well-posedness will be given in Chapter 3.

Theorem B gives a necessary and sufficient condition for the well-posedness of RFDEs with general delay structure which can be considered as an ex-tension of earlier results of Hale & Kato [28]. Furthermore, our main result

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clarifies the meaning of their axioms, because Theorem B reveals that the continuity of the solution semi-group (S0(t))t≥0 is essential for the

well-posedness, and the Hale–Kato axiom is a sufficient condition for this con-tinuity, as shown in a corollary of Theorem B in Section 7.1. See also Section 2.3 for comparison between our formulation with previous works. We refer the reader to Hale [23, 24], Hino [34] for related references.

Since the continuity of (S0(t))t≥0is a purely topological concept, we can

apply Theorem B to a Fr´echet space C(R−, E)co with the compact-open

topology, which appears DDEs with unbounded time- and state-dependent delays. The use of the compact-open topology originates from Walther [77]. We mention that Walther [77] developed his C1-theory by using the solu-tion manifold. For the differentiable dynamics on solusolu-tion manifold, see Walther [71, 72, 73]. In Section 7.2, we prove the well-posedness of some DDEs with unbounded time- and state-dependent delays by applying The-orem B.

1.6

Organization

In Chapter 2, we introduce the RFDEs with general delay structure. In Section 2.1, we consider continuations of a pair (t0, φ0) of initial time and

initial history and define the continuability of X by using the continuations. In Section 2.2, we consider the IVP of the trivial RFDE ˙x = 0 and the solution semi-group generated by this IVP. In Section 2.3, we compare the continuability with the properties introduced by Schumacher [61] and Hale & Kato [28].

In Chapter 3, we investigate the uniqueness of a maximal solution of IVP (1.19).

In Chapter 4, we study the continuity of local semi-flows and local pro-cesses on uniform spaces.

In Chapter 5, we prove theorems about the existence and uniqueness including Theorem A. For this purpose, we introduce the notion of Lips-chitzian about continuations and about memories.

In Chapter 6, we study the well-posedness of IVP (1.19) and prove The-orem B.

In Chapter 7, we apply Theorems A and B to the RFDEs with infinite delay and to DDEs with time- and state-dependent delays.

Finally, in Chapter 8, we conclude the thesis by giving some discussions and future works.

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Chapter 2

General delay structure and

related concepts

The following is the notation used throughout this thesis.

Notation. Let I denote a bounded interval [−r, 0] for some r > 0 or an unbounded interval R− := (−∞, 0]. We consider the linear space Map(I, E)

of all not necessarily continuous maps from I to a Banach space E = (E, k · k). The zero element 0 of Map(I, E) is the constant map whose value is identically equal to 0 ∈ E. X denotes a linear subspace of Map(I, E).

We introduce RFDEs with general delay structure, where the topology of X is not specified. We adopt the notation

[a, b] + I := { t + θ : t ∈ [a, b], θ ∈ I } =

(

[a − r, b], I = [−r, 0],

(−∞, b], I = R−

which is convenient because the domain of x consists of t + θ for t ∈ [a, b] and θ ∈ I. This notation [a, b] + I originates from Walther [77].

Definition I. 1. For real numbers a < b and for a map x : [a, b] + I → E, we call

xt: I 3 θ 7→ x(t + θ) ∈ E

the history of x at t ∈ [a, b] on I.

2. Let F : D ⊂ R×X → E be a map. We call a differential equation (1.18) the RFDE with history interval I, initial value space X, and history functional F . For real numbers a < b, we say that a map x : [a, b]+I → E is a solution of RFDE (1.18) if

(i) x|[a,b]: [a, b] → E is differentiable,

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(ii) (t, xt) ∈ D holds for all t ∈ [a, b],

(iii) x satisfies (1.18) for t ∈ [a, b]. ˙

x(a) and ˙x(b) denote the right derivative at a and the left derivative at b, respectively.

3. For (t0, φ0) ∈ D and β > 0, we say that x : [t0, t0+ β] + I → E is

a solution of (1.19) if (i) x is a solution of RFDE (1.18), and (ii) xt0 = φ0. We call the problem to find a solution of (1.19) for each

(t0, φ0) ∈ D the initial value problem (IVP) of RFDE (1.18).

Remark 1. In the same way, for a < b ≤ +∞, we define a solution x : [a, b) + I → E of RFDE (1.18) by conditions (i) x|[a,b) is differentiable, (ii) (t, xt) ∈

D holds for all t ∈ [a, b), and (iii) x satisfies (1.18) for t ∈ [a, b). Then x : [a, b) + I → E is a solution of RFDE (1.18) if and only if the restriction x|[a,c]+I: [a, c] + I → E is a solution of (1.18) for any c ∈ (a, b).

Let C([−r, 0], E)u denote the Banach space C([−r, 0], E) of continuous

maps with the supremum norm. The RFDEs with general delay

struc-ture contains the RFDEs with finite delay by choosing I = [−r, 0] and X = C([−r, 0], Rn)u for any r > 0. We refer the reader to Hale &

Ver-duyn Lunel [30] and Diekmann et al. [13] as general references of RFDEs with finite delay. For RFDEs with finite delay, the uniform topology on C([−r, 0], Rn) and the continuity of F are usually assumed. RFDEs with

general delay structure formally contains ODEs by choosing I = {0}, where C({0}, E)u ' E. When I = R−, the RFDEs becomes the RFDEs with

infi-nite delay, where it is usually assumed that X is a seminormed space. We also refer the reader to Hino et al. [35] as a general reference of RFDEs with infinite delay.

2.1

Continuations and continuability

2.1.1 Continuations

Definition II. Let (t0, φ0) ∈ R × Map(I, E) be given.

1. For a left-closed interval J with left end point t0, we say that a map

x : J + I → E is a continuation of (t0, φ0) if (i) xt0 = φ0 and (ii) x is

continuous on J . When t0 = 0, we simply say that x is a continuation

of φ0. Furthermore, if x is continuously differentiable on J , we say

that x is a C1-continuation. 2. For b > 0, let Ct0,φ0(b) = { x ∈ Map([t0, t0+ b] + I, E) : x is a continuation of (t0, φ0) }, Ct100(b) = { x ∈ Ct0,φ0(b) : x is of class C 1 on [t 0, t0+ b] }.

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CHAPTER 2. GENERAL DELAY STRUCTURE 16

Ct0,φ0(b) can be considered as a metric space with the following metric

d(x1, x2) = kx1− x2k∞, x1, x2 ∈ Ct0,φ0(b). Here kx1− x2k ∞= sup t∈[t0,t0+b] kx1(t) − x2(t)k < +∞

holds because x1t0 = x2t0 = φ0. We call Ct0,φ0(b) the continuation space

with length b at (t0, φ0). Furthermore, for c > 0, let

Ct0,φ0(b, c) = x ∈ Ct0,φ0(b) : supt∈[t0,t0+b]kx(t) − φ0(0)k ≤ c ,

Ct100(b, c) = Ct0,φ0(b, c) ∩ C

1 t0,φ0(b).

Ct0,φ0(b, c) is a closed set of Ct0,φ0(b).

3. For φ ∈ Map(I, E), let φ : R++ I → E denote the continuation of φ

by the constant:

φ(s) = (

φ(s), s ∈ I, φ(0), s ∈ R+.

For b > 0, define a normalization operatorN: Ct0,φ0(b) → C0,0(b) as

Nx(s) = x(t0+ s) − φ0(s), s ∈ [0, b] + I. (2.1)

By definition, N is bijective.

2.1.2 Continuability

We introduce the following properties of a topological linear space X which is not necessarily Hausdorff. In this thesis, we say that N is a neighborhood of 0 in X if there exists an open set U of X such that 0 ∈ U ⊂ N .

Definition III (cf. Hale & Kato [28], Kato [36], Schumacher [61]). Assume that X is a topological linear space. We say that X is continuable if the following conditions are satisfied:

(C1) For all ξ ∈ C0,φ(b) for any b > 0 and φ ∈ X,

(i) ξt∈ X holds for all t ∈ [0, b],

(ii) [0, b] 3 t 7→ ξt∈ X is continuous.

(C2) For any b > 0 and for any neighborhood N of 0 in X, there exists γ > 0 such that for all ξ ∈ C0,0(b), kξk∞≤ γ implies ξt∈ N holds for

all t ∈ [0, b].

When (i) of (C1) is satisfied, we say that X is closed under continuations. When (C1) is satisfied, we say that X is topologically closed under continu-ations. We say X is closed under C1-continuations if (i) of (C1) is satisfied for C1-continuations.

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Under the condition (C2) of the definition of continuability, the topology of X is characterized by the topology of the continuation space in the sense that the supremum norm of the continuation ξ of 0 is used.

Proposition 2.1. Let I = [−r, 0]. If X is closed under continuations, then an inclusion C([−r, 0], E) ⊂ X holds.

Proof. Let φ ∈ C([−r, 0], E). For b > r, one can take a continuous map ξ : [−r, b] → E such that ξ0 = 0 and ξb = φ because E is connected. By

assumption, we have φ = ξb ∈ X.

It is not obvious whether C([−r, 0], E) ⊂ X implies that X is closed under continuations.

Lemma 2.1. Let (t0, φ0) ∈ R × Map(I, E) and b > 0 be given. We consider

the normalization operatorN: Ct0,φ0(b) → C0,0(b). Then y =Nx if and only

if ys= xt0+s− (φ0)s holds for all s ∈ [0, b].

Proof. (If-part). By evaluating the value at 0 in both sides, we have y(s) = x(t0+ s) − φ0(s) for all s ∈ [0, b]. This equality also holds for s ∈ I because

y0 = xt0 − φ0= 0.

This means that y =Nx by definition.

(Only-if-part). Fix s ∈ [0, b]. For θ ∈ I, we have

ys(θ) = y(s + θ) = x(t0+ s + θ) − φ0(s + θ)

= xt0+s(θ) − (φ0)s(θ)

because t0+ s ∈ [t0, t0+ b] and s ∈ R+. This shows that ys= xt0+s− (φ0)s

holds for all s ∈ [0, b].

Lemma 2.2. Suppose that X is topologically closed under continuations. Then for each (t0, φ0) ∈ R × X and each b > 0, every continuation x ∈

Ct0,φ0(b) satisfies the following conditions:

(i) (t, xt) ∈ R × X holds for all t ∈ [t0, t0+ b],

(ii) [t0, t0+ b] 3 t 7→ (t, xt) ∈ R × X is continuous.

Proof. Let N: Ct0,φ0(b) → C0,0(b) be the normalization operator. We

con-sider y =Nx. Let t ∈ [t0, t0+ b]. From Lemma 2.1, we have

xt= yt−t0 + (φ0)t−t0.

Since X is closed under continuations, xt∈ X by the linear structure of X.

The continuity is translation-invariant. Therefore, the continuity [t0, t0+b] 3

t 7→ xt∈ X also holds by because [0, b] 3 s 7→ ys∈ X and R+3 s 7→ φs∈ X

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CHAPTER 2. GENERAL DELAY STRUCTURE 18

Proposition 2.2. Let (t0, φ0) ∈ R×Map(I, E). For each b > 0, (Ct0,φ0(b), d)

is a complete metric space.

Proof. C0,0(b) has the linear structure by the point-wise addition and the

point-wise scalar multiplication. Since all elements of C0,0(b) are continuous

and bounded, C0,0(b) becomes a normed linear space with supremum norm.

Let

Yb := { y ∈ C([0, b], E) : y(0) = 0 },

which is a closed subspace of a Banach space C([0, b], E)u. Since C0,0(b) is

isometrically isomorphic to Yb by the restriction, C0,0(b) is a Banach space.

N: Ct0,φ0(b) → C0,0(b) is isometry because

kNx1−Nx2k∞= sup s∈[0,b]+I

kx1(t0+ s) − x2(t0+ s)k

= kx1− x2k∞

holds for all x1, x2 ∈ Ct00(b). Therefore, (Ct0,φ0(b), d) is isometrically

homeomorphic to C0,0(b). This shows that (Ct0,φ0(b), d) is complete.

2.1.3 Examples

We give examples of a continuable space and a non-continuable space. A Banach space C([−r, 0], E)u is an example of a continuable space. This fact

is a basis of the theory of RFDEs with finite delay (e.g., see [30]).

Notation. Let C(R−, E)co denote the topological linear space of all

con-tinuous maps from R− to E endowed with the compact-open topology.

C(R−, E)co is a Fr´echet space. Let Lip(I, E) denote the linear space of

all locally Lipschitz continuous maps from I to E. We notice that when I = [−r, 0], φ ∈ Lip(I, E) implies the (global) Lipschitz continuity of φ. Example 1. X = C(R−, E)co is continuable.

Proof. The conditions (i) of (C1) and (C2) are trivial. We check that X satisfies (ii) of (C1).

Take a continuation ξ : (−∞, b] → E of an element of X for some b > 0. Let N be a neighborhood of 0 ∈ X. For this N , take an integer j ≥ 1 so that

Nj := { φ ∈ X : kφ|[−j,0]k∞< 1/j } ⊂ N.

ξ is uniformly continuous on [−j − 1, b]. Therefore, one can take δ > 0 so that for all s, s0∈ [−j − 1, b], |s − s0| < δ implies kξ(s) − ξ(s0)k < 1/j. This

means that for all t, t0 ∈ [0, b], |t − t0| < δ implies sup θ∈[−j,0] kξt(θ) − ξt0(θ)k = sup θ∈[−j,0] kξ(t + θ) − ξ(t0+ θ)k < 1/j,

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which shows ξt− ξt0 ∈ N .

For a given neighborhood N of 0 ∈ X, one can choose δ > 0 so that for all t, t0 ∈ [0, b], |t − t0| < δ implies ξt− ξt0 ∈ N . Therefore, [0, b] 3 t 7→ ξt∈ X

is uniformly continuous. Thus, (ii) of (C1) holds.

In the above, we use the fact that {Nj}j≥1 is a neighborhood base of 0.

Example 2. The topological linear space X = Lip(I, E) endowed with any topology is not continuable.

Proof. A continuation ξ : [0, b] + I → E of an element of X is not necessarily locally Lipschitz on [0, b], therefore, X is not closed under continuations.

2.2

Trivial RFDE

We consider the IVP

( ˙

x(t) = 0, t ≥ 0,

x0 = φ, φ ∈ X

(2.2) of the trivial RFDE ˙x = 0 with initial value space X.

Proposition 2.3. If X is closed under continuations, then IVP (2.2) has a unique solution φ : R++ I → E for each φ ∈ X.

Proof. By definition, (φ)0(t) = 0 for t ≥ 0, and φ0 = φ. Since φ is constant on R+, φ is a continuation of φ. Therefore, φt ∈ X holds for all t ≥ 0 by

assumption. This shows that φ is a solution of (2.2). The uniqueness is obvious.

Notation. When IVP (2.2) has a unique solution for each φ ∈ X, we denote by (S0(t))t≥0 the solution semi-group on X determined by this IVP. S0(t)

is expressed by

S0(t) : X → X, S0(t)φ = φt.

If a topological linear space X is topologically closed under continuations,

then for each φ ∈ X, IVP (2.2) has a unique solution, and R+ 3 t 7→

S0(t)φ ∈ X is continuous. We say that (S0(t))t≥0 is continuous if the

semi-flow R+× X 3 (t, φ) 7→ S0(t)φ ∈ X is continuous. For each (t0, φ0) ∈ R × X,

we define a map τt0,φ0: R+× X → [t0, +∞) × X as

τt0,φ0(t, φ) = (t0+ t, S0(t)φ0+ φ).

Lemma 2.3. Let (t0, φ0) ∈ R × X be fixed. If R+ 3 t 7→ S0(t)φ0 ∈ X is

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CHAPTER 2. GENERAL DELAY STRUCTURE 20

Proof. For (s, ψ) ∈ [t0, +∞) × X, (s, ψ) = τt0,φ0(t, φ) is equivalent to

t = −t0+ s,

φ = −S0(t)φ0+ ψ = −S0(−t0+ s)φ0+ ψ.

Therefore, the inverse τt−1

0,φ0: [t0, +∞) × X → R+× X is

τt−1

0,φ0(s, ψ) = (−t0+ s, −S0(−t0+ s)φ0+ ψ).

The assumption means that τt0,φ0 and τ

−1

t0,φ0 are continuous. Therefore, τt0,φ0

is a homeomorphism.

2.3

Comments on preceding results about

continu-ability

2.3.1 Axiom by Hale & Kato

For a seminormed space B = (B, | · |B) where B ⊂ Map(R−, Rn), Hale &

Kato [28] formalized conditions for B under which they showed the well-posedness of the IVP of an RFDE

˙

x(t) = F (t, [xt]) (2.3)

for a map F : dom(F ) ⊂ R × (B/| · |B) → Rn. Here B/| · |B denote the

quotient space with the equivalence relation induced by the seminorm, and [φ] denotes the equivalence class containing φ ∈ B. However, the IVP of RFDE (2.3) is not the same problem as IVP (1.19). Based on the conditions introduced in [28], Kato [36] reformulated the axiom for a seminormed space B, and consider the IVP (1.19) for an RFDE (1.18) for the case E = Rn. On his axioms, Kato did not give the proof of the theorems about well-posedness which are associated to those given in [28]. The proof of the existence and uniqueness part can be obtained by Theorems 1.1 and 1.2 of Hino et al. [35, Chapter 2]. The continuous dependence is discussed in Theorem 2.1 and the succeeding remark of [35, Chapter 2]. However, it is not apparent that this is the same as the continuity of the solution process. See Section 3.2 for the definition of the solution process.

Kato [36] considered the following conditions (i) and (ii) for this B: (i) For each σ ∈ R and A > σ, every map x : (−∞, A) → Rn with

prop-erties that x is continuous on [σ, A) and xσ ∈ B satisfies

• xt∈ B for t ∈ [σ, A),

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(ii) There exist M0 > 0 and continuous maps K, M : R+ → R+ with

the following property: For each σ ∈ R and A > σ, every map x : (−∞, A) → Rn such that x is continuous on [σ, A) and xσ ∈ B

satisfies

M0kx(t)k ≤ |xt|B,

|xt|B ≤ K(t − σ) · sup s∈[σ,t]

kx(s)k + M (t − σ)|xσ|B

for all t ∈ [σ, A).

Here k · k is a norm on Rn. We call this set of conditions the Hale–Kato axiom for the phase space B. From Lemma 2.2, B satisfies the above (i) if and only if B is topologically closed under continuations. For the above (ii), it is sufficient to consider continuations of each φ ∈ B by translation.

We investigate the relationship between the condition (C2) and (ii) of the Hale–Kato axiom. The first inequality in (ii) of the Hale–Kato axiom only means that the projection B 3 φ 7→ φ(0) ∈ Rn is continuous.

Definition 2.4. For a seminorm p on X, we consider the following condi-tion: There are continuous maps K, M : R+→ R+ such that for each b > 0

and φ ∈ X,

p(ξt) ≤ K(t) · sup s∈[0,t]

kξ(s)k + M (t)p(φ) (2.4)

holds for all ξ ∈ C0,φ(b) and for all t ∈ [0, b]. We say that p has the

contin-uation inequality if the above condition on p is satisfied.

Proposition 2.4. Suppose that X is a seminormed linear space with a seminorm p on X. If p has the continuation inequality, then X satisfies (C2).

Proof. Fix b > 0 and a neighborhood N = { φ ∈ X : p(φ) < δ } of 0 ∈ X for some δ > 0. Choose γ > 0 as

γ < δ ·supt∈[0,b]K(t)−1.

When supt∈[0,b]K(t) = 0, the right-hand side is interpreted as +∞. For all ξ ∈ C0,0(b, γ) and for all t ∈ [0, b],

p(ξt) ≤supt∈[0,b]K(t) · γ + M (t) · 0 < δ,

which is equivalent to ξt∈ N . Therefore, (C2) holds.

2.3.2 Hypotheses by Schumacher

Let I = R−. Suppose that X is a topological linear space having an

equiv-alence relation ∼ on X such that the quotient space X/∼ is Hausdorff. Schumacher [61] laid the following hypotheses A1–A6 for such an X:

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CHAPTER 2. GENERAL DELAY STRUCTURE 22

A1 For all φ ∈ X and for all s ∈ R+, φs∈ X.

A2 For all φ, ψ ∈ X, φ ∼ ψ implies φs ∼ ψs for all s ∈ R+. A3 For all φ, ψ ∈ X, φ ∼ ψ implies φ(0) = ψ(0).

A4 For all φ ∈ X, R+3 s 7→ φs∈ X is continuous.

A5 For τ > 0, let Xτ be the set of all continuous maps φ : R−→ E with

supp(φ) ⊂ [−τ, 0]. Then for all τ > 0, Xτ ⊂ X holds.

A6 For all τ > 0, the inclusion Xτ ⊂ X is continuous with respect to the

maximum norm on Xτ.

A1 is equivalent to the existence of a solution of IVP (2.2) (see Propo-sition 2.3). This is also mentioned in [61]. We give an example of X which satisfies A3 but does not satisfy A2.

Example 3. Let X be the seminormed space C(R−, E) with a seminorm p

given by

p(φ) = kφ(−r)k + kφ(0)k

for some r > 0. We consider the equivalence relation ∼ induced by this norm: φ ∼ ψ if and only if p(φ − ψ) = 0. Let [φ] denote the equivalence class containing φ. Then X/∼ = { [φ] : φ ∈ X } becomes a normed linear space with the linear operations [φ] + [ψ] = [φ + ψ], α · [φ] = [αφ] and with the norm k[φ]k = p(φ). Therefore, X/∼ is Hausdorff.

A3: φ ∼ ψ if and only if φ(−r) = ψ(−r) and φ(0) = ψ(0). This means that A3 holds.

A2: Take φ ∈ X such that φ(θ) = 0 for θ 6∈ (−r, 0), and φ(θ) 6= 0 for θ ∈ (−r, 0). Then φ ∼ 0. However, φs 6∼ 0 for 0 < s < r because p(φs) = kφ(s − r)k > 0.

We study the relationship between these hypotheses A1, A4–A6 and the continuability. Schumacher stated, without proof, (b) of Proposition 2.5 given below.

Lemma 2.5. Let b > 0 and ξ ∈ C0,0(b). Then for each s ∈ [0, b], kξt− ξsk∞

converges to 0 as t → s in [0, b]. Furthermore, the map [0, b] 3 t 7→ kξtk∞ is

continuous. Proof. Since

kξt− ξsk∞= sup θ∈I

kξ(t + θ) − ξ(s + θ)k,

the first part of the statement follows if ξ is uniformly continuous.

We show the uniform continuity of ξ. If I = [−r, 0] for some r > 0, the uniform continuity holds because [0, b] + I = [−r, b]. Assume I = R−. Let

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ε > 0 be given. Since ξ is uniformly continuous on [0, b], there is δ > 0 such that for all t, t0 ∈ [0, b],

|t − t0| < δ =⇒ kξ(t) − ξ(t0)k < ε. (2.5)

For any t, t0 ∈ R−, we have kξ(t) − ξ(t0)k = 0 since ξ0 = 0. For all t ∈ [0, b]

and t0 ∈ R− satisfying |t − t0| < δ, we have |t − 0| = |t| ≤ |t − t0| < δ, and

kξ(t) − ξ(t0)k = kξ(t) − ξ(0)k < ε

holds from (2.5). Therefore, (2.5) holds for all t, t0∈ (−∞, b], which implies that ξ : (−∞, b] → E is uniformly continuous. Thus, the proof of the first part is obtained.

The second part follows from

tk∞− kξsk∞

≤ kξt− ξsk∞

because the right-hand side converges to 0 as t → s.

Proposition 2.5. Suppose that X is a topological linear space having an equivalence relation ∼ on X such that the quotient space X/∼ is Hausdorff. Then the following statements hold:

(a) A1 and A5 hold if and only if X is closed under continuations. (b) Under the assumptions that X is closed under continuations and X

satisfies (C2), A4 is equivalent to (ii) of (C1). (c) A6 is equivalent to (C2).

As a result, A1, A4–A6 hold if and only if X is continuable.

Proof. (a). Suppose that X is closed under continuations. Then A1 is trivial. We check A5. Take φ ∈ Xτ and consider the map ξ : (−∞, τ ]+I → E defined

as

ξ(s) = φ(s − τ ).

Then ξ is a continuation of 0. Therefore, φ = ξτ ∈ X holds. Suppose that

A1 and A5 hold. Let ξ ∈ C0,φ(b) for b > 0 and φ ∈ X. Then for η = Nξ

whereN: C0,φ(b) → C0,0(b) is a normalization operator,

ηs= ξs− φs

holds for s ∈ [0, b]. Since ηs ∈ Xs, we have ξs= ηs+ φs∈ X. Therefore, X

is closed under continuations.

(b). The implication (ii) of (C1) ⇒ A4 is trivial. Suppose that A4 holds. Let ξ ∈ C0,φ(b) for b > 0 and φ ∈ X. In the same way as the proof of (a),

we have the decomposition

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CHAPTER 2. GENERAL DELAY STRUCTURE 24

From A4, [0, b] 3 s 7→ φs is continuous. The continuity of [0, b] 3 s 7→ ηs

follows from Lemma 2.5 because X satisfies (C2). Here (C2) is needed

because Lemma 2.5 only says that kηs− ηs0k∞ converges to 0 as s → s0.

(c). Xτ introduced in A5 is a Banach space with supremum norm. Since

the inclusion Xτ 3 φ 7→ φ ∈ X is a linear operator, the inclusion is

continu-ous if and only if the inclusion is continucontinu-ous at 0. This shows the equivalence A6 ⇔ (C2).

Let [φ] denote the equivalence class containing φ ∈ X. On the

as-sumptions on X and on the hypotheses A1–A6, Schumacher considered an RFDE (2.3) for a map F : dom(F ) ⊂ R × (X/∼) → E.

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Uniqueness and maximal

solutions

Let F : D ⊂ R × X → E be a map, and we consider RFDE (1.18). In this chapter, we investigate the relationship between uniqueness and maximality of a solution of the corresponding IVP (1.19) on the assumption of existence of a solution. Here we do not assume the continuity of F and the openness of D. See Chapter 5 for the existence of a solution.

Definition 3.1. For (t0, φ0) ∈ D, we say that IVP (1.19) has a locally

unique solution if the following conditions are satisfied: (i) (1.19) has a solution,

(ii) If x,x : [te 0, t0+ β] + I → E are solutions of (1.19), there exists β0< β such that x(t) ≡ex(t) holds for t ∈ [t0, t0+ β0].

3.1

Maximality

Definition 3.2. Let (t0, φ0) ∈ D be given. For a left-closed interval J with

left end point t0, a solution x : J + I → E of IVP (1.19) is said to be maximal

if for all solutions x : ee J + I → E of (1.19) where eJ is a left-closed interval with left end point t0, eJ ⊃ J and x =ex on J implies eJ = J .

Domain of maximal solutions

Proposition 3.1. Suppose that IVP (1.19) has a locally unique solution for each (t0, φ0) ∈ D. Then for each (t0, φ0) ∈ D, a solution x : [t0, t0+ β] + I →

E of (1.19) is not maximal for any β > 0.

Proof. Let (t1, φ1) = (t0+ β, xt0+β) ∈ D. We consider

( ˙

y(t) = F (t, yt), t ≥ t1,

yt1 = φ1.

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CHAPTER 3. UNIQUENESS AND MAXIMAL SOLUTIONS 26

Since the IVP of RFDE (1.18) has a locally unique solution, the above system has a solution y : [t1, t1+ β0] + I → E for some β0 > 0. Define a map

e x : [t0, t0+ β + β0] + I → E as xet0 = φ0 and e x(t) = ( x(t) t ∈ [t0, t1], y(t) t ∈ [t1, t1+ β0]. Then for t ∈ [t1, t1+ β0], ˙ e x(t) = ˙y(t) = F (t, yt) = F (t,ext) because yt1 = xt1 = φ1.

Unique existence of a maximal solution

Lemma 3.3. Suppose that IVP (1.19) has a locally unique solution for each (t0, φ0) ∈ D. For each (t0, φ0) ∈ D, if x,ex : [t0, t0+ β] + I → E are solutions of (1.19), then x =ex for any β > 0.

Proof. Let (t0, φ0) ∈ D be fixed. Suppose that x,x : [te 0, t0+ β] + I → E are solutions of (1.19). We consider the supremum of the set

J := { t ∈ [t0, t0+ β] : x =x on [te 0, t] }.

Since (1.19) has a locally unique solution, x = x holds on [te 0, t0 + δ] for some δ > 0. Therefore, J 6= ∅, and T := sup J exists. By definition, t0< T ≤ t0+ β. By the minimality of the supremum, for each ε > 0, there

is t ∈ J such that t > T − ε. This implies that x = ex on [t0, T − ε] for all ε > 0. Therefore, x =x on [te 0, T ] by the continuity of x,ex on [t0, t0+ β].

When T = t0+β, we have the conclusion. Suppose T < t0+β and derive

the contradiction. Let φ := xT =xeT. Then (T, φ) ∈ D, and x,ex : [T, t0+ β] + I → E are solutions of the system

( ˙

x(t) = F (t, xt), t ≥ T,

xT = φ.

By the local uniqueness of a solution of the IVP of RFDE (1.18), x =ex on some interval [T, T + δ0]. This contradicts to the choice of T . Therefore, T = t0+ β.

Theorem 3.4. IVP (1.19) has a locally unique solution for each (t0, φ0) ∈ D

if and only if IVP (1.19) has a unique maximal solution for any (t0, φ0) ∈ D.

Proof. If-part is clear. We show only-if-part. Let (t0, φ0) ∈ D be fixed.

(Existence). We consider

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Definex : J + I → E ase J = [ x∈S [t0, t0+ βx], e x(t) = x(t) if t ∈ [t0, t0+ βx].

This is well-defined from Lemma 3.3. In the same way as the proof of Propo-sition 3.1, one can show thatx is a solution of (1.19). By the construction,e e

x is a maximal solution of (1.19).

(Uniqueness). From Proposition 3.1, the domain of maximal solutions are right-open. Letx : [tb 0, t0+ b) + I → E be an another maximal solution of (1.19). For any b0< b, the restriction bx : [t0, t0+ b0] + I → E is a solution of (1.19). By the construction of ex, we have

J ⊃ [t0, t0+ b) =

[

b0<b

[t0, t0+ b0],

and x =e bx on [t0, t0 + b0] for any b0 < b. By the maximality of bx, J = [t0, t0+ b) andx =e x.b

3.2

Solution process

From Theorem 3.4, the local unique existence of a solution of IVP (1.19) is a necessary and sufficient condition for the unique existence of a maximal solution of this IVP. Under this condition, we obtain the solution process as introduced below.

Definition IV. Suppose that the IVP of RFDE (1.18) has a locally unique solution. Let

x(·, t0, φ0) : [t0, t0+ b(t0, φ0)) + I → E, b(t0, φ0) > 0

denote the maximal solution of (1.19) for each (t0, φ0) ∈ D. We define a

map UF: dom(UF) ⊂ R+× D → X, where R+ := [0, +∞), as

dom(UF) =

[

(t0,φ0)∈D

[0, b(t0, φ0)) × {(t0, φ0)},

UF(τ, t0, φ0) = x(·, t0, φ0)t0+τ.

We call UF the solution process generated by RFDE (1.18).

The term “process” comes from the following definition.

Definition 3.5. LetX be a topological space. A continuous map U : R+×

R × X → X is said to be a process if U has the following properties: (i) U (0, s, x) = x holds for all (s, x) ∈ R × X,

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CHAPTER 3. UNIQUENESS AND MAXIMAL SOLUTIONS 28

(ii) For all τ1, τ2 ∈ R+ and for all (s, x) ∈ R × X,

U (τ1+ τ2, s, x) = U (τ2, s + τ1, U (τ1, s, x))

holds.

The concept of processes was originally introduced by Dafermos [12] and was developed by Hale [26], which is a mathematical formulation of non-autonomous dynamical systems. We refer the reader to Carvalho et al. [6] and Kloeden & Rasmussen [41] for general references of non-autonomous dynamical systems.

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Local semi-flows and

local semi-dynamical systems

Let G = R or Z. A topological dynamical system is a triple (X, G, ϕ) where • X is a topological space,

• ϕ : G × X → X is a continuous map with the properties that (i) ϕ0 is the identity map,

(ii) ϕt+s = ϕt◦ ϕs,

where ϕt(x) := ϕ(t, x) for (t, x) ∈ G × X.

X is called the phase space (or the state space), G is called the time set, and ϕ is called the flow. When G = R, (X, G, ϕ) is called a dynamical system with continuous time, and when G = Z, (X, G, ϕ) is called a dynamical system with discrete time.

Sometimes a dynamical system with continuous time is specified by a flow with the time set R, and a dynamical system with discrete time is specified by a map because a flow ϕ : Z × X → X is generated by the map f :X → X defined by

f (x) := ϕ1(x) = ϕ(1, x). Then we have

ϕn(x) = ϕ(n, x) = fn(x),

where fn= f ◦ · · · ◦ f stands for the n-th composition of f .

The concept of dynamical systems originally comes from autonomous

ODEs on the Euclidean space Rn, of course. For an ODE with smooth

vector field f on Rn

˙

x = f (x)

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CHAPTER 4. LOCAL SEMI-FLOWS 30

whose IVP with initial condition x(0) = x0 generates an entire solution

x(·, x0) : R → Rn, the dynamical system (Rn, R, ϕ) is obtained by the flow

ϕ : R × Rn→ Rn defined by

ϕt(x0) := ϕ(t, x0) := x(t, x0).

However, solutions of ODEs often blow up in finite time.

Example 4. An example of an ODE which has a blow up solution in finite time is

˙

x = x2, x ∈ R.

The unique solution x(·, x0) that satisfies an initial condition x(0) = x0 is

given by

x(t, x0) =

x0

1 − tx0

. The maximal domain of definition is

     (−∞, 1/x0), x0 > 0, R, x0 = 0, (1/x0, +∞), x0 < 0, and for x0 6= 0, x(t, x0) → +∞ as t → 1/x0.

Therefore, the notions of local (semi-) flows and local (semi-) dynami-cal systems with continuous time are needed, which will be treated in the following sections.

Let R+ denote the interval [0, +∞).

4.1

Local semi-flows on topological spaces

4.1.1 Local semi-flows and local processes

Definition 4.1. LetX be a set. We call a map Φ: dom(Φ) ⊂ R+×X → X

a local semi-flow if the following conditions are satisfied: (i) For each x ∈X, T (x) ∈ (0, +∞] exists, and

dom(Φ) = [

x∈X

[0, T (x)) × {x}

holds,

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(iii) For each x ∈ X, the following statement holds: For all t, s ∈ R+,

t ∈ [0, T (x)) and s ∈0, T (Φ(t, x)) imply t + s ∈ [0, T (x)), Φ(t + s, x) = Φ(s, Φ(t, x)). For a local semi-flow Φ, we write

Φt(x) = Φ(t, x) for (t, x) ∈ dom(Φ).

Remark 2. The terminology of local semi-flow is related to the notion “local dynamical systems” introduced by Sell [62]. When T (x) = +∞ for every x ∈X, Φ is a semi-flow in the usual sense. In this case, dom(Φ) = R+×X.

Definition 4.2. LetX be a set. We call a map U : dom(U) ⊂ R+×R×X →

X a local process if the following conditions (i) and (ii) are satisfied:

(i) There exists a subset D ⊂ R×X such that for each (t, x) ∈ D, T (t, x) ∈ (0, +∞] exists, and

dom(U ) = [

(t,x)∈D

[0, T (t, x)) × {(t, x)}

holds,

(ii) The map Φ : dom(Φ) ⊂ R+× D → D defined as

dom(Φ) = dom(U ),

Φ(τ, (t, x)) = (t + τ, U (τ, t, x)) is a local semi-flow.

We call the corresponding local semi-flow Φ introduced in (ii) the extended local semi-flow of U . For a local process U , we write

Uτ(t, x) = U (τ, t, x)

for (τ, t, x) ∈ dom(U ). We call τ and t the elapsed time and the initial time for Uτ(t, x), respectively.

The following lemma gives a characterization of local processes.

Lemma 4.3. Let X be a set. We consider a map U : dom(U) ⊂ R+× R ×

X → X with the domain of definition

dom(U ) = [

(t,x)∈D

[0, T (t, x)) × {(t, x)},

where D ⊂ R × X and T (t, x) ∈ (0, +∞] for each (t, x) ∈ D. Then U is a local process if and only if U satisfies the following conditions:

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CHAPTER 4. LOCAL SEMI-FLOWS 32

(i) For all (t, x) ∈ D, U (0, t, x) = x,

(ii) For each (t, x) ∈ D, the following statement holds: For all τ, σ ∈ R+,

τ ∈ [0, T (t, x)) and σ ∈0, T (t + τ, U (τ, t, x)) imply τ + σ ∈ [0, T (t, x)),

U (τ + σ, t, x) = U σ, (t + τ, U (τ, t, x)).

Proof. Let Φ : dom(Φ) ⊂ R+× D → D be the extended local semi-flow of

U .

(i). Let (t, x) ∈ D. Then

Φ(0, (t, x)) = (t, x) ⇐⇒ U (0, t, x) = x.

(ii). Let (t, x) ∈ D, τ ∈ [0, T (t, x)) and σ ∈ 0, T (t + τ, U (τ, t, x)). By definition,

Φ σ, Φ(τ, (t, x)) = Φ σ, (t + τ, U (τ, t, x))

= (t + τ ) + σ, U σ, (t + τ, U (τ, t, x)). Therefore, Φ(τ + σ, (t, x)) = Φ(σ, Φ(τ, (t, x))) if and only if

U (τ + σ, t, x) = U σ, (t + τ, U (τ, t, x)). This completes the proof.

This lemma shows that the notion of local processes is a non-autonomous version of that of local semi-flows, because

(τ, x) 7→ U (τ, t, x)

is a local semi-flow for a local process U if U (τ, t, x) does not depend on the initial time t.

Remark 3. When T (t, x) = +∞ for every (t, x) ∈ D and D = R × X, U is a process introduced by Hale et al. [29]. Originally, the notion of processes was introduced by Dafermos [12].

4.1.2 Continuity of local semi-flows and local processes

Definition 4.4. LetX be a topological space, and let Φ: dom(Φ) ⊂ R+×

X → X be a semi-flow with the domain of definition

dom(Φ) = [

x∈X

[0, T (x)) × {x},

where T (x) ∈ (0, +∞]. We say that Φ is a continuous local semi-flow if conditions

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(i) Φ is a continuous map,

(ii) X 3 x 7→ T (x) ∈ (0, +∞] is lower semi-continuous

are satisfied. We call a local process U : dom(U ) ⊂ R+× R × X → X a

continuous local process if the extended local semi-flow of U is a continuous local semi-flow.

Definition 4.5. LetX be a topological space, and let Y be a uniform space. We consider a family (fλ)λ∈Λof maps fromX to Y, where Λ is a set. (fλ)λ∈Λ

is said to be equi-continuous at x0 ∈ X if for any entourage V on Y, there

exists a neighborhood N of x0 inX such that for all x ∈ N and λ ∈ Λ,

(fλ(x), fλ(x0)) ∈ V

holds. We further assume that Λ is a topological space. We say that (fλ)λ∈Λ

is locally equi-continuous at x0 if for each λ0 ∈ Λ, there is a neighborhood

W of λ0 such that (fλ)λ∈W is equi-continuous at x0.

Theorem 4.6. Let X be a uniform space. We consider a local semi-flow

Φ : dom(Φ) ⊂ R+×X → X, where

dom(Φ) = [

x∈X

[0, T (x)) × {x}, and the map

X 3 x 7→ T (x) ∈ (0, +∞]

is lower semi-continuous. Then for every x ∈ X and for every interval

I ⊂ [0, T (x)), there exists a neighborhood W of x such that the following statements are equivalent:

(a) Φ|I×W is continuous,

(b) For each x0 ∈ W , I 3 t 7→ Φt(x0) ∈X is continuous, and (Φt|W)t∈I is

locally equi-continuous at x0.

Proof. Take T > 0 so that I ⊂ [0, T ]. We notice that the lower semi-continuity ofX 3 x 7→ T (x) ∈ (0, +∞] ensures that there is a neighborhood W of x0 such that

I × W ⊂ [0, T ] × W ⊂ dom(Φ).

(b) ⇒ (a). Fix (t0, x0) ∈ I × W . Let U0 be an entourage on X, and take

a symmetric entourage U onX so that U ◦ U ⊂ U0.

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CHAPTER 4. LOCAL SEMI-FLOWS 34

Since the map I 3 t 7→ Φt(x0) ∈ X is continuous at t0, there is δ > 0 such

that for all t ∈ I,

|t − t0| < δ =⇒ Φt(x0) ∈ U [Φt0(x0)].

By the local equi-continuity of (Φt|W)t∈I at x0, there are δ0 > 0 and a

neighborhood N ⊂ W of x0 such that for all (t, x) ∈ I × W ,

|t − t0| < δ0, x ∈ N =⇒ (Φt(x), Φt(x0)) ∈ U.

Let δ0 := min{δ, δ0}. The above argument shows that for all (t, x) ∈

I × W ,

|t − t0| < δ0, x ∈ N =⇒ Φ(t, x) ∈ U0[Φ(t0, x0)]

holds. This implies that Φ|I×W is continuous at (t0, x0).

(a) ⇒ (b). The continuity of the map I 3 t 7→ Φt(x0) ∈X

follows by assumption. We show that (Φt|W)t∈[a,b] is equi-continuous at x0

for every compact interval [a, b] ⊂ I by a contradiction. Suppose that (Φt|

W)t∈[a,b]is not equi-continuous at x0. Then there is an

entourage U0 on X such that for every neighborhood N ⊂ W of x0, there is

(tN, xN) ∈ [a, b] × N such that

(ΦtN(x

N), ΦtN(x0)) 6∈ U0.

LetN denote the directed set of all neighborhoods of x0 with a preorder

≤ defined as

N1 ≤ N2 ⇐⇒ N1⊃ N2.

Then the net (xN)N ∈N converges to x0 by the choice of xN. By the

com-pactness of [a, b], there is a convergent subnet th(j)



j∈J, where h : J →N is

a monotone final map. Let t∗∈ [a, b] be the limit of this convergent subnet:

t∗ := lim j th(j).

We notice that the limit t∗ is unique since [a, b] is Hausdorff. Then th(i)

converges to t∗, and xh(i) converges to x0.

We choose a symmetric entourage U onX as U ◦ U ⊂ U0. By the above argument, there is j0∈ J such that for all j ≥ j0

Φth(j)(x

h(j)) ∈ U [Φt∗(x0)], Φth(j)(x0) ∈ U [Φt∗(x0)]

by the continuity of Φ. This contradicts to Φth(j)(x

h(j)), Φth(j)(x0) 6∈ U0.

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