# Unbounded Perturbation of Linear Partial Functional Differential Equations
via Yosida Distance

Xuan-Quang Bui  Faculty of Fundamental Sciences, PHENIKAA University, Hanoi
12116, Vietnam<EMAIL_ADDRESS>and  Nguyen Van Minh
Department of Mathematics and Statistics, University of Arkansas at Little
Rock, 2801 S University Ave, Little Rock, AR 72204, USA<EMAIL_ADDRESS>
###### Abstract.

This paper is concerned with perturbation of evolution equation with delay
$u^{\prime}(t)=Au(t)+Bu_{t}\ (*)$ where $A,B$ are linear operators in a Banach
space $\mathbb{X}$ and from $C([-r,0],\mathbb{X})$ to $\mathbb{X}$,
respectively; $r>0$ is a given constant. To measure the size of unbounded
perturbation we introduce the so-called Yosida distance of linear operators
$U$ and $V$ in a Banach space $\mathbb{X}$ as
$d_{Y}(U,V):=\limsup_{\mu\to+\infty}\|U_{\mu}-V_{\mu}\|$, where $U_{\mu}$ and
$V_{\mu}$ are the Yosida approximations of $U$ and $V$, respectively. The
obtained results states that if $d_{Y}(A,A_{1})$ and $d_{Y}(B,B_{1})$ are
sufficiently small, then the equation $u^{\prime}(t)=A_{1}u(t)+B_{1}u_{t}\
(**)$ also has an exponential dichotomy, if (*) has it. The proofs are based
on estimates of the Yosida distance of the generators of the solution
semigroups generated by (*) and (**) in the phase space $C([-r,0],\mathbb{X})$
whose domains have no relations at all. The obtained results seem to be new.

###### Key words and phrases:

Yosida distance, exponential dichotomy, roughness, perturbation, partial
functional differential equation

###### 2020 Mathematics Subject Classification:

34G10, 37D10, 34D20, 34C45, 34D09

The work of the first author is partly supported by the Vietnam Institute for
Advanced Study in Mathematics (VIASM)

##  1\. Introduction

In this paper we are interested in the unbounded perturbation of delayed
evolution equations, that serve as abstract models of partial functional
differential equations (PFDE), of the form

$u^{\prime}(t)=Au(t)+Bu_{t},\quad t\geq 0,$ (1.1)

where $A$ generates a $C_{0}$-semigroup in a Banach space $\mathbb{X}$ and
$B\in\mathcal{L}(\mathcal{C},\mathbb{X})$,
$\mathcal{C}:=C([-r,0],\mathbb{X})$, $r>0$, is a given positive constant; if
$u(\cdot)$ is a function from $[-r,\varepsilon)$ to $\mathbb{X}$ with
$\varepsilon>0$, then for $t\in[0,\varepsilon)$, we denote by
$u_{t}\in\mathcal{C}$ the function $u_{t}(\theta):=u(t+\theta)$,
$\theta\in[-r,0]$. As is well known, for reasonable conditions on $A$ and $B$
(see Travis-Webb [25]) Eq. (1.1) generates a $C_{0}$-semigroup in
$\mathcal{C}$ that we will call ”solution semigroup”. We refer the reader to
the monograph by Hale [16] and Wu [28] for a complete account of the theory
and applications of functional differential equations to real world problems
(see also [5, 11, 12]). The interest in the functional differential equations
has been growing recently as they are better models for real world processes
that are determined by delay factors.

It is well known in the theory of dynamical systems that for a linear systems
of the form

$x^{\prime}(t)=Ax(t),\quad x(t)\in\mathbb{R}^{n}$

under ”small perturbation” some asymptotic behavior like exponential dichotomy
persists (see e.g. [8, 10, 15], see also [18, 19, 21]), that is, the perturbed
system

$x^{\prime}(t)=(A+B)x(t)$

also has an exponential dichotomy if $B$ is small. In the finite-dimensional
case the small perturbation’s size can be determined by the norm of $\|B\|$.
Many results from the dynamical systems in the finite-dimensional case could
be extended to the infinite-dimensional case. In this paper, assuming that the
evolution equation

$u^{\prime}(t)=Au(t),\quad u(t)\in\mathbb{X},$ (1.2)

where $\mathbb{X}$ is a Banach space, generates a $C_{0}$-semigroup in
$\mathbb{X}$ that has an exponential dichotomy, we consider the perturbed
equation

$u^{\prime}(t)=(A+B)u(t),$

where $B$ is (generally unbounded) linear operator in $\mathbb{X}$.

One of the most complicated steps in studying this equation is related to
understanding how to measure the size of the perturbation $B$. It is naturally
easy to extend the persistence of exponential dichotomy under small
perturbation $B$ in the sense of its norm, that is, $\|B\|$ is sufficiently
small. A more general class of unbounded perturbation $B$ was discussed in
Chow-Leiva [7], Prüss [24] (see also Dunford-Schwartz [13, p. 630-631], Da
Prato-Lunardi [9], Yang-Gimeno-de la Llave [29], Zhou-Lu-Zhang [31]).

As for delayed equation Eq. (1.1), under reasonable conditions on $A$ and $B$,
it generates a $C_{0}$-semigroup (called solution semigroup) in the phase
space $\mathcal{C}$, so its asymptotic behavior is studied via this semigroup.
It is expected that many results of the non-delayed equation (1.2) could be
easily extended to the delayed equations (1.1). That means, if the unperturbed
equation (1.1) has an exponential dichotomy, then under ”small perturbation”
$A_{1}$ and $B_{1}$, the perturbed equation

$u^{\prime}(t)=(A+A_{1})u(t)+(B+B_{1})u_{t}$ (1.3)

also has an exponential dichotomy. Even when $A_{1}=0$ and $B_{1}$ is bounded
and $\|B_{1}\|$ is small, the study of the persistence of exponential
dichotomy, to the best of our knowledge, has yet to be complete. In fact, the
problem is that the Variation-of-Constants Formula (VCF) in the phase space
$\mathcal{C}$ is not available for the solution semigroups by Eq.(1.1) and
Eq.(1.3). It turns out that in the infinite-dimensional case, the familiar
form of VCF in the phase space $\mathcal{C}$ is not valid as shown in Hino-
Murakami-Naito-Minh [14]. Instead, a weaker version of VCF was proposed. With
a weaker VCF, as shown in [2, 3, 4, 22], some bounded (or nonlinear Lipchitz)
perturbation $\|B_{1}\|$ of the operator $B$ could be studied.

To our knowledge, the persistence of exponential dichotomy of Eq. (1.3) under
unbounded perturbations $A_{1}$ and $B_{1}$ is not been studied yet. The idea
of applying the method by Chow-Leiva [7] or Dunford-Schwartz [13, pp. 630-631]
to study the persistence of exponential dichotomy of the solution semigroup
generated by Eq.(1.1) in the phase space $\mathcal{C}$ is impossible. In fact,
the very first condition on the domain for the generator of solution
semigroups is not satisfied for delayed equations as shown in our Corollary
4.2 and Example 4.3.

In our paper we will take an attempt to study the persistence of exponential
dichotomy of the solution semigroup generated by Eq. (1.1) where both $A$
and/or $B$ are perturbed by unbounded operators $A_{1}$ and $B_{1}$ from some
classes. We will approach this problem via the concept of Yosida distance as
defined below. The idea of using the Yosida distance arises from the Theory of
Semigroups of Bounded Operators, where Yosida approximation $A_{\lambda}$ is
used to construct the semigroup generated by a linear equation
$u^{\prime}(t)=Au(t)$. The advantage of using Yosida distance between two
unbounded operators is that we can free the domain conditions. Our Lemma 3.3
below shows that this concept is a natural extension of the concept of
distance between bounded operators to some common classes of unbounded
operators, so it is suitable to measure the size of perturbation even in the
unbounded case.

Our paper is organized as follows: after Section 2 where we list some
notations to be used the paper as well as some know results, we present the
main results of the papers in Sections 3, 4 and 5. In the last section
(Section 6) we give two examples illustrating how the paper’s results could be
applied to PFDE under unbounded perturbation. Our key concept of Yosida
distance is given in Definition 3.1 and Theorem 3.2 on persistence under small
perturbation in the sense of Yosida distance. In Lemma 3.3 we visit several
frequently met classes linear operators and their perturbation. The perturbed
equations Eq. (1.3) with unbounded $A_{1}$ and bounded $B_{1}$ is studied in
Section 4, while the case of both unbounded $A_{1}$ and $B_{1}$ is studied in
Section 5. Example 4.8 shows that the domain condition on the perturbation
operators is not satisfied in the case of the solution semigroups generated by
PFDE. This is a justification for a different approach, the Yosida distance
approach, to the perturbation theory of PFDE is needed. The estimates of
Yosida distance of the generators of solution semigroups generated by the
unperturbed and perturbed equations are given in Lemma 4.5 and Lemma 5.4 that
are the key for us to apply Theorem 3.2. Our main results are stated in
Theorems 4.7 and 5.5. To the best of our knowledge, these results are new.
Parts of the concepts and results of this paper was announced in Bui-Minh [6].

##  2\. Preliminaries

###  2.1. Notations

In this paper we will denote by $\mathbb{X},\,\mathbb{Y}$ Banach spaces with
corresponding norms. The symbols $\mathbb{R}$ and $\mathbb{C}$ stand for the
fields of real and complex numbers, respectively. Denote by
$\mathcal{L}(\mathbb{X},\mathbb{Y})$ the Banach space of all bounded linear
operators from a Banach space $\mathbb{X}$ to a Banach space $\mathbb{Y}$. We
will denote the domain of an operator $T$ by $D(T)$ and its range by $R(T)$.
The resolvent set of a linear operator $T$ in a Banach space will be denoted
by $\rho(T)$ while its spectrum is denoted by $\sigma(T)$. For
$\lambda\in\rho(T)$, we denote the inverse $(\lambda I-T)^{-1}$ by
$R(\lambda,T)$ and call it the resolvent (at $\lambda$) of linear operator
$T$. Let $P$ be a linear operator in a Banach space $\mathbb{X}$, as usual,
$\ker(P)$ and $\mathrm{Im}(P)$ are the notations of kernel and image of $P$.
For a complex number $z$, the real part is denoted by $\Re z$ and the
imaginary part is denoted by $\Im z$.

###  2.2. Exponential Dichotomy

######  Definition 2.1 (Exponential dichotomy).

A linear semigroup $(T(t))_{t\geq 0}$ in a Banach space $\mathbb{X}$ is said
to have an exponential dichotomy or to be hyperbolic if there exist a bounded
projection $P$ on $\mathbb{X}$ and positive constants $N$ and $\alpha$
satisfying

  1. (1)

$T(t)P=PT(t)$, for $t\geq 0$;

  2. (2)

$T(t)\big{|}_{\ker(P)}$ is an isomorphism from $\ker(P)$ onto $\ker(P)$, for
all $t\geq 0$, and its inverse on $\ker(P)$ is defined by
$T(-t):=\left(T(t)\big{|}_{\ker(P)}\right)^{-1}$;

  3. (3)

the following estimates hold

$\displaystyle\|T(t)x\|\leq Ne^{-\alpha t}\|x\|,\quad\mbox{ for all }t\geq
0,\quad x\in\mathrm{Im}(P),$ (2.1) $\displaystyle\|T(-t)x\|\leq Ne^{-\alpha
t}\|x\|,\quad\mbox{ for all }t\geq 0,\quad x\in\ker(P).$ (2.2)

The projection $P$ is called the dichotomy projection for the hyperbolic
semigroup $(T(t))_{t\geq 0}$, and the constants $N$ and $\alpha$ are called
dichotomy constants.

The following result is well known:

######  Lemma 2.2.

Let $(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup. Then, it has an exponential
dichotomy if and only if
$\sigma(T(1))\cap\\{z\in\mathbb{C}:|z|=1\\}=\emptyset$.

###### Proof.

See Engel-Nagel [15, 1.17 Theorem]. ∎

From Lemma 2.2, it is easy to prove the following result:

######  Lemma 2.3.

Let $(T(t))_{t\geq 0}$ be a $C_{0}$-semigroup that has an exponential
dichotomy. Then, $(S(t))_{t\geq 0}$ has an exponential dichotomy provided that
$S(1)$ is sufficiently close to $T(1)$.

###  2.3. Fractional Powers of Closed Operators

In this subsection, we present some results on fractional powers of closed
linear operators.

######  Definition 2.4 (Sectorial operator, see Henry [17, Definition 1.3.1]).

We call a linear operator $A$ in a Banach space $\mathbb{X}$ a sectorial
operator if it is a closed densely defined operator such that, for some $\phi$
in $\left(0,\frac{\pi}{2}\right)$ and some $M\geq 1$, and real $a$, the sector

$S_{a,\phi}:=\left\\{\lambda:\phi\leq|\arg(\lambda-a)|\leq\pi,\,\lambda\neq
a\right\\}$ (2.3)

is in the resolvent set of $A$ and

$\left\|(\lambda-A)^{-1}\right\|\leq\frac{M}{|\lambda-a|},\quad\mbox{ for all
}\lambda\in S_{a,\phi}.$ (2.4)

If $-A$ is the generator of an analytic semigroup we will use the fractional
powers as defined in Henry [17, 1.4, p. 24] and Pazy [23, 2.6]. To be
specific, we assume that $0\leq\alpha\leq 1$ we will denote
$\mathbb{X}^{\alpha}:=D(A^{\alpha}_{1})$, where $A_{1}:=A+aI$ with $a$ chosen
so that $\Re\sigma(A_{1})>0$ with the graph norm $\|x\|_{\alpha}$ defined as

$\|x\|_{\alpha}:=\|A^{\alpha}_{1}x\|,\quad x\in\mathbb{X}^{\alpha},$ (2.5)

Since different $a$ will determine equivalent different norms we can fix a
number $a$. As is well known (see Henry [17, Theorem 1.4.8]) if $-A$ is the
generator of an analytic semigroup, then $\mathbb{X}^{\alpha}$ with the norm
defined above is a Banach space. Moreover, by Henry [17, Theorem 1.4.3], Pazy
[23, Theorem 6.13, p. 74],

######  Theorem 2.5.

Let $-A$ be the infinitesimal generator of an analytic semigroup
$(T(t))_{t\geq 0}$. If $\Re\sigma(A)>\delta$, then,

  1. (1)

$T(t):\mathbb{X}\to D(A^{\alpha})$ for every $t>0$ and $\alpha\geq 0$.

  2. (2)

For every $x\in D(A^{\alpha})$ we have $T(t)A^{\alpha}x=A^{\alpha}T(t)x$;

  3. (3)

For every $t>0$ the operator $A^{\alpha}T(t)$ is bounded and

$\|A^{\alpha}T(t)\|\leq M_{\alpha}t^{-\alpha}e^{-\delta t}.$ (2.6)

  4. (4)

Let $0<\alpha\leq 1$ and $x\in D(A^{\alpha})$, then

$\|T(t)x-x\|\leq C_{\alpha}t^{\alpha}\|A^{\alpha}x\|.$ (2.7)

######  Theorem 2.6 (see Henry [17, Theorem 1.4.8]).

If $A$ is sectorial in a Banach space $\mathbb{X}$, then $\mathbb{X}^{\alpha}$
is a Banach space in the norm $\|\cdot\|_{\alpha}$, for $\alpha\geq 0$, and
for $\alpha\geq\beta\geq 0$, $\mathbb{X}^{\alpha}$ is a dense subspace of
$\mathbb{X}^{\beta}$ with continuous inclusion. If $A$ has compact resolvent,
the inclusion $\mathbb{X}^{\alpha}\subset\mathbb{X}^{\beta}$ is compact when
$\alpha>\beta\geq 0$.

If $A_{1}$ and $A_{2}$ are sectorial operators in $\mathbb{X}$ with the same
domain and $\Re\sigma(A_{j})>0$, for $i=1,2$, and if
$(A_{1}-A_{2})A_{1}^{-\alpha}$ is a bounded operator for some $\alpha<1$, then
with $\mathbb{X}_{j}^{\beta}=D(A_{j}^{\beta})$,
$\mathbb{X}_{1}^{\beta}=\mathbb{X}_{2}^{\beta}$ with equivalent norms for
$0\leq\beta\leq 1$.

###  2.4. Partial Functional Differential Equations

Let us consider partial functional differential equations (PFDE) of the form

$u^{\prime}(t)=Au(t)+Bu_{t},\quad t\geq 0,$ (2.8)

where $A$ generates a $C_{0}$-semigroup in a Banach space $\mathbb{X}$ and
$B\in\mathcal{L}(\mathcal{C},\mathbb{X})$. In this paper we will use the
standard theory on PFDE developed in Travis-Webb [25] for Eq. (1.1). For
simplicity, we may assume that the $C_{0}$-semigroup $(T(t))_{t\geq 0}$
generated by $A$ satisfies $\|T(t)\|_{\mathbb{X}}\leq e^{\omega t}$ for some
fixed $\omega$ and all $t\geq 0$. Otherwise, a new equivalent norm of
$\mathbb{X}$ could be used for this purpose. When $A$ is the generator of an
analytic semigroup, we may consider fractional powers $A^{\alpha}$ of the
operator $A$ with $0<\alpha\leq 1$. In this case, for simplicity, we will
assume that $\sigma(A)$ is in the left half of the complex plane, so the
Banach space $\mathbb{X}^{\alpha}:=D(A^{\alpha})$ will be equipped with the
norm $\|x\|_{\alpha}:=\|A^{\alpha}x\|_{\mathbb{X}}$ for all
$x\in\mathbb{X}^{\alpha}$. As shown in Travis-Webb [26], if
$B:C([-r,0],\mathbb{X}^{\alpha})\to\mathbb{X}$ is a bounded linear operator,
then Eq. (1.1) generates a $C_{0}$-semigroup in
$\mathcal{C}_{\alpha}:=C([-r,0],\mathbb{X}^{\alpha})$. We will use the formula
for the generator of the solution semigroup generated by Eq. (1.1) from
Travis-Webb [25, (3.3), p.402] or Travis-Webb [26, (4.3), p. 136] accordingly.
We also use the notations and results by Travis-Webb in [27].

##  3\. Yosida Distance and Linear Perturbation of Linear (Non-Delayed)
Evolution Equations

We begin this section with the concept of Yosida distance between two linear
operators. To this end, we recall the concept of Yosida approximation of a
linear operator (see e.g. [23, 30]). Given operator $A$ in a Banach space
$\mathbb{X}$ with $\rho(A)\supset[\omega,\infty)$, where $\omega$ is a given
number, the Yosida approximation $A_{\lambda}$ is defined as
$A_{\lambda}:=\lambda^{2}R(\lambda,A)-\lambda I$ for sufficiently large
$\lambda$.

######  Definition 3.1 (Yosida distance).

The Yosida distance between two linear operators $A$ and $B$ satisfying
$\rho(A)\supset[\omega,\infty)$ and $\rho(B)\supset[\omega,\infty)$, where
$\omega$ is a given number, is defined to be

$d_{Y}(A,B):=\limsup_{\mu\to+\infty}\left\|A_{\mu}-B_{\mu}\right\|.$ (3.1)

The following theorem is the main result of this subsection on the roughness
of exponential dichotomy under Yosida perturbation.

######  Theorem 3.2.

Let $A$ be the generator of a $C_{0}$-semigroup that has an exponential
dichotomy. Then, the $C_{0}$-semigroup generated by an operator $B$ also has
an exponential dichotomy, provided that $d_{Y}(A,B)$ is sufficiently small.

###### Proof.

First, we assume that both semigroups generated by $A$ and $B$ are contraction
$C_{0}$-semigroups. Then, by Pazy [23, Lemma 3.4], $e^{tA_{\lambda}}$ is a
$C_{0}$-semigroup of contractions.

Let $C$ and $D$ be two bounded linear operators in a Banach space
$\mathbb{X}$. We will estimate the growth of $e^{tC}-e^{tD}$. By the
Variation-of-Constants Formula that is applied to the equation
$x^{\prime}(t)=Cx(t)+(D-C)x(t)$ and by setting $x(t)=e^{tD}x$ we have

$x(t)=e^{tC}x+\int\limits^{t}_{0}e^{(t-s)C}(D-C)x(s)\mathrm{d}s.$

For each $t\geq 0$, we have

$\displaystyle\left\|e^{tC}x-e^{tD}x\right\|$
$\displaystyle\leq\int\limits^{t}_{0}\left\|e^{(t-s)C}(D-C)e^{sD}x\right\|\mathrm{d}s$
$\displaystyle\leq t\|C-D\|\left\|e^{tC}\right\|\left\|e^{tD}\right\|\|x\|.$

Therefore,

$\left\|e^{tA_{\lambda}}-e^{tB_{\lambda}}\right\|\leq
t\left\|A_{\lambda}-B_{\lambda}\right\|\left\|e^{tA_{\lambda}}\right\|\left\|e^{tB_{\lambda}}\right\|.$
(3.2)

Now we assume that $(T(t))_{t\geq 0}$ and $(S(t))_{t\geq 0}$ are the
$C_{0}$-semigroups generated by $A$ and $B$ that satisfy

$\|T(t)\|\leq Me^{\omega t},\quad\|S(t)\|\leq Me^{\omega t}$

for certain positive numbers $M$ and $\omega$. As is well known, for the
Yosida approximation $A_{\lambda}$ of the generator $A$ of a $C_{0}$-semigroup
$T(t)$ that satistifies $\|T(t)\|\leq Me^{\omega t}$ the following estimate of
the growth is valid (see e.g. Pazy [23, (5.25)])

$\left\|e^{tA_{\lambda}}\right\|\leq Me^{2\omega
t},\quad\left\|e^{tB_{\lambda}}\right\|\leq Me^{2\omega t}.$

Hence, we have

$\left\|e^{tA_{\lambda}}-e^{tB_{\lambda}}\right\|\leq
tM^{2}\left\|A_{\lambda}-B_{\lambda}\right\|e^{4t\omega}.$ (3.3)

By Pazy [23, Theorem 5.5], for each $x\in\mathbb{X}$, we have

$\displaystyle\|T(t)x-S(t)x\|$
$\displaystyle=\lim_{\lambda\to\infty}\left\|e^{tA_{\lambda}}x-e^{tB_{\lambda}}x\right\|$
$\displaystyle\leq tM^{2}e^{4\omega
t}\limsup_{\lambda\to\infty}\left\|A_{\lambda}-B_{\lambda}\right\|$
$\displaystyle=tM^{2}e^{4\omega t}d_{Y}(A,B).$ (3.4)

Finally, if $d_{Y}(A,B)$ is sufficiently small, $\|T(1)-S(1)\|$ is
sufficiently small as well, and thus, $(S(t))_{t\geq 0}$ has an exponential
dichotomy. ∎

In the perturbation theory of linear equations $u^{\prime}(t)=Au(t)$ by an
operator $C$, one considers the equation $u^{\prime}(t)=(A+C)u(t)$ a
persistent domain condition that $D(A)\subset D(C)$ is a primary one in many
researches, see for instance, [7, 13, 20]. The novelty in this paper is the
persistent domain condition is not primary in the assumption as demonstrated
later. The following lemma will show that the concept of Yosida distance
allows us to recover all important cases of perturbation under this persistent
domain condition.

######  Lemma 3.3.

The following assertions are valid:

  1. (1)

Let $A,\,B$ be the generators of contraction semigroups. Assume further that
$D(A)=D(B)$. Then, $A=B$, provided that $d_{Y}(A,B)=0$.

  2. (2)

Let $A,\,B\in\mathcal{L}(\mathbb{X})$. Then

$d_{Y}(A,B)=\|A-B\|.$ (3.5)

  3. (3)

If $A$ is the generator of a $C_{0}$-semigroup $T(t)$ such that $\|T(t)\|\leq
Me^{\omega t}$, and $C$ is a bounded operator, then $d_{Y}(A,A+C)$ is finite.
Moreover,

$d_{Y}(A,A+C)\leq M^{2}\|C\|$ (3.6)

  4. (4)

Let $A$ be the generator of an analytic semigroup $(T(t))_{t\geq 0}$ such that
$\|T(t)\|\leq Me^{\omega t}$, and $C$ be an $A$-bounded operator, that is,
$D(C)\supset D(A)$, and there are positive constants $a,\,c$ such that

$\|Cx\|\leq a\|Ax\|+c\|x\|.$ (3.7)

Then, there exists a constant $\delta>0$ such that if $0\leq a\leq\delta$, the
distance $d_{Y}(A,A+C)$ is finite. Moreover,

$d_{Y}(A,A+C)\leq aKM+cM^{2},$ (3.8)

where $K$ and $M$ are positive constants that depend only on $A$.

###### Proof.

  1. (1)

From the semigroup theory (see Pazy [23, Lemma 1.3.3]), if $A$ is the
generator of a contraction $C_{0}$-semigroup, then

$\lim_{\mu\to+\infty}A_{\mu}x=Ax,\quad x\in D(A).$ (3.9)

Therefore, for $x\in D(A)=D(B)$, $Ax=Bx$.

  2. (2)

Firstly, we have

$\displaystyle R(\mu,A)-R(\mu,B)$
$\displaystyle=(\mu-B)R(\mu,B)R(\mu,A)-R(\mu,B)R(\mu,A)(\mu-A).$

Set

$C_{\mu}:=R(\mu,B)R(\mu,A).$

Then

$\displaystyle R(\mu,A)-R(\mu,B)$
$\displaystyle=(\mu-B)C_{\mu}-C_{\mu}(\mu-A)$ $\displaystyle=\mu
C_{\mu}+BC_{\mu}-C_{\mu}\mu-C_{\mu}A$ $\displaystyle=BC_{\mu}-C_{\mu}A.$

Thus,

$\mu^{2}\|R(\mu,A)-R(\mu,B)\|=\mu^{2}\left\|BC_{\mu}-C_{\mu}A\right\|.$

We will show that for a bounded linear operator $A$ in $\mathbb{X}$ the
following is valid

$\lim_{\mu\to+\infty}R(\mu,A)=0.$ (3.10)

In fact, for sufficiently large $\mu$, say $\mu>\|A\|$, using the Neuman
series, for large $\mu$, we have

$\displaystyle\|R(\mu,A)\|=\frac{1}{\mu}\left\|R\left(1,\frac{1}{\mu}A\right)\right\|$
$\displaystyle\leq\frac{1}{\mu}\left\|\sum_{n=0}^{\infty}\left(\frac{1}{\mu}A\right)^{n}\right\|$
$\displaystyle\leq\frac{1}{\mu}\frac{1}{1-\frac{1}{\mu}\|A\|}.$

This proves (3.10). Next, by (3.10) and the identity $(\mu-A)R(\mu,A)=I$ we
have that

$\lim_{\mu\to+\infty}\mu R(\mu,A)=I,$

so

$\lim_{\mu\to+\infty}\mu^{2}C_{\mu}=I,$

Finally, we have

$d_{Y}(A,B):=\limsup_{\mu\to+\infty}\mu^{2}\left\|BC_{\mu}-C_{\mu}A\right\|=\|B-A\|,$

and this finishes the proof.

  3. (3)

By a simple computation we have

$R(\mu,A+C)-R(\mu,A)=R(\mu,A+C)CR(\mu,A).$ (3.11)

It is known (see e.g. Pazy [23, Theorem 1.1, p. 76] that linear operator $A+C$
with $D(A+C)=D(A)$ generates a $C_{0}$-semigroup $(S(t))_{t\geq 0}$ satisfying

$\|S(t)\|\leq Me^{(\omega+M\|C\|)t},$ (3.12)

so by the Hille-Yosida Theorem

$\displaystyle\|R(\mu,A)\|\leq\frac{M}{\mu-\omega},$
$\displaystyle\|R(\mu,A+C)\|\leq\frac{M}{\mu-(\omega+M\|C\|)},$

for certain positive constants $M$ and $\omega$. Therefore,

$\displaystyle\limsup_{\mu\to\infty}\mu^{2}\|R(\mu,A)-R(\mu,A+C)\|$
$\displaystyle\leq\limsup_{\mu\to\infty}\frac{\mu^{2}M^{2}\|C\|}{(\mu-\omega)(\mu-(\omega+M\|C\|)}$
$\displaystyle=M^{2}\|C\|<\infty.$ (3.13)

  4. (4)

By Pazy [23, Theorem 2.1, p. 80] and the remark that follows it, there exists
a positive constant $\delta$ such that if $0\leq a\leq\delta$, then, $A+C$
generates an analytic semigroup $(S(t))_{t\geq 0}$ that satisfies

$\|S(t)\|\leq Me^{(\omega+\Lambda(c))t},$

where $\lim_{c\to 0}\Lambda(c)=0$. Therefore, by (3.11), as $A$ generates an
analytic semigroup there are positive constants $K$ and $N$ (see Pazy [23,
Theorem 5.5, p. 65]) such that if $\mu>N$, then

$\|AR(\mu,A)\|\leq\frac{K}{\mu}.$

Hence,

$\displaystyle d_{Y}(A,A+C)$
$\displaystyle=\limsup_{\mu\to\infty}\mu^{2}\|R(\mu,A+C)-R(\mu,A)\|$
$\displaystyle\leq\limsup_{\mu\to\infty}\frac{\mu^{2}M}{\mu-\omega-\Lambda(c)}\left(a\|AR(\mu,A)\|+c\|R(\mu,A)\|\right)$
$\displaystyle\leq\limsup_{\mu\to\infty}\frac{\mu^{2}M}{\mu-\omega-\Lambda(c)}\left(\frac{aK}{\mu}+\frac{cM}{\mu-\omega}\right)$
$\displaystyle\leq aKM+cM^{2}.$

The proof is completed. ∎

In view of Lemma 3.3 we will extend the concept of Yosida distance of two
bounded linear operators $A,\,B\in\mathcal{L}(\mathbb{X},\mathbb{Y})$ as
below:

######  Definition 3.4.

If $A,\,B\in\mathcal{L}(\mathbb{X},\mathbb{Y})$, then

$d_{Y}(A,B):=\|A-B\|.$ (3.14)

######  Corollary 3.5.

Let $A$ be the generator of an exponentially dichotomous $C_{0}$-semigroup
$(T(t))_{t\geq 0}$ in $\mathbb{X}$ and $C$ be a bounded linear operator in
$\mathbb{X}$. Then, the operator $A+C$ with domain $D(A+C)=D(A)$ generates an
exponentially dichotomous $C_{0}$-semigroup, provided that $\|C\|$ is
sufficiently small.

###### Proof.

Let $A$ generate a $C_{0}$-semigroup $(T(t))_{t\geq 0}$ that satisfies
$\|T(t)\|\leq Me^{\omega t}$. By (3.13) if $\|C\|$ is sufficiently small, then
$d_{Y}(A,A+C)$ is sufficiently small as well, so by Theorem 3.2, the semigroup
generated by $A+C$ also has an exponential dichotomy. ∎

######  Corollary 3.6.

Let $A$ be the generator of a hyperbolic analytic semigroup $(T(t))_{t\geq 0}$
in $\mathbb{X}$ satisfying $\|T(t)\|\leq Me^{\omega t}$. Then, if $C$ is an
$A$-bounded linear operator in $\mathbb{X}$, that is, it satisfies for all
$x\in D(A)$

$\|Cx\|\leq a\|Ax\|+c\|x\|$

for certain positive constants $a$ and $c$. Then, $A+C$ generates an
exponentially dichotomous analytic semigroup, provided that $a$ and $c$ are
sufficiently small.

###### Proof.

By assumption for each $\ x\in D(A)$ we have

$\displaystyle\|Cx\|$
$\displaystyle\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|C\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\cdot{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
$\displaystyle\leq{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|C\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\cdot\|Ax\|+{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|C\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}\cdot{\left|\kern-1.07639pt\left|\kern-1.07639pt\left|x\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}.$
(3.15)

As is well known (see e.g. Pazy [23, Theorem 2.1, p. 80]) for sufficiently
small
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|C\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$,
the operator $A+C$ generates an analytic semigroup. Moreover, by (3.8) if
${\left|\kern-1.07639pt\left|\kern-1.07639pt\left|C\right|\kern-1.07639pt\right|\kern-1.07639pt\right|}$
is sufficiently small, $d_{Y}(A,A+C)$ is sufficiently small, so by Theorem
3.2, the semigroup generated by $A+C$ has an exponential dichotomy. ∎

######  Remark 3.7.

Lemma 3.3 shown that the Yosida distance can measure the perturbation of a
closed operator $A$ by an operator $C$ such that the domain of $A+C$ contains
that of $A$. Below we will present an example to show that our concept of
Yosida distance can actually be applied to larger classes of perturbation in
which the requirement that $D(A)\subset D(A+C)$ is freed.

##  4\. Perturbation of Partial Functional Differential Equation
$u^{\prime}(t)=Au(t)+Bu_{t}$

Below we will demonstrate an application of Yosida distance to an important
case of evolution equations where the domain condition mentioned in Remarks
3.7 is generally unsatisfied. Consider PFDEs of the form

$u^{\prime}(t)=A_{i}u(t)+B_{i}u_{t},$ (4.1)

where $A_{i}$ generates a $C_{0}$-semigroup $(T_{i}(t))_{t\geq 0}$ in
$\mathbb{X}$, and $B_{i}$ is a bounded operator from
$\mathcal{C}:=C([-1,0],\mathbb{X})$ to $\mathbb{X}$ and $i=0,\,1$ and
$u_{t}(\theta):=u(t+\theta),\theta\in[-1,0]$. By the standard theory of
partial functional differential equation, this equation generates a solution
semigroup $\mathcal{T}_{i}(t))_{t\geq 0}$ that is strongly continuous in the
phase space $\mathcal{C}$ with the generator $\mathcal{G}_{i}$ defined as (see
Travis-Webb [25, p. 402])

$\displaystyle G_{i}\phi$ $\displaystyle=\phi^{\prime},$ (4.2) $\displaystyle
D(\mathcal{G}_{i})$ $\displaystyle=\\{\phi\in\mathcal{C}:\phi(0)\in
D(A_{i}),\,\phi^{\prime-}(0)=A_{i}\phi(0)+B_{i}\phi\\}.$ (4.3)

The investigation of Eq. (4.1) is, therefore, reduced to that of the evolution
equation (without delay) $w^{\prime}(t)=\mathcal{G}_{i}w(t)$ in the Banach
space $\mathcal{C}$.

As shown below, the primary domain condition used in the perturbation theory
as discussed above is the inclusion of the domains of the operators
$D(\mathcal{G}_{0})\subset D(\mathcal{G}_{1})$. We will see that, generally,
this never happens to partial functional differential equations. However, the
Yosida distance between $\mathcal{G}_{0},\mathcal{G}_{1}$ can be determined.
That means, we can still study the perturbation of the system.

Below, for our convenience, we will drop the index $i$ in the above problem to
get a general formula for the domain $D(\mathcal{G}_{i})$. Suppose that $x\in
D(A)$ is an arbitrary element that is fixed. Assume that $BD(A)\subset D(A)$.
We define

$\phi(t):=T(t)x+\int\limits^{t}_{0}T(t-\xi)Bx\mathrm{d}\xi.$ (4.4)

Then,

$\phi(1)=T(1)x+\int\limits^{1}_{0}T(1-\xi)Bx\mathrm{d}\xi\in D(A).$

Moreover, by Pazy [23, Theorem 2.4, p. 4],

$(T(t)x)^{\prime}=AT(t)x$

and

$\frac{\mathrm{d}}{\mathrm{d}t}\left(\int\limits^{t}_{0}T(t-\xi)Bx\mathrm{d}\xi\right)=Bx+\int\limits^{t}_{0}AT(t-\xi)Bx\mathrm{d}\xi.$

Therefore,

$\displaystyle\phi^{\prime}(1)$
$\displaystyle=A\left(T(1)x+\int\limits^{1}_{0}T(1-\xi)Bx\mathrm{d}\xi\right)+Bx$
(4.5) $\displaystyle=A\phi(1)+B\phi(0).$ (4.6)

Consequently, if we set $\varphi(t)=\phi(t+1)$, $t\in[-1,0]$, then the
following is valid:

######  Lemma 4.1.

Assume that for $i=0,1$, $B_{i}D(A_{i})\subset D(A_{i})$. Then, for any $x\in
D(A_{i})$ the function

$\varphi(t):=T(t+1)x+\int\limits^{t+1}_{0}T(t+1-\xi)B_{i}x\mathrm{d}\xi,\quad
t\in[-1,0]$ (4.7)

is an element in $D(G_{i})$.

######  Corollary 4.2.

Assume that $A_{0}=A_{1}=A$, and $B_{i}D(A)\subset D(A)$, for $i=0,\,1$. Then,
$D(\mathcal{G}_{0})\subset D(\mathcal{G}_{1})$ if and only if $B_{1}=B_{0}$.

###### Proof.

If we choose any $x\in D(A_{0})$, then $\varphi(t)$ as defined in (4.7) is an
element of $D(G_{0})$. Hence, $\varphi\in D(G_{1})$. This yields that

$\displaystyle\varphi^{\prime}(0)$
$\displaystyle=A\varphi(0)+B_{0}\varphi(-1)$
$\displaystyle=A\varphi(0)+B_{1}\varphi(-1).$

Therefore, $B_{0}\varphi(-1)=B_{1}\varphi(-1)$. In other words,
$B_{0}x=B_{1}x$ for any given $x\in D(A)$. As $D(A)$ is dense everywhere in
$\mathbb{X}$ and $B_{0}$ and $B_{1}$ are bounded linear operators,
$B_{0}=B_{1}$. ∎

######  Example 4.3.

Consider functional differential equations of the form

$x^{\prime}(t)=b_{i}x(t-1),\quad x(t)\in\mathbb{R},$ (4.8)

where $b_{i}\in\mathbb{R}$ is a given number for $i=0,1$. As shown in Hale
[16] this functional differential equation generates a $C_{0}$-semigroup in
the phase space $\mathbb{X}:=C([-1,0],\mathbb{R})$ with the generator $G_{i}$
defined as

$[G_{i}\phi](t)=\begin{cases}\phi^{\prime}(t)&\mbox{ if }t\in[-1,0),\\\
b_{i}\phi(-1)&\mbox{ if }t=0,\end{cases}$

with

$D(G_{i}):=\left\\{\varphi\in
C^{1}([-1,0],\mathbb{R})\mid\varphi^{\prime}(0)=b_{i}\varphi(-1)\right\\}.$

In this example, $A=0$ and $B_{i}=b_{i}\cdot$. Obviously, $B_{i}D(A)\subset
D(A)=\mathbb{R}$. We have $\varphi(t)=(1+(1+t)b_{i})x$ with $t\in[-1,0]$,
where $x\in\mathbb{R}$ is any given number. We can see that
$\varphi^{\prime}(0)=b_{i}x=b_{i}\varphi(0)$, so $\varphi\in
D(\mathcal{G}_{i})$. Assuming that $D(\mathcal{G}_{0})\subset
D(\mathcal{G}_{1})$ as $x\in\mathbb{R}$ is arbitrary, we can choose a number
$x=1$ to see that $b_{0}=b_{1}$.

Below we will demonstrate the usefulness of the concept of Yosida distance as
it still works in this case when the persistent domain condition is not
satisfied to apply the previously known perturbation theory. In fact we study
the Yosida distance between the generators of the solution semigroups of
partial functional differential equations of the form Eq. (4.1).

Consider partial functional differential equations of the form

$u^{\prime}(t)=A_{i}u(t)+B_{i}u_{t},$ (4.9)

where $i=0,1$, $A$ is the generator of a $C_{0}$-semigroup $(T_{i}(t))_{t\geq
0}$ satisfying $\|T_{i}(t)\|\leq e^{\omega t}$ and
$B_{i}\in\mathcal{L}(\mathbb{X})$ for a certain (fixed) positive constant
$\omega$. We will estimate the Yosida distance between the generators
$\mathcal{G}_{0}$ and $\mathcal{G}_{1}$ of the two solution semigroups
generated $(\mathcal{T}_{i}(t))_{t\geq 0}$ by Eq. (4.9) in the phase space
$\mathcal{C}$. For simplicity we will drop the index ”$i$” in the following
paragraph.

Below we will derive a general formula for the generators of the solution
semigroups generated by Eq. (4.9), where we will drop the index ”$i$” as they
are similar.

######  Proposition 4.4.

The resolvent $R(\lambda,\mathcal{G})$ of the generator of the solution
semigroup $(\mathcal{S}(t))$ in $\mathcal{C}$ is determined by the equation
(in $\varphi$)

$\varphi(t)=e^{\lambda
t}\left[R(\lambda,A)\left(B\varphi+\psi(0)\right)\right]-\int\limits_{0}^{t}e^{\lambda(t-\xi)}\psi(\xi)\mathrm{d}\xi,$
(4.10)

where $\varphi:=R(\lambda,\mathcal{G})\psi$ if $\psi$ is a given element of
$\mathcal{C}$.

###### Proof.

To find resolvent $R(\lambda,\mathcal{G})$ for $\lambda>0$, we will solve the
following equation in $\mathcal{C}$

$\lambda\varphi-\mathcal{G}\varphi=\psi\quad\mbox{ for every given
}\psi\in\mathcal{C}.$ (4.11)

As $\varphi\in D(\mathcal{G})$, $\mathcal{G}\varphi(t)=\dot{\varphi}(t)$. From
(4.11), we have

$\lambda\varphi(t)-\dot{\varphi}(t)=\psi(t),$

equivalently

$\dot{\varphi}(t)=\lambda\varphi(t)-\psi(t),\quad\mbox{ for all }t\in[-1,0].$

By the Variation-of-Constants Formula, we have

$\varphi(t)=e^{\lambda(t)}\varphi(0)-\int\limits_{0}^{t}e^{\lambda(t-\xi)}\psi(\xi)\mathrm{d}\xi,\quad\mbox{
for all }t\in[-1,0].$ (4.12)

As $\varphi\in D(\mathcal{G})$, we have

$\lambda\varphi(0)-\psi(0)=\dot{\varphi}(0)=A\varphi(0)+B\varphi.$

Hence, for large $\lambda$ we have

$\varphi(0)=R(\lambda,A)\left(B\varphi+\psi(0)\right),$

so (4.12), for $t\in[-1,0]$ we have

$\displaystyle\left(R(\lambda,\mathcal{G})\psi\right)(t)$
$\displaystyle=\varphi(t)$ $\displaystyle=e^{\lambda
t}\left[R(\lambda,A)\left(B\varphi+\psi(0)\right)\right]-\int\limits_{0}^{t}e^{\lambda(t-\xi)}\psi(\xi)\mathrm{d}\xi.$

The proof is completed. ∎

If we define the operator $\varphi\mapsto\mathcal{F}\varphi(A,B)$ as

$\displaystyle\mathcal{F}_{\lambda}(A,B)\varphi(t)$ $\displaystyle:=e^{\lambda
t}R(\lambda,A)B\varphi,$

and $\psi\mapsto\mathcal{J}_{\lambda}(A)\psi$ as

$\mathcal{J}_{\lambda}(A)\psi(t):=e^{\lambda
t}R(\lambda,A)\psi(0)-\int\limits_{0}^{t}e^{\lambda(t-\xi)}\psi(\xi)\mathrm{d}\xi,$

then, for each $\psi\in\mathcal{C}$, we have

$(I-\mathcal{F}_{\lambda}(A,B))\varphi=\mathcal{J}_{\lambda}(A)\psi.$ (4.13)

Note that as

$\displaystyle\limsup_{\lambda\to+\infty}\|\mathcal{F}_{\lambda}\|$
$\displaystyle\leq\limsup_{\lambda\to+\infty}\|R(\lambda,A)\|\|B\|$
$\displaystyle=\limsup_{\lambda\to+\infty}\frac{\|B\|}{\lambda-\omega}=0.$
(4.14)

the operator $R(1,\mathcal{F}_{\lambda})(A,B)$ exists, so

$\left(R(\lambda,\mathcal{G})\psi\right)=\varphi=R(1,\mathcal{F}_{\lambda})\mathcal{J}_{\lambda}\psi.$
(4.15)

######  Lemma 4.5.

Let $\mathcal{G}_{i}$ be the generators of the solution semigroups generated
by Eq. (4.9) in the phase space $\mathcal{C}:=C([-1,0],\mathbb{X})$. Assume
further that

$d_{Y}(A_{0},A_{1})<\infty,\quad\|B_{i}\|<\infty$ (4.16)

for $i=0,1$. Then,

$d_{Y}(\mathcal{G}_{0},\mathcal{G}_{1})\leq
2d_{Y}(B_{0},B_{1})+d_{Y}(A_{0},A_{1}).$ (4.17)

###### Proof.

By (4.15) we have

$\displaystyle\left\|R(\lambda,\mathcal{G}_{0})-R(\lambda,\mathcal{G}_{1})\right\|$
$\displaystyle\leq\|R(1,\mathcal{F}_{\lambda}(A_{0},B_{0}))\mathcal{J}_{\lambda}(A_{0})-R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\mathcal{J}_{\lambda}(A_{1})\|$
$\displaystyle\leq\|R(1,\mathcal{F}_{\lambda}(A_{0},B_{0}))\mathcal{J}_{\lambda}(A_{0})-R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\mathcal{J}_{\lambda}(A_{0})\|$
$\displaystyle\qquad+\|R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\mathcal{J}_{\lambda}(A_{0})-R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\mathcal{J}_{\lambda}(A_{1})\|$
$\displaystyle\leq\|R(1,\mathcal{F}_{\lambda}(A_{0},B_{0}))-R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\|\cdot\|\mathcal{J}_{\lambda}(A_{0})\|$
$\displaystyle\qquad+\|R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\|\cdot\|\mathcal{J}_{\lambda}(A_{1})-\mathcal{J}_{\lambda}(A_{0})\|.$
(4.18)

We will estimate each term in (4.18). For the second term, note that for
sufficiently large $\lambda$,

$\displaystyle\|R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\|$
$\displaystyle\leq\frac{1}{1-\|\mathcal{F}_{\lambda}(A_{1},B_{1})\|},$ (4.19)
$\displaystyle\|\mathcal{J}_{\lambda}(A_{1})-\mathcal{J}_{\lambda}(A_{0})\|$
$\displaystyle\leq\|R(\lambda,A_{1})-R(\lambda,A_{0})\|,$ (4.20)

so by (4.14)

$\limsup_{\lambda\to+\infty}\lambda^{2}\|R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\|\cdot\|\mathcal{J}_{\lambda}(A_{1})-\mathcal{J}_{\lambda}(A_{0})\|\leq
d_{Y}(A_{0},A_{1}).$ (4.21)

Now we estimate the first term. Using the identity for any
$U,\,V\in\mathcal{L}(\mathbb{X})$ with sufficiently large $\lambda$

$R(\mu,U)-R(\mu,V)=R(\mu,V)R(\mu,U)U-VR(\mu,V)R(\mu,U)$

and notation

$C_{\lambda}:=R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))R(1,\mathcal{F}_{\lambda}(A_{0},B_{0})),$

we have

$\|R(1,\mathcal{F}_{\lambda}(A_{0},B_{0}))-R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\|\leq
C_{\lambda}\mathcal{F}_{\lambda}(A_{0},B_{0})-\mathcal{F}_{\lambda}(A_{1},B_{1})C_{\lambda}.$
(4.22)

By (4.14) using the identity $R(1,U)-I=R(1,U)U$ we see that

$\displaystyle\|C_{\lambda}-I\|$
$\displaystyle\leq\|(R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))-I)R(1,\mathcal{F}_{\lambda}(A_{0},B_{0}))\|$
$\displaystyle\qquad+\|R(1,\mathcal{F}_{\lambda}(A_{0},B_{0}))-I\|$
$\displaystyle=\|R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\|\cdot\|\mathcal{F}_{\lambda}(A_{1},B_{1})\|\cdot\|\mathcal{F}_{\lambda}(A_{0},B_{0}))\|$
$\displaystyle\qquad+\|R(1,\mathcal{F}_{\lambda}(A_{0},B_{0}))\|\cdot\|\mathcal{F}_{\lambda}(A_{0},B_{0}))\|.$

Consequently, since

$\displaystyle\|\mathcal{J}_{\lambda}(A_{0})\|$
$\displaystyle\leq\frac{2}{\lambda-\omega}$ (4.23)

we can easily show that

$\displaystyle\limsup_{\lambda\to+\infty}\lambda^{2}(C_{\lambda}-I)\mathcal{F}_{\lambda}(A_{0},B_{0}))\mathcal{J}_{\lambda}(A_{0})=0,$
(4.24)
$\displaystyle\limsup_{\lambda\to+\infty}\lambda^{2}\mathcal{F}_{\lambda}(A_{1},B_{1})(C_{\lambda}-I)\mathcal{J}_{\lambda}(A_{0})=0.$
(4.25)

Hence, by (4.22)

$\displaystyle\limsup_{\lambda\to+\infty}\lambda^{2}\|\left(R(1,\mathcal{F}_{\lambda}(A_{0},B_{0}))-R(1,\mathcal{F}_{\lambda}(A_{1},B_{1}))\right)\mathcal{J}_{\lambda}(A_{0})\|$
$\displaystyle\leq\limsup_{\lambda\to+\infty}\lambda^{2}\left\|\mathcal{F}_{\lambda}(A_{0},B_{0})-\mathcal{F}_{\lambda}(A_{1},B_{1})\right\|\cdot\|\mathcal{J}_{\lambda}(A_{0})\|$
$\displaystyle\leq\limsup_{\lambda\to+\infty}\lambda^{2}\|R(\lambda,A_{0})B_{0}-R(\lambda,A_{0})B_{1}+R(\lambda,A_{0})B_{1}-R(\lambda,A_{1})B_{1}\|\cdot\|\mathcal{J}_{\lambda}(A_{0})\|$
$\displaystyle\leq\limsup_{\lambda\to+\infty}\left(\frac{\lambda^{2}}{\lambda-\omega}\|B_{0}-B_{1}\|\cdot\frac{2}{\lambda-\omega}+\lambda^{2}\|R(\lambda,A_{0})-R(\lambda,A_{1})\|\cdot\frac{2}{\lambda-\omega}\right)$
$\displaystyle\leq 2\|B_{0}-B_{1}\|.$

Finally,

$\displaystyle\limsup_{\lambda\to+\infty}\lambda^{2}\|R(\lambda,\mathcal{G}_{0})-R(\lambda,\mathcal{G}_{1})\|\leq
2d_{Y}(B_{0},B_{1})+d_{Y}(A_{0},A_{1}),$ (4.26)

finishing the proof. ∎

######  Definition 4.6.

Let $A$ generate a $C_{0}$-semigroup in a Banach space $\mathbb{X}$ and
$B\in\mathcal{L}(\mathbb{X})$. Then, the partial functional differential
equation $u^{\prime}(t)=Au(t)+Bu_{t}$, $t\geq 0$, is said to have an
exponential dichotomy if the solution semigroup generated by this equation in
$\mathcal{C}$ has an exponential dichotomy.

As an immediate consequence of Lemma 4.5 we have the following:

######  Theorem 4.7.

Let $A_{i}$ generate a $C_{0}$-semigroup $(T_{i}(t))$, $i=0,1$, in a Banach
space $\mathbb{X}$ that satisfies $\|T_{i}(t)\|\leq Me^{\omega t}$, $t\geq 0$,
and $B_{0},\,B_{1}\in\mathcal{L}(\mathcal{C},\mathbb{X})$. Assume that the
partial functional differential equation

$u^{\prime}(t)=A_{0}u(t)+B_{0}u_{t},\quad t\geq 0,$ (4.27)

has an exponential dichotomy. Then, the partial functional differential
equation

$u^{\prime}(t)=A_{1}u(t)+B_{1}u_{t},\quad t\geq 0,$ (4.28)

also has an exponential dichotomy if the Yosida distance $d_{Y}(A_{0},A_{1})$
and $d_{Y}(B_{0},B_{1})$ are sufficiently small.

######  Remark 4.8.

In the infinite-dimensional case Theorem 4.7 seems to be new as the
exponential dichotomy of the perturbed equation has been known in the
literature only when $A_{0}=A_{1}$. In the infinite-dimensional case, the
”standard” Variation-of-Constants Formula is not available, making the study
of perturbed equation much harder (see e.g. [1, 2, 14]).

##  5\. Perturbation in $\alpha$-Norm of Partial Functional Differential
Equations

In this section we consider partial functional differential equations of the
form

$u^{\prime}(t)=A_{0}u(t)+B_{0}u_{t},\quad t\geq 0,$ (5.1)

where $u(t)\in\mathbb{X}$, $\mathbb{X}$ is a Banach space, $A_{0}$ is the
generator of an analytic semigroup in $\mathbb{X}$ such that $\Re\sigma(A)<0$.
When such an operator $A_{0}$ is given we consider the operator
$(-A_{0})^{\alpha}$ as defined in Pazy [23] with $0\leq\alpha<1$. We will
denote $\mathbb{X}_{0}^{\alpha}:=D(A_{0}^{\alpha})$ and
$C_{0,\alpha}:=C([-r,0];\mathbb{X}_{0}^{\alpha})$. We assume that
$B_{0}:C_{0,\alpha}\to\mathbb{X}$ is a bounded linear operator. We will use
some standard results in the theory of partial functional differential
equations in Travis-Webb [25]:

######  Theorem 5.1.

Under the above mentioned assumptions on $A_{0}$ and $B_{0}$, Eq. (5.1)
generates a strongly semigroup of bounded linear operators $(S(t))_{t\geq 0}$
in $C_{\alpha}$ defined as

$S(t)\phi(\theta):=u(t+\theta),\quad\theta\in[-r,0],$

where $u(\cdot)$ is the unique solution of the Cauchy Problem associated with
Eq. (5.1) with initial condition $u_{0}=\phi$. Moreover, the infinitesimal
generator $\mathcal{G}$ of the semigroup $(S(t))_{t\geq 0}$ is defined as

$\displaystyle D(\mathcal{G})$ $\displaystyle=\\{\phi\in C_{\alpha}:\
\phi^{\prime}\in C_{\alpha},\phi(0)\in
D(A),\phi^{\prime-}(0)=A_{0}\phi(0)+B_{0}\phi\\},$ (5.2)
$\displaystyle(\mathcal{G}\phi)(\theta)$
$\displaystyle=\phi^{\prime}(\theta),\quad\theta\in[-r,0].$ (5.3)

As a perturbed equation of Eq. (5.1) we consider partial functional
differential equations of the form

$u^{\prime}(t)=A_{1}u(t)+B_{1}u_{t},\quad t\geq 0$ (5.4)

under the following assumptions:

######  Assumption 1.

Throughout this section with a given fixed $\alpha\in[0,1)$ we assume that

  1. (A1)

$A_{0}$ is the generator of an analytic $C_{0}$-semigroup $(T(t))_{t\geq 0}$
such that there are positive constants $M$ and $\omega$ for which the
following is valid for $t\geq 0$

$\|T(t)\|\leq e^{-\omega t};$ (5.5)

  2. (A2)

$D(A_{0})\subset D(A_{1})$ and $A_{1}$ is $A_{0}$-bounded, that is, there are
positive constants $\delta$ and $\gamma$ such that

$\|A_{1}x\|\leq\delta\|x\|+\gamma\|A_{0}x\|,\quad\mbox{ for all }x\in
D(A_{0});$ (5.6)

  3. (A3)

$(A_{1}-A_{0})A_{0}^{-\alpha}\in\mathcal{L}(\mathbb{X},\mathbb{X})$;

  4. (B1)

$B_{0}\in\mathcal{L}(C_{0,\alpha},\mathbb{X})$ and
$B_{1}\in\mathcal{L}(C_{1,\alpha},\mathbb{X})$.

######  Remark 5.2.

  1. (1)

Under Assumption (A1), $A^{-\alpha}\in\mathcal{L}(\mathbb{X},\mathbb{X})$ and
the space $\mathbb{X}_{0}^{\alpha}:=D(A_{0}^{\alpha})$ with norm
$\|x\|_{\alpha}:=\|A_{0}^{\alpha}x\|$ is well-defined (see [17, 23]).
Moreover, for sufficiently small $\delta$ and $\gamma$ in Assumption (A2), the
operator $A_{1}$ is also the generator of an analytic $C_{0}$-semigroup
satisfying a similar inequality as (5.5), so there is another $\alpha$ space
$\mathbb{X}^{\alpha}_{1}:=D(A_{1}^{\alpha})$ with norm
$\|x\|_{1,\alpha}:=A_{1}^{\alpha}$.

  2. (2)

By Assumption (A3) the vector spaces $\mathbb{X}^{\alpha}_{0}$ and
$X_{1}^{\alpha}$ are the same, and the norms $\|x\|_{\alpha}$ and
$\|x\|_{1,\alpha}$ are equivalent (see Henry [17, Theorem 1.4.8, p. 29], see
also Theorem 2.6).

  3. (3)

As a consequence of the above remark, under Assumption (B1), the operator
$B_{1}$ is actually an element of $\mathcal{L}(C_{0,\alpha},\mathbb{X})$.

Therefore, the following claim is true:

######  Proposition 5.3.

Under Assumption 1, the Yosida distances $d_{Y}(A_{0},A_{1})$ and
$d_{Y}(B_{0},B_{1})$ are well-defined and finite.

######  Lemma 5.4.

Under Assumption 1 for the partial functional differential equations

$u^{\prime}(t)=A_{i}u(t)+B_{i}u_{t},\quad t\geq 0,$ (5.7)

where $i=0,1$, generate $C_{0}$-semigroups in $C_{\alpha}$ with the generators
$\mathcal{G}_{i}$, $i=0,1$. Moreover,

$d_{Y}(\mathcal{G}_{0},\mathcal{G}_{1})\leq
2d_{Y}(B_{0},B_{1})+d_{Y}(A_{0},A_{1}).$ (5.8)

###### Proof.

The proof can be done in the same way as in that of Lemma 4.5. In fact, under
Assumptions 1 and by Travis-Webb [26], Eq. (5.4) generates a solution
semigroup $(S_{1}(t))_{t\geq 0}$ in the same space $C_{\alpha}$ as does the
unperturbed equation Eq. (5.1). If we denote their generators by
$\mathcal{G}_{0}$ and $\mathcal{G}_{1}$, respectively, then following the same
lines as in the proof of Lemma 4.5 we can prove (5.8). ∎

######  Theorem 5.5.

Let Assumptions 1 be made. Moreover, assume that the $C_{0}$-semigroup with
the generator $\mathcal{G}_{0}$ has an exponential dichotomy. Then, for
$A_{1}$ and $B_{1}$ sufficiently close to $A_{0}$ and $B_{0}$ in the Yosida
distance’s sense, the semigroup with the generator $\mathcal{G}_{1}$ has also
an exponential dichotomy.

##  6\. Examples

In this section we consider reaction-diffusion equations with finite delay of
the form

$\left\\{\begin{aligned} \frac{\partial w(x,t)}{\partial
t}&=\frac{\partial^{2}w(x,t)}{\partial x^{2}}-aw(x,t)\\\ &\qquad-
bw(x,t-r),&&0\leq x\leq\pi,&&t\geq 0,\\\ w(0,t)&=w(\pi,t)=0,&&&&t\geq 0,\\\
w(x,t)&=\varphi(t)(x),&&0\leq x\leq\pi,&&-r\leq t\leq 0,\end{aligned}\right.$
(6.1)

where $a$, $b$, and $r$ are given positive constants.

We choose the Hilbert space $\mathbb{X}=L^{2}[0,\pi]$ and consider the linear
operator

$\displaystyle A_{T}y$ $\displaystyle=\ddot{y},$ $\displaystyle D(A_{T})$
$\displaystyle=\\{y\in\mathbb{X}:\mbox{ $y$ and $\dot{y}$ are absolutely
continuous, }\ddot{y}\in\mathbb{X},\,y(0)=y(\pi)=0\\}.$

and

$A_{0}:=A_{T}-aI,\quad B_{0}:=bI.$ (6.2)

Consider the evolution equation

$u^{\prime}(t)=A_{0}u(t)+B_{0}u(t-r)$ (6.3)

in the Hilbert space $\mathbb{X}$. Note that $A_{0}$ generates an analytic
compact semigroup in $\mathbb{X}$ (see Travis-Webb [26]) and Eq. (6.3)
generates a compact semigroup $(\mathcal{S}_{0}(t))_{t\geq 0}$ in
$\mathcal{C}=C([-r,0],\mathbb{X})$. By the Spectral Mapping Theorem as
$(\mathcal{S}_{0}(t))_{t\geq 0}$ is compact, the exponential dichotomy of this
semigroup can be determined by the fact that the spectrum of its generator
does not intersect with the imaginary axis. In turn, the spectrum of its
generator consists of only eigenvalues, so the spectrum of $A_{T}$ consists of
eingenvalues $\sigma(A_{T})=\left\\{-n^{2}:n\in\mathbb{N}\right\\},$ they are
determined by the characteristic equation

$\lambda+a+be^{-\lambda r}=-n^{2}.$ (6.4)

Semigroup $(\mathcal{S}_{0}(t))_{t\geq 0}$ has an exponential dichotomy if and
only if Eq. (6.4) has no root of the form $i\xi$, where $\xi\in\mathbb{R}$,
for all $n=1,2,\dots$. For instance, $(\mathcal{S}_{0}(t))_{t\geq 0}$ has an
exponential dichotomy if either $b<1/r$ or $a+1>b$. So, below we will make the
following assumption on Eq. (6.1):

######  Assumption 2.

Either $b<1/r$ or $a+1>b$.

######  Example 6.1.

As a perturbed equation of Eq. (6.3) we consider the following reaction-
diffusion equations with finite delay

$\left\\{\begin{aligned} \frac{\partial w(x,t)}{\partial
t}&=(1+\varepsilon_{1})\frac{\partial^{2}w(x,t)}{\partial
x^{2}}+\varepsilon_{2}\frac{\partial w(x,t)}{\partial x}\\\
&\qquad-(a+\varepsilon_{3}(x))w(x,t)\\\ &\qquad-bw(x,t-r)\\\
&\qquad+\int\limits^{0}_{-r}[\mathrm{d}\varepsilon_{4}(\xi)]w(x,t+\xi)\mathrm{d}\xi,&&0\leq
t,&&0\leq x\leq\pi,\\\ w(0,t)&=w(\pi,t)=0,&&0\leq t,\\\
w(x,t)&=\varphi(t)(x),&&-r\leq t\leq 0,&&0\leq x\leq\pi,\end{aligned}\right.$
(6.5)

where $\varepsilon_{1}$ and $\varepsilon_{2}$ are small constant,
$\varepsilon_{3}(\cdot)$ are smooth functions, $\varepsilon_{4}(\cdot)$ is a
left continuous function from $[-r,0]$ to
$\mathcal{L}(\mathcal{C},\mathbb{X})$ with bounded variation
$\operatorname{Var}(\varepsilon_{4})$.

If we re-write the equation (6.5) in the abstract form we have

$u^{\prime}(t)=(A_{0}+A_{1})u(t)+(B_{0}+B_{1})u(t-r),$ (6.6)

where

$\displaystyle A_{1}y$
$\displaystyle:=\varepsilon_{1}\ddot{y}+\varepsilon_{2}\dot{y}-\varepsilon_{3}(\cdot)y,\quad
y\in D(A_{T}),$ (6.7) $\displaystyle B_{1}v$
$\displaystyle:=\int\limits^{0}_{-r}[\mathrm{d}\varepsilon_{4}(\xi)]v(t+\xi)\mathrm{d}\xi,\quad
v\in\mathcal{C}.$ (6.8)

######  Claim 6.2.

Operator $A_{1}$ is an $A_{0}$-bounded operator. More precisely, for each
$y\in D(A_{T})$,

$\|A_{1}y\|\leq\left(\varepsilon_{1}+\frac{\varepsilon_{2}}{2}\right)A_{0}y+\sup_{s\in[0,\pi]}|\varepsilon_{3}(s)|\|y\|.$
(6.9)

###### Proof.

By Engel-Nagel [15, Example 2.2, p. 169-170], if we choose $\varepsilon=1/2$,
then

$\|\dot{y}\|_{2}\leq 18\|y\|_{2}+\frac{1}{2}\|\ddot{y}\|_{2}.$

Therefore, (6.9) follows. ∎

######  Corollary 6.3.

Under Assumption 2, for sufficiently small $\varepsilon_{1}$,
$\varepsilon_{2}$, $\sup_{s\in[0,\pi]}|\varepsilon_{3}(s)|$, and
$\operatorname{Var}(\varepsilon_{4})$, the solution semigroup generated by Eq.
(6.5) in $\mathcal{C}$ has an exponential dichotomy.

###### Proof.

The claim of the corollary follow immediately from Claim 6.2, Lemma 3.3, Lemma
4.5 and Theorem 3.2. ∎

######  Example 6.4.

As an example of a perturbed equation in $\mathcal{C}_{\alpha}$ of Eq. (6.3)
with $\alpha=1/2$ we consider

$\left\\{\begin{aligned} \frac{\partial w(x,t)}{\partial
t}&=(1+\varepsilon_{1})\frac{\partial^{2}w(x,t)}{\partial x^{2}}\\\
&\qquad-(a+\varepsilon_{3}(x))w(x,t)\\\ &\qquad-bw(x,t-r)\\\
&\qquad+\varepsilon_{5}\frac{\partial w(x,t-r)}{\partial x}\\\
&\qquad+\int\limits^{0}_{-r}[\mathrm{d}\varepsilon_{4}(\xi)]w(x,t+\xi),&&0\leq
t,&&0\leq x\leq\pi,\\\ w(0,t)&=w(\pi,t)=0,&&0\leq t,\\\
w(x,t)&=\varphi(t)(x),&&-r\leq t\leq 0,&&0\leq x\leq\pi,\end{aligned}\right.$
(6.10)

where $\varepsilon_{1}$, $\varepsilon_{3}$, and $\varepsilon_{5}$ are small
constants, and $\varepsilon_{4}(\cdot)$ is a left continuous function with
bounded variation on $[-r,0]$ with values in $\mathcal{L}(\mathbb{X})$.

We will re-write Eq. (6.10) in an abstract form as in Travis-Webb [26]

$u^{\prime}(t)=(A_{0}+A_{2})u(t)+(B_{0}+B_{2})u(t-r),$ (6.11)

where

$\displaystyle A_{2}u$
$\displaystyle:=\varepsilon_{1}\ddot{u}-\varepsilon_{3}(\cdot)u,\quad u\in
D(A_{2}):=D(A_{T}),$ $\displaystyle B_{2}v$
$\displaystyle:=\varepsilon_{5}\dot{v}(-r)+\int\limits^{0}_{-r}[\mathrm{d}\varepsilon_{4}(\xi)]v(\xi),\quad
v\in C_{\alpha}=C([-r,0],\mathbb{X}^{\alpha}_{2}),$

where $\mathbb{X}_{2}^{\alpha}$ is defined by $(A_{0}+A_{2})$ as the generator
of an analytic semigroup,

Here we denote by $(\mathbb{X}_{1}^{\alpha},\|\cdot\|_{1,\alpha})$ (by
$(\mathbb{X}_{2}^{\alpha},\|\cdot\|_{1,\alpha})$, respectively) the Banach
space determined by $A_{0}$ (by $(A_{0}+A_{2})$, respectively) as the
generator of an analytic semigroup in $\mathbb{X}$. As shown in Travis-Webb
[26, Example 5.1], $B_{0}+B_{2}$ defines a bounded linear operator from
$C([-r,0],\mathbb{X}_{2}^{\alpha})$ to $\mathbb{X}$. Moreover, by Theorem 2.6,
the spaces $\mathbb{X}^{\alpha}_{1}=\mathbb{X}^{\alpha}_{2}$ with equivalent
norms, so we can identify $\mathbb{X}^{\alpha}_{2}$ as
$\mathbb{X}^{\alpha}_{1}$ and $B_{0}+B_{2}$ is a bounded linear operator from
$C([-r,0],\mathbb{X}_{1}^{\alpha})$ to $\mathbb{X}$. That yields that the
Yosida distance $d_{Y}(B_{0},B_{0}+B_{2})$ is defined and finite. We will show
that this Yosida distance can be made as small as we like if $\varepsilon_{5}$
and $\varepsilon_{6}$ are taken sufficiently small. In fact, we have

######  Claim 6.5.

Under the above notations,

$d_{Y}(B_{0},B_{0}+B_{2})\leq\varepsilon_{5}+\operatorname{Var}(\varepsilon_{4}).$
(6.12)

###### Proof.

By Travis-Webb [26, Example 5.1], it is shown that

$\displaystyle\|\phi(-r)\|_{\mathbb{X}}\leq\|\phi\|_{1,\alpha},$
$\displaystyle\|\phi^{\prime}(-r)\|_{\mathbb{X}}\leq\|\phi\|_{1,\alpha}.$

Therefore, $B_{0}$ and $B_{2}$ are bounded operators from
$C([-r,0],\mathbb{X}_{1}^{\alpha})$ to $\mathbb{X}$, so we have

$d_{Y}(B_{0},B_{0}+B_{2})=\|B_{2}\|_{1,\alpha}.$

Therefore, for $\phi\in C_{\alpha}$

$\displaystyle\|B_{2}\phi\|_{\mathbb{X}}$
$\displaystyle=\left\|\varepsilon_{5}\dot{\phi}(-r)+\int\limits^{0}_{-r}[\mathrm{d}\varepsilon_{4}(\xi)\phi(\xi)\right\|_{\mathbb{X}}$
$\displaystyle\leq\varepsilon_{5}\left\|\dot{\phi}\right\|_{\mathbb{X}}+\operatorname{Var}(\varepsilon_{4})\|\phi\|_{\mathbb{X}}$
$\displaystyle\leq\varepsilon_{5}\|\phi\|_{1,\alpha}+\operatorname{Var}(\varepsilon_{4})\|\phi\|_{1,\alpha}.$

Consequently,

$d_{Y}(B_{0},B_{0}+B_{2})\leq\varepsilon_{5}\|\phi\|_{1,\alpha}+\operatorname{Var}(\varepsilon_{4})\|\phi\|_{1,\alpha}.$

This finishes the proof. ∎

######  Corollary 6.6.

Under Assumption 2 for sufficiently small $\varepsilon_{1}$,
$\varepsilon_{3}$, $\operatorname{Var}(\varepsilon_{4})$, and
$\varepsilon_{5}$, the solution semigroup generated by Eq. (6.10) in
$C([-r,0],\mathbb{X}^{\alpha}_{2})$ has an exponential dichotomy.

###### Proof.

For sufficiently small $\varepsilon_{1}$ and $\varepsilon_{3}$ as the spaces
$(\mathbb{X}^{\alpha}_{2},\|\cdot\|_{2,\alpha})$ and
$(\mathbb{X}^{\alpha}_{1},\|\cdot\|_{1,\alpha})$ are equivalent. We are going
to show that Eq. (5.1) generates a solution semigroup
$(\mathcal{S}_{2}(t))_{t\geq 0}$ in $C([-r,0],\mathbb{X}^{\alpha}_{1})$ that
has an exponential dichotomy. Then, by the equivalence of the norms this
yields that this solution semigroup has an exponential dichotomy in
$C([-r,0],\mathbb{X}^{\alpha}_{2})$ as well. For sufficiently small
$\varepsilon_{1}$ and $\varepsilon_{3}$ by Lemma 3.3 the Yosida distance
$d_{Y}(A_{0},A_{0}+A_{2})$ could be made as small as we like. Next, the Yosida
distance $d_{Y}(B_{0},B_{0}+B_{2})$ could be made as small as we like by
taking $\varepsilon_{5}$ and $\varepsilon_{6}$ sufficiently small. Finally,
for the Yosida distance $d_{Y}(\mathcal{G}_{0},\mathcal{G}_{2})$, where
$\mathcal{G}_{2}$ is the generator of the solution semgroup
$(\mathcal{S}_{2}(t))$, could be made as small as we like by taking
$\varepsilon_{1}$, $\varepsilon_{3}$, $\operatorname{Var}(\varepsilon_{4})$,
and $\varepsilon_{5}$ sufficiently small. By Theorem 3.2 this yields that for
sufficiently small $\varepsilon_{1}$, $\varepsilon_{3}$,
$\operatorname{Var}(\varepsilon_{4})$, and $\varepsilon_{5}$ the solution
semigroup $(\mathcal{S}_{2}(t))_{t\geq 0}$ has an exponential dichotomy in
$C([-r,0],\mathbb{X}^{\alpha}_{1})$. Since
$(\mathbb{X}^{\alpha}_{2},\|\cdot\|_{2,\alpha})$ and
$(\mathbb{X}^{\alpha}_{1},\|\cdot\|_{1,\alpha})$ are equivalent this implies
that $(\mathcal{S}_{2}(t))_{t\geq 0}$ has an exponential dichotomy in
$C([-r,0],\mathbb{X}^{\alpha}_{2})$. ∎

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