#  On asymptotic periodic solutions of fractional differential equations and
applications

Vu Trong Luong  VNU University of Education, Vietnam National University at
Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam<EMAIL_ADDRESS>,
Nguyen Duc Huy  VNU University of Education, Vietnam National University at
Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam<EMAIL_ADDRESS>,  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>and  Nguyen Ngoc Vien  Faculty of Foundations, Hai Duong
University, Hai Duong City, Vietnam<EMAIL_ADDRESS>
###### Abstract.

In this paper we study the asymptotic behavior of solutions of fractional
differential equations of the form
$D^{\alpha}_{C}u(t)=Au(t)+f(t),u(0)=x,0<\alpha\leq 1,(*)$ where
$D^{\alpha}_{C}u(t)$ is the derivative of the function $u$ in the Caputo’s
sense, $A$ is a linear operator in a Banach space $\mathbb{X}$ that may be
unbounded and $f$ satisfies the property that
$\lim_{t\to\infty}(f(t+1)-f(t))=0$ which we will call asymptotic
$1$-periodicity. By using the spectral theory of functions on the half line we
derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give
sufficient conditions in terms of spectral properties of the operator $A$ for
all asymptotic mild solutions of Eq. (*) to be asymptotic $1$-periodic, or
there exists an asymptotic mild solution that is asymptotic $1$-periodic.

###### Key words and phrases:

Asymptotic behavior, polynomial boundedness, stability, spectrum of a function
on the half line

###### 2010 Mathematics Subject Classification:

Primary: 34K37, 34G10; Secondary: 34K30, 45J05

A part of this article was written while the first author was visiting Vietnam
Institute for Advanced Study in Mathematics (VIASM). He would like to thank
the Institute for its hospitality.

The authors are grateful to the anonymous referee for his carefully reading
the manuscript and suggestions to improve its presentation

##  1\. Introduction

We are interested in the asymptotic behavior of solutions to linear fractional
differential equations of the form

(1.1) $D^{\alpha}_{C}u(t)=Au(t)+f(t),u(0)=x,0<\alpha\leq 1,$

where $D^{\alpha}_{C}u(t)$ is the derivative of the function $u$ in the
Caputo’s sense, $A$ is a linear operator in a Banach space $\mathbb{X}$ that
may be unbounded and $f$ satisfies the property that
$\lim_{t\to\infty}(f(t+1)-f(t))=0$ which we will call ”asymptotic
$1$-periodicity”.

In recent decades fractional differential equations are of growing interests
as this kind of equations allows us to model more complex processes. We refer
the reader to the monographs [11, 15] for an account of applications in
Physics and Engineering. For general results and concepts in abstract spaces
the reader is referred to [3, 6, 7].

Asymptotic behavior of solutions of differential equations is a central topic
in the qualitative theory of differential equations and dynamical systems that
is rich of results and ideas. The very first behavior of solutions that are of
interest to mathematicians because of its applicability is the periodicity of
solutions. There are many classic results in this direction that are models
for subsequent researches. Among many such works we refer the reader to very
well known results by Massera [20] and Katznelson-Tsafriri [13] due to their
very different techniques of study. These results are recently extended to
wider classes of equations and solutions in [19] and [22]. It is natural to
ask if the concepts and results in these recent works could be extended to a
larger class of Eq.(1.1). In this paper we will take an attempt to address
this question.

More specifically, we look for conditions under which some (or all) solutions
$x(\cdot)$ are asymptotic $1$-periodic. We will prove an analog of the
Katznelson-Tzafiri Theorem and Massera Theorem for this kind of solutions for
Eq.(1.1). Our method is to use the spectral theory of functions and the
technique of decomposition rather than the period map as it is often used in
the area since the equation is actually not periodic and its homogeneous
equation does not generate a dynamical system even in the autonomous case. In
the recent paper [19] similar problems in case $\alpha=1$ were studied for
evolution equations with periodic coefficients based on a spectral theory of
functions. Namely, in [19] the spectrum of a function $f\in
BUC(\mathbb{R}^{+},\mathbb{X})$ (the space of all uniformly continuous and
bounded functions on $[0,\infty)$) is defined to be the subset $\sigma(f)$ of
the unit circle $\Gamma$ in the complex plane consisting of all points
$\xi_{0}$ such that the complex function $R(\lambda,\bar{S})\bar{f}$ has no
analytic extension to any neighborhood of $\xi_{0}$. Here $S$ is the
translation by $1$ and $\bar{S}$ is the induced operator by $S$ in the
quotient space
$\mathbb{Y}:=BUC(\mathbb{R}^{+},\mathbb{X})/C_{0}(\mathbb{R}^{+},\mathbb{X})$
(the space of continuous functions approaching zero at infinity and
$R(\lambda,\bar{S})$ stands for the resolvent of $\bar{S}$. In [19], the
asymptotic $1$-periodicity of a function $f\in BUC(\mathbb{R}^{+},\mathbb{X})$
is characterized by the property $\sigma(f)\subset\\{1\\}$. Note that for
fractional equations (1.1), using the transform $R(\lambda,\bar{S})\bar{f}$ to
capture the spectrum of an asymptotic mild solution of (1.1) does not work. As
shown in [18], the spectrum $sp(f)$ determined by the transform
$R(\lambda,\bar{\mathcal{D}})\bar{f}$, where ${\mathcal{D}}$ is the
differential operator $d/dt$ and $\bar{\mathcal{D}}$ stands for the induced
operator by ${\mathcal{D}}$ in $\mathbb{Y}$, can be used to study the
asymptotic behavior of Eq.(1.1). Our next step is to characterize the
asymptotic $1$-periodicity in terms of $sp(f)$. This will be done by proving
the identity $\sigma(f)=\overline{e^{i\cdot sp(f)}}$ that is based on the Weak
Spectral Mapping Theorem for $C_{0}$-groups of isometries. This
characterization of the asymptotic $1$-periodicity allows us to study the
asymptotic $1$-periodicity of all asymptotic mild solutions existence of
Eq.(1.1), an analog of the Katznelson-Tzifriri for fractional equations. We
also prove an analog of Massera Theorem for this class of equations based on
the technique of decomposition. Our main results are stated in Theorems 3.9,
3.12 and 3.16.

##  2\. Preliminaries and Notations

###  2.1. Notations

Throughout this paper we will denote by $\mathbb{R},\mathbb{R}^{+},\mathbb{C}$
the real line $(-\infty,\infty)$, half line $[0,+\infty)$ and the complex
plane. For $J$ being either $\mathbb{R}$ or $\mathbb{R}^{+}$, the notation
$BUC(J,\mathbb{X})$ stands for the function space of all bounded and uniformly
continuous functions taking values in a (complex) Banach space $\mathbb{X}$
with sup-norm. Throughout the paper we will use the following notations

$\displaystyle C_{0}(\mathbb{R}^{+},\mathbb{X})$ $\displaystyle:=\\{f\in
BUC(\mathbb{R}^{+},\mathbb{X}):\ \lim_{t\to\infty}\|f(t)\|=0\\}$
$\displaystyle g_{\alpha}(t)$
$\displaystyle:=\frac{t^{\alpha-1}}{\Gamma(\alpha)},t>0,\alpha>0.$

For a complex number $z$, $\Re z$ denotes its real part and $\Im z$ stands for
its imaginary part. In this paper the single valued power function
$\lambda^{\alpha}$ of the complex variable $\lambda$ is uniquely defined as
$\lambda^{\alpha}=|\lambda|^{\alpha}e^{i\alpha\cdot arg(\lambda)}$, with
$-\pi<arg(\lambda)<\pi$.

###  2.2. Fractional differentiation in Caputo’s sense

Let $\alpha>0,t\geq a,$ and $a$ is a fixed number. Then, the fractional
operator

(2.1) $\displaystyle J^{\alpha}_{a}u(t):=(g_{\alpha}\ast
u)(t)=\displaystyle\int_{a}^{t}g_{\alpha}(t-\tau)u(\tau)d\tau$

is called fractional Riemann-Liouville integral of degree $\alpha$. The
function

$D^{\alpha}_{C}u(t):=\begin{cases}J^{n-\alpha}u^{(n)}(t)=\dfrac{1}{\Gamma(n-\alpha)}\displaystyle\int_{a}^{t}\dfrac{u^{(n)}(\tau)}{(t-\tau)^{\alpha+1-n}}d\tau,&{n-1}<{\alpha}<n\in\mathbb{N},\\\
u^{(n)}(t),&\alpha=n\in\mathbb{N},\end{cases}$

is called the fractional derivative in Caputo’s sense of degree $\alpha$. By
this notation we have for $0<\alpha\leq 1$

$J^{\alpha}_{a}D^{\alpha}_{C}u(t)=u(t)-u(a).$

###  2.3. Cauchy Problem

For a fixed $0<\alpha\leq 1,$ consider the Cauchy problem

(2.2) $D^{\alpha}_{C}u(t)=Au(t),u(0)=x,$

where $A$ is generally an unbounded linear operator. The well-posedness of
(2.2) is equivalent to that of the problem

(2.3) $u(t)=x+\int_{0}^{t}g_{\alpha}(t-s)Au(s)ds.$

The reader is referred to the monograph [26] for an extensive study of the
well-posedness of this kind of equations when $A$ is generally an unbounded
operator. Recent extensions for more general equations can be found in [14]
and their references for more details.

Let us consider inhomogeneous linear equations of the form

(2.4) $D^{\alpha}_{C}u(t)=Au(t)+f(t),t\geq 0,$

where $f\in BUC(\mathbb{R}^{+},\mathbb{X})$ is given.

######  Definition 2.1.

A mild solution $u$ of Eq.(2.4) on $\mathbb{R}^{+}$ is a continuous function
on $\mathbb{R}^{+}$ such that, for each $t\in\mathbb{R}^{+}$,
$J^{\alpha}u(t)\in D(A)$ and

$u(t)=AJ^{\alpha}u(t)+J^{\alpha}f(t)+u(0).$

###  2.4. A spectral theory of bounded Functions

In this paper we will use the spectral theory of functions defined on the half
line as it is outlined in [18]. The spectrum of a bounded function on the half
line is defined as below:

Let $\mathcal{D}$ denote the differentiation operator $d/dt$ in
$BUC(\mathbb{R}^{+},\mathbb{X})$ with domain

$D(\mathcal{D})=\\{f\in BUC(\mathbb{R}^{+},\mathbb{X}):\exists f^{\prime},\
f^{\prime}\in BUC(\mathbb{R}^{+},\mathbb{X})\\}.$

######  Lemma 2.2.

The translation semigroup $(S(t)_{t\geq 0})$ in
$BUC(\mathbb{R}^{+},\mathbb{X})$ is strongly continuous with ${\mathcal{D}}$
as its infinitesimal generator.

###### Proof.

This is a well known result (see e.g. [10]). ∎

###  2.5. Operator $\tilde{\mathcal{D}}$

In this subsection we recall some construction made in [18]. Since
$C_{0}(\mathbb{R}^{+},\mathbb{X})$ is a closed subspace
$BUC(\mathbb{R}^{+},\mathbb{X})$, and is invariant under the translation
semigroup $(S(t)_{t\geq 0})$. In the space $BUC(\mathbb{R}^{+},\mathbb{X})$ we
introduce the following relation $R$:

(2.5) $f\ R\ g\ \mbox{if and only if}\ \ f-g\in
C_{0}(\mathbb{R}^{+},\mathbb{X}).$

This is an equivalence relation and the quotient space
$\mathbb{Y}:=BUC(\mathbb{R}^{+},\mathbb{X})/R$ is a Banach space. We will also
denote the norm in this quotient space $\mathbb{Y}$ by $\|\cdot\|$ if it does
not cause any confusion.

The class containing $f\in BUC(\mathbb{R}^{+},\mathbb{X})$ will be denoted by
$\tilde{f}$. Define $\tilde{\mathcal{D}}$ in
$\mathbb{Y}=BUC(\mathbb{R}^{+},\mathbb{X})/R$ as follows:

(2.6) $\displaystyle D(\tilde{\mathcal{D}})$ $\displaystyle:=$
$\displaystyle\\{\tilde{f}\in\mathbb{Y}:\exists u\in\tilde{f},u\in
D(\mathcal{D})\\}.$

If $f\in D(\tilde{\mathcal{D}})$, we set

(2.7) $\tilde{D}\tilde{f}:=\widetilde{\mathcal{D}u}$

for some $u\in\tilde{f}$. The following lemma will show that this $\tilde{D}$
is well defined as an operator in $\mathbb{Y}$.

######  Lemma 2.3.

With the above notations, $\tilde{\mathcal{D}}$ is a well defined single
valued linear operator in $\mathbb{Y}$.

###### Proof.

For the proof see [18]. ∎

For each given $f\in BUC(\mathbb{R}^{+},\mathbb{X})$ consider the following
complex function $\hat{f}(\lambda)$ in $\lambda$ defined as

(2.8) $\hat{f}(\lambda):=(\lambda-\tilde{\mathcal{D}})^{-1}\tilde{f}.$

######  Definition 2.4.

The set of all points $\xi_{0}\in\mathbb{R}$ such that $\hat{f}(\lambda)$ has
no analytic extension to any neighborhood of $i\xi_{0}$ is defined to be the
spectrum of $f$, denoted by $sp(f)$.

##  3\. Main results

###  3.1. Spectral characterization of asymptotic $1$-periodicity

######  Definition 3.1.

Let $p$ be a given real number in $[0,2\pi)$. A function $g\in
BUC(\mathbb{R}^{+},\mathbb{X})$ is said to be an asymptotic Bloch $1$-periodic
function of type $p$ if

$\lim_{t\to\infty}(g(t+1)-e^{ip}g(t))=0.$

If $p=0$, an asymptotic Bloch $1$-periodic function $g$ of type $p$ will be
called an asymptotic $1$-periodic function. When $p=\pi$ we call the function
asymptotic anti $1$-periodic.

Below we will show that the spectrum of a function $f\in
BUC(\mathbb{R}^{+},\mathbb{X})$ can be determined as the spectrum of a linear
operator. Let us define ${\mathcal{M}}_{f}$ as the closure of the linear space
spanned by the set $\\{\tilde{S}(t)f:\ t\in\mathbb{R}\\}\subset\mathbb{Y}$.
Obviously $\bar{S}(t)$ leaves ${\mathcal{M}}_{f}$ invariant for every
$t\in\mathbb{R}$. We consider the $C_{0}$-group of isometries
$\\{\tilde{S}|_{{\mathcal{M}}_{f}},t\in\mathbb{R}\\}$. The generator of this
group will be denoted by $\tilde{D}|_{{\mathcal{M}}_{f}}$.

######  Lemma 3.2.

For each $f\in BUC(\mathbb{R}^{+},\mathbb{X})$, we have

(3.1) $i\cdot sp(f)=\sigma(\tilde{\mathcal{D}}|_{{\mathcal{M}}_{f}}).$

###### Proof.

If $\lambda_{0}\in\rho(\tilde{D}|_{{\mathcal{M}}_{f}})$, then, for $\lambda\in
U(\lambda_{0})\backslash i\mathbb{R}$, where $U(\lambda_{0})$ is a small
neighborhood of $\lambda_{0}$

$\hat{f}(\lambda)=(\lambda-\tilde{\mathcal{D}})^{-1}\tilde{f}=(\lambda-\tilde{\mathcal{D}}_{{\mathcal{M}}_{f}})^{-1}\tilde{f}.$

Therefore, $\hat{f}(\lambda)$ has
$(\lambda-\tilde{\mathcal{D}}_{{\mathcal{M}}_{f}})^{-1}\tilde{f}$ as an
analytic extension to a neighborhood of $\lambda_{0}$. That means
$\lambda_{0}\not\in i\cdot sp(f)$.

Conversely, let $\lambda_{0}\in i\mathbb{R}\backslash i\cdot sp(f)$. Then,
$\hat{f}(\lambda)$ has an analytic extension to a connected neighborhood
$U(\lambda_{0})$ of $\lambda_{0}$. We will show that the equation

(3.2) $\lambda_{o}y-\tilde{\mathcal{D}}y=w$

has a unique solution $y\in{\mathcal{M}}_{f}$ for each
$w\in{\mathcal{M}}_{f}$. First, we show that Eq.(3.2) has a solution. For
$\lambda\in U(\lambda_{0})\backslash i\mathbb{R}$ such that $\Re\lambda>0$,
using the formula

$R(\lambda,\tilde{\mathcal{D}})\tilde{f}=\int^{\infty}_{0}e^{-\lambda
t}\tilde{S}(t)\tilde{f}dt,$

we see that
$\hat{f}(\lambda)=R(\lambda,\tilde{D})\tilde{f}\in{\mathcal{M}}_{f}$.
Subsequently, $\hat{f}(\lambda_{0})\in{\mathcal{M}}_{f}$. As the operator
$\tilde{D}$ is a closed operator and $\hat{f}(\lambda)$ is analytic at
$\lambda_{0}$, we see that

(3.3)
$\lambda_{0}\hat{f}(\lambda_{0})-\tilde{\mathcal{D}}\hat{f}(\lambda_{0})=w.$

Next, we will show that the solution of Eq.(3.2) is unique for each given
$w\in{\mathcal{M}}_{f}$. This is equivalent to show that the homogeneous
equation has only trivial solution. Indeed, suppose
$y_{0}\in{\mathcal{M}}_{f}$ is a solution of

$\lambda_{0}y_{0}-\tilde{\mathcal{D}}y_{0}=0.$

Using the maximum modulus principle of holomorphic functions as in [5, Proof
of Theorem 2.2,p. 2074] and then the Vitali’s Theorem on convergence of
sequences of holomorphic functions we can show that for each
$w\in{\mathcal{M}}_{f}$, the function $\hat{w}(\lambda)$ has an analytic
extension in the connected neighborhood $U(\lambda_{0})$ of $\lambda_{0}$.
Next, for $\lambda\in U(\lambda_{0})\backslash i\mathbb{R}$ by the identity

$R(\lambda,\tilde{\mathcal{D}})(\lambda-\tilde{\mathcal{D}})y_{0}=y_{0}$

we have

$\lambda
R(\lambda,\tilde{\mathcal{D}})y_{0}-y_{0}=R(\lambda,\tilde{\mathcal{D}})\tilde{\mathcal{D}}y_{0}=R(\lambda,\tilde{\mathcal{D}})\lambda_{0}y_{0}.$

Hence

(3.4)
$\hat{y}_{0}(\lambda)=R(\lambda,\tilde{\mathcal{D}})y_{0}=\frac{y_{0}}{\lambda-\lambda_{0}}.$

This function has an analytic extension to a neighborhood of $\lambda_{0}$ if
and only if $y_{0}=0$. That is, $\lambda_{0}$ is in
$\rho(\tilde{\mathcal{D}}|_{{\mathcal{M}}f})$. This completes the proof of the
lemma. ∎

Recall that in [22] the ”circular” spectrum $\sigma(f)$ of $f\in
BUC(\mathbb{R}^{+},\mathbb{X})$ is defined to be the set of
$\lambda_{0}\in\Gamma$ such that the complex function
$R(\lambda,\tilde{S}(1))\tilde{f}$ has no analytic extension to any
neighborhood of $\lambda_{0}$. In the same way as [21, Lemma 2.7] we can show
that

######  Lemma 3.3.

For $f\in BUC(\mathbb{R}^{+},\mathbb{X})$, the following is valid

(3.5) $\sigma(f)=\sigma(\tilde{S}(1)|_{{\mathcal{M}}_{f}}).$

######  Corollary 3.4.

For $f\in BUC(\mathbb{R}^{+},\mathbb{X})$ we have

(3.6) $\sigma(f)=\overline{e^{i\cdot{\rm sp}(f)}},$

where the overlining means the closure in the topology of the complex plane.

###### Proof.

Since $\tilde{S}(t),t\in\mathbb{R}$ is a $C_{0}$-group of isometries, by the
Weak Spectral Mapping Theorem (see e.g. [10, Theorem 3.16, p. 283]) the
identity (3.6) is valid. ∎

In [19] it is proved that a function $g\in BUC(\mathbb{R}^{+},\mathbb{X})$ is
an asymptotic Bloch $1$-periodic function of type $p$ if and only if
$\sigma(g)\subset\\{e^{ip}\\}$. In the following another characterization is
given in term of spectrum $sp(f)$ as in many circumstances it is easier to
estimate this spectrum than $\sigma(f)$.

######  Theorem 3.5.

Let $g\in BUC(\mathbb{R}^{+},\mathbb{X})$. Then,

  1. i)

If $\xi_{0}$ is an isolated point in $sp(g)$, then $i\xi_{0}$ is either
removable or a pole of ${\hat{g}(\lambda)}$ of order less than $1$;

  2. ii)

If $sp(g)=\emptyset$, then $g\in C_{0}(\mathbb{R}^{+},\mathbb{X})$;

  3. iii)

$sp(g)$ is a closed subset of $\mathbb{R}$;

  4. iv)

$g$ is anasymptotic Bloch $1$-periodic function of type $p$ if and only if
$sp(g)\subset p+2\pi\mathbb{Z}$.

###### Proof.

For the proofs of (i), (ii) and (iii) see [18]. For (iv) note that as shown in
[19], $f$ is an asymptotic Bloch $1$-periodic function of type $p$ if and only
if $\sigma(f)\subset\\{e^{ip}\\}$. By Collorary 3.4, this is equivalent to
$sp(f)\subset p+2\pi\mathbb{Z}$. ∎

###  3.2. Applications to the asymptotic behavior of solutions of Eq(1.1)

We are going to apply the spectral theory of bounded functions in the previous
section to study the asymptotic behavior of mild solutions of Eq.(2.2) with
$0<\alpha\leq 1$.

######  Definition 3.6.

A function $u\in BUC(\mathbb{R}^{+},\mathbb{X})$ is said to be an asymptotic
mild solution of Eq.(2.2) if there exists a function $\epsilon(\cdot)\in
C_{0}(\mathbb{R}^{+},\mathbb{X})$ such that $u$ is a mild solution of the
equation

(3.7) $D^{\alpha}_{C}u(t)=Au(t)+f(t)+\epsilon(t),t\geq 0.$

Below we will denote by $\rho(A,\alpha)$ the set of all $\xi_{0}\in\mathbb{C}$
such that $(\xi_{0}^{\alpha}-A)$ has an inverse $(\xi_{0}^{\alpha}-A)^{-1}$
that is analytic in a neighborhood of $\xi_{0}$, and by
$\Sigma_{i}(A,\alpha):=i\mathbb{R}\backslash\rho(A,\alpha)$.

We will use a new notation

(3.8) $R_{\alpha}(\lambda,A):=\lambda^{\alpha-1}(\lambda^{\alpha}-A)^{-1}.$

The lemma below is actually stated in [1, Lemma 2.2]. Below is an adaptation
made in [18, Lemma 4.1]

######  Lemma 3.7.

Let $u\in BUC(\mathbb{R}^{+},\mathbb{X})$, $\xi_{0}\in\mathbb{R}$, and let the
function $G(\lambda)$ (in $\lambda$) be an analytic extension of the function
$\hat{u}(\lambda)=(\lambda-\tilde{\mathcal{D}})^{-1}\tilde{u}$ with
$\Re\lambda>0$ on the open disk $B(i\xi_{0},r)$ with some positive $r$. Then,
$G(\lambda)=\hat{u}(\lambda)$ for $\Re\lambda<0$ on a disk $B(i\xi_{0},r)$.

######  Corollary 3.8.

Let $u\in BUC(\mathbb{R}^{+},\mathbb{X})$ be an asymptotic mild solution of
Eq.(2.2), where $f\in BUC(\mathbb{R}^{+},\mathbb{X})$. Then,

(3.9) $i\cdot sp(u)\subset\Sigma_{i}(A,\alpha)\cup i\cdot sp(f).$

###### Proof.

See [18, Lemma 4.2]. ∎

The main results of this paper are the following:

######  Theorem 3.9.

Assume that $\Sigma_{i}(A,\alpha)\subset 2i\pi\mathbb{Z}$
($\Sigma_{i}(A,\alpha)\subset(2\mathbb{Z}+1)i\pi$, respectively) and
$sp(f)\subset 2\pi\mathbb{Z}$ ($sp(f)\subset(2\mathbb{Z}+1)\pi$,
respectively). Then, every asymptotic mild solution of Eq.(2.2) is an
asymptotic $1$ -periodic solution (an asymptotic anti $1$ -periodic solution,
respectively).

###### Proof.

By Corollary 3.8 for every asymptotic mild solution $u\in
BUC(\mathbb{R}^{+},\mathbb{R})$ has the property that

$i\cdot sp(u)\subset\Sigma_{i}(A,\alpha)\cup i\cdot sp(f).$

Therefore, $sp(u)\subset 2\pi\mathbb{Z}$ if both $\Sigma_{i}(A,\alpha)\cup
i\cdot sp(f)$ are parts of $2i\pi\mathbb{Z}$ , so by Theorem 3.5, $u$ is
asymptotic $1$-periodic. The case of asymptotic anti $1$-periodicity is
treated in the same manner. ∎

######  Lemma 3.10.

For $\alpha\in[0,1)$ the following assertions are valid

  1. i)

The operator $J^{\alpha}$ maps $C_{0}(\mathbb{R}^{+},\mathbb{X})\to
C_{0}(\mathbb{R}^{+},\mathbb{X})$, so it induces a linear bounded operator
$\tilde{J}^{\alpha}$ from $\mathbb{Y}\to\mathbb{Y}$.

  2. ii)

Let $u\in BUC(\mathbb{R}^{+},\mathbb{X})$ be an asymptotic mild solution of
Eq.(2.2) and $A$ be bounded linear operator. Then,

(3.10)
$\tilde{S}\tilde{u}=\tilde{A}\tilde{J^{\alpha}}\tilde{S}\tilde{u}+\tilde{S}\tilde{J}^{\alpha}\tilde{f}.$

###### Proof.

Part (i): The proof is obvious.
Part (ii):

$\displaystyle\left[Su\right](t)$ $\displaystyle=u(t+1)$
$\displaystyle=A\int^{t+1}_{0}g_{\alpha}(t+1-\tau)u(\tau)d\tau+\int^{t+1}_{0}g_{\alpha}(t+1-\tau)[f(\tau)+\epsilon(\tau)]d\tau$
$\displaystyle=A\left[\int^{t+1}_{1}g_{\alpha}(t+1-\tau)u(\tau)d\tau+\int^{1}_{0}g_{\alpha}(t+1-\tau)u(\tau)d\tau\right]$
$\displaystyle\hskip
14.51074pt+\int^{t+1}_{1}g_{\alpha}(t+1-\tau)f(\tau)d\tau+\int^{1}_{0}g_{\alpha}(t+1-\tau)[f(\tau)+\epsilon(\tau)]d\tau$
$\displaystyle=A\left[\left[J^{\alpha}Su\right](t)+\int^{1}_{0}\frac{(t+1-\tau)^{\alpha-1}}{\Gamma(\alpha)}u(\tau)d\tau\right]$
$\displaystyle\hskip
28.45274pt+J^{\alpha}S[f+\epsilon](t)+\int^{1}_{0}\frac{(t+1-\tau)^{\alpha-1}}{\Gamma(\alpha)}[f(\tau)+\epsilon(\tau)]d\tau$

Since $A$ is bounded and

$\displaystyle\lim_{t\to\infty}\int^{1}_{0}\frac{(t+1-\tau)^{\alpha-1}}{\Gamma(\alpha)}[f(\tau)+\epsilon(\tau)]d\tau=0,$
$\displaystyle\lim_{t\to\infty}J^{\alpha}S\epsilon(t)=0.$

(3.10) follows. ∎

######  Lemma 3.11.

Let $u\in BUC(\mathbb{R}^{+},\mathbb{X})$ and $A$ be bounded linear operator.
Assume further that $u$ is an asymptotic mild solution of Eq.(2.2) if and only
if

(3.11)
$\tilde{u}=\tilde{A}\tilde{J^{\alpha}}\tilde{u}+\tilde{J}^{\alpha}\tilde{f}.$

###### Proof.

It suffices to show that if $u\in BUC(\mathbb{R}^{+},\mathbb{X})$ satisfies
(3.11), then it must be an asymptotic mild solution of Eq.(2.2). In fact,
there are functions $\epsilon_{i}(\cdot)\in C_{0}(\mathbb{R}^{+},\mathbb{X})$
such that

(3.12) $\displaystyle u(t)+\epsilon_{1}(t)$
$\displaystyle=AJ^{\alpha}[u+\epsilon_{1}](t)+\epsilon_{2}(t)+J^{\alpha}f(t)+\epsilon_{3}(t)$

Therefore, as $A$ is a bounded linear operator,

(3.13) $\displaystyle u(t)+$
$\displaystyle=AJ^{\alpha}u(t)+J^{\alpha}f(t)+\epsilon_{4}(t),$

where
$\epsilon_{4}(t):=AJ^{\alpha}\epsilon_{1}(t)-\epsilon_{1}(t)+\epsilon_{2}(t)+\epsilon_{3}(t)$.
as $A$ is a bounded linear operator, and $\tilde{J}^{\alpha}$ are bounded,
$\epsilon_{4}\in C_{0}(\mathbb{R}^{+},\mathbb{X})$. ∎

######  Theorem 3.12.

Assume that $A$ is a bounded operator,
$e^{\Sigma_{i}(A,\alpha)}\backslash\\{1\\}$ is closed, and $f$ is asymptotic
$1$-periodic. Then, if Eq.(2.2) has an asymptotic 1-perioidc solution if and
only if it has a bounded uniformly continuous asymptotic solution on
$\mathbb{R}^{+}$.

###### Proof.

Consider the operator

$L\tilde{u}:=\tilde{u}-(\tilde{A}\tilde{J^{\alpha}}\tilde{u}+\tilde{J}^{\alpha}\tilde{f}).$

It is clear that $u\in BUC(\mathbb{R}^{+},\mathbb{X})$ is an asymptotic mild
solution of Eq.(2.2) if and only if $L\tilde{u}=0$. Set
$\Lambda:=e^{\Sigma_{i}(A,\alpha)}\cup\\{1\\}$,
$\Lambda_{1}:=e^{\Sigma_{i}(A,\alpha)}\backslash\\{1\\}$ and
$\Lambda_{2}:=\\{1\\}$. Then, by the assumption $\Lambda_{1}$ and
$\Lambda_{2}$ are two disjoint closed subset of the unit circle $\Gamma$. By
Corollary 3.8, $\sigma(u)\subset\Lambda$. Moreover, by [19, Assertion (iii),
Proposition 3.4] there exists a projection
$P:\mathbb{Y}_{\Lambda}\to\mathbb{Y}_{\Lambda_{2}}$ that commutes with any
bounded operator that commutes with $\tilde{S}$. therefore, as
$\tilde{S}L=L\tilde{S}$, we have $PL=LP$. Consequently,

$\displaystyle 0$ $\displaystyle=PL\tilde{u}$ $\displaystyle=LP\tilde{u}$
$\displaystyle=P\tilde{u}-(\tilde{A}\tilde{J^{\alpha}}P\tilde{u}+\tilde{J}^{\alpha}\tilde{f}).$

That means, $P\tilde{u}$ contains an asymptotic mild solution of Eq.(2.2).
Since $P\tilde{u}\in\mathbb{Y}_{\Lambda_{2}}$ the spectrum of this solution
must be part of $\Lambda_{2}=\\{1\\}$. By [19, Proposition 3.4], this solution
is asymptotic $1$-periodic. ∎

######  Remark 3.13.

The above theorem is an analog version of the famous result in [20] for
fractional differential equations. Another analog for asymptotic
anti-$1$-periodic solutions can be obtained by using the set $\\{-1\\}$
instead of $\\{1\\}$ in the statements of the theorem.

###  3.3. The case with unbounded $A$

Now we can state an analog of Theorem 3.12 in the case of $A$ is an unbounded
operator. We assume that $A:D(A)\subset X\to X$ is a closed linear operator
which generates a $C_{0}$-semigroup $\\{T(t)\\}_{t\geq 0}$ in $X$ and
$\\{\lambda^{\alpha}:Re\lambda>0\\}\subset\rho(A)$. Taking the Laplace
transforms of both sides of (2.2) gives

(3.14) $\displaystyle\hat{u}(\lambda)$ $\displaystyle=$
$\displaystyle\hat{S}_{\alpha}(\lambda)x+\widehat{\big{(}(\cdot)^{\alpha-1}P_{\alpha}\big{)}}(\lambda)\cdot\hat{f}(\lambda).$

where
$\widehat{(\cdot)^{\alpha-1}P_{\alpha}}(\lambda)=(\lambda^{\alpha}-A)^{-1},\hat{S}_{\alpha}(\lambda)=\lambda^{\alpha-1}(\lambda^{\alpha}-A)^{-1}$.
The inversion of the Laplace transform shows that $u(t)$ has the following
form

$u(t)=S_{\alpha}(t)x+\int_{0}^{t}(t-s)^{\alpha-1}P_{\alpha}(t-s)f(s)ds.$

Hence, asymptotic mild solutions of (2.2) are defined as functions $u\in
BUC(\mathbb{R}^{+},\mathbb{X})$ satisfy

(3.15)
$u(t)=S_{\alpha}(t)x+\int_{0}^{t}(t-s)^{\alpha-1}P_{\alpha}(t-s)(f(s)+\epsilon(s))ds,\;t\geq
0.$

By the subordination principle (see [6]), $S_{\alpha}$ and
$P_{\alpha},\alpha\in(0,1]$, exist if $A$ generates a $C_{0}$\- semigroup
$\\{T(t)\\}_{t\geq 0}$. The explicit formulas of $S_{\alpha}$ and $P_{\alpha}$
were given in [6, 27]:

$\displaystyle S_{\alpha}(t)$
$\displaystyle=\int_{0}^{\infty}\Phi_{\alpha}(s)T(t^{\alpha}s)ds,$
$\displaystyle P_{\alpha}(t)$
$\displaystyle=\alpha\int_{0}^{\infty}s\Phi_{\alpha}(s)T(t^{\alpha}s)ds,$

where $\Phi_{\alpha}$ is a probability density function defined on
$(0,\infty)$, that is, $\Phi_{\alpha}(t)\geq 0$ and
$\int_{0}^{\infty}\Phi_{\alpha}(t)dt=1$. Assume that the semigroup
$\\{T(t)\\}_{t\geq 0}$ generated by $A$ is exponentially stable, i.e., there
are positive numbers $a,M$ such that

$\|T(t)\|\leq Me^{-at},t\geq 0.$

By the fact that (see [28])

$\int_{0}^{\infty}\Phi_{\alpha}(\theta)e^{-z\theta}d\theta=E_{\alpha,1}(-z),$
$\int_{0}^{\infty}\alpha\theta\Phi_{\alpha}(\theta)e^{-z\theta}d\theta=E_{\alpha,\alpha}(-z),$

we have

$\displaystyle\|S_{\alpha}(t)\|$
$\displaystyle\leq\int_{0}^{\infty}\Phi_{\alpha}(\theta)\|T(\theta
t^{\alpha})\|d\theta$
$\displaystyle\leq\int_{0}^{\infty}\Phi_{\alpha}(\theta)e^{-at^{\alpha}\theta}d\theta=E_{\alpha,1}(-at^{\alpha}),$

and

$\displaystyle\|P_{\alpha}(t)\|$
$\displaystyle\leq\alpha\int_{0}^{\infty}\theta\Phi_{\alpha}(\theta)\|T(\theta
t^{\alpha})\|d\theta$ $\displaystyle\leq
M\alpha\int_{0}^{\infty}\theta\Phi_{\alpha}(\theta)e^{-at^{\alpha}\theta}d\theta$
$\displaystyle=ME_{\alpha,\alpha}(-at^{\alpha}).$

where $E_{\alpha,\beta}$, the Mittag-Leffler function, is defined as follows

$E_{\alpha,\beta}(z)=\sum\limits_{n=0}^{\infty}\frac{z^{n}}{\Gamma(\alpha
n+\beta)},\;\alpha,\beta>0,\;z\in\mathbb{C}.$

For $\alpha\in(0,1),t\in\mathbb{R}$, these Mittag-Leffler functions have

$\displaystyle\lim\limits_{t\to-\infty}E_{\alpha,1}(t)=\lim\limits_{t\to-\infty}E_{\alpha,\alpha}(t)=0.$

From the estimates above, we obtain

$\|S_{\alpha}(t)\|,\|P_{\alpha}(t)\|\to 0,t\to\infty.$

and

$\displaystyle\int_{-\infty}^{t}(t-s)^{\alpha-1}\|P_{\alpha}(t-s)\|ds$
$\displaystyle\leq
M\int_{-\infty}^{t}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-a(t-s)^{\alpha})ds$
$\displaystyle\leq\frac{M}{a}E_{\alpha,1}(-a(t-s)^{\alpha})\bigg{|}_{-\infty}^{t}=\frac{M}{a}.$

It means that $S_{\alpha}(t)x\in C_{0}(\mathbb{R}^{+},\mathbb{X})$. By the
same argument as in the proof Lemma 3.11, we have the following result.

######  Lemma 3.14.

Assume that $A$ generate a exponentially stable semigroup and
$\\{\lambda^{\alpha}:Re\lambda>0\\}\subset\rho(A)$. Then, $u\in
BUC(\mathbb{R}^{+},\mathbb{X})$ is an asymptotic mild solution of Eq.(2.2) if
and only if

(3.16) $\tilde{u}=\tilde{F}_{\alpha}\tilde{f},$

where
$F_{\alpha}f(t)=\int\limits_{0}^{t}(t-s)^{\alpha-1}P_{\alpha}(t-s)f(s)ds,\;t>0.$

In the following we assume that $e^{\Sigma_{i}(A,\alpha)}\backslash\\{1\\}$ is
closed, and $f$ is asymptotic $1$-periodic. Then, if we use the notations in
the proof of Theorem 3.12 with $\Lambda:=e^{\Sigma_{i}(A,\alpha)}\cup\\{1\\}$,
$\Lambda_{1}:=e^{\Sigma(A,\alpha)}\backslash\\{1\\}$ and
$\Lambda_{2}:=\\{1\\}$ and the projection
$P:\mathbb{Y}_{\Lambda}\to\mathbb{Y}_{\Lambda_{2}}$.

######  Lemma 3.15.

Under the assumption of Lemma 3.14 we assume that
$e^{\Sigma_{i}(A,\alpha)}\backslash\\{1\\}$ is closed, and $f$ is asymptotic
$1$-periodic. Then, if $u\in BUC(\mathbb{R}^{+},\mathbb{X})$ is an asymptotic
mild solution of Eq.(2.2),

(3.17) $P\tilde{u}=\tilde{F}_{\alpha}\tilde{f}.$

###### Proof.

The operator $F_{\alpha}$ maps $C_{0}(\mathbb{R}^{+},\mathbb{X})\to
C_{0}(\mathbb{R}^{+},\mathbb{X})$, so it induces a linear bounded operator
$\tilde{F}_{\alpha}$ from $\mathbb{Y}\to\mathbb{Y}$. If $g\in
C_{0}(\mathbb{R}^{+},\mathbb{X})$, then, for each $\epsilon>0$, there exists a
number $T>0$ such that $\|g\|\leq\epsilon$ for all $s\geq T.$ Then for all
$t\geq 2T$, we have

$\displaystyle\|F_{\alpha}f(t)\|\leq$
$\displaystyle\|g\|\int_{0}^{t/2}(t-s)^{\alpha-1}\|P_{\alpha}(t-s)\|ds+\epsilon\int_{t/2}^{t}(t-s)^{\alpha-1}\|P_{\alpha}(t-s)\|ds$
$\displaystyle\leq$
$\displaystyle\|g\|\int_{0}^{t/2}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-a(t-s)^{\alpha})ds+\epsilon\frac{M}{a}$
$\displaystyle=$
$\displaystyle\|g\|(E_{\alpha,1}(-at^{\alpha})-E_{\alpha,1}(-a(t/2)^{\alpha}))+\epsilon\frac{M}{a}.$

Moreover,

$\displaystyle SF_{\alpha}f(t)=$
$\displaystyle\int_{0}^{t+1}(t+1-s)^{\alpha-1}P_{\alpha}(t+1-s)f(s)ds$
$\displaystyle=$
$\displaystyle\int_{1}^{t+1}(t+1-s)^{\alpha-1}P_{\alpha}(t+1-s)f(s)ds+\int_{0}^{1}(t+1-s)^{\alpha-1}P_{\alpha}(t+1-s)f(s)ds$
$\displaystyle=$ $\displaystyle F_{\alpha}Sf(t)+h(t).$

Note that

$\|h(t)\|=\|\int_{0}^{1}(t+1-s)^{\alpha-1}P_{\alpha}(t+1-s)f(s)ds\|\leq\|f\|(E_{\alpha,1}(-a(t+1)^{\alpha})-E_{\alpha,1}(-a(t)^{\alpha})),$

it means that $h\in C_{0}(\mathbb{R}^{+},\mathbb{X})$ and $\tilde{F}_{\alpha}$
commutes with $\tilde{S}$. Using notations in Theorem 3.12, it easy to see
that if $u\in BUC(\mathbb{R}^{+},\mathbb{X})$ is asymptotic mild solution of
(2.2), then

$\displaystyle 0=$ $\displaystyle P(\tilde{u}-\tilde{F}_{\alpha}\tilde{f})$
$\displaystyle=$ $\displaystyle P\tilde{u}-P\tilde{F}_{\alpha}\tilde{f}$
$\displaystyle=$ $\displaystyle P\tilde{u}-\tilde{F}_{\alpha}\tilde{f}.$

∎

As a consequence of Lemma 3.15 we note that
$\sigma(F_{\alpha}f)\subset\sigma(f).$ From arguments above, we also obtain
the following theorem.

######  Theorem 3.16.

Assume that $A$ generate a exponentially stable semigroup and
$\\{\lambda^{\alpha}:Re\lambda>0\\}\subset\rho(A)$,
$e^{\Sigma_{i}(A,\alpha)}\backslash\\{1\\}$ is closed, and $f$ is asymptotic
$1$-periodic. Then, if Eq.(2.2) has an asymptotic 1-periodic solution if and
only if it has a bounded uniformly continuous asymptotic solution on
$\mathbb{R}^{+}$.

###  3.4. Examples

######  Example 3.17.

Consider the initial value problem

(3.18) $\begin{cases}D^{\alpha}_{t}u(x,t)=au_{xx}(x,t)+f(x,t),\ \
x\in\Omega=(0,\pi),\ t>0,\cr u(0,t)=u(\pi,t)=0,\ \ t>0,\cr
u(x,0)=u_{0}(x),\;x\in\Omega,\end{cases}$

where $D^{\alpha}_{t}u(x,t)$ is the fractional derivative in Caputo’s sense in
$t$ of degree $\alpha\in(0,1)$, $u(x,t),f(x,t)$ are scalar-valued functions
such that $f(\cdot,t)\in L^{2}(\Omega)$ for each $t\in\mathbb{R}^{+}$,
$f:\mathbb{R}^{+}\ni t\mapsto f(\cdot,t)\in L^{2}(\Omega)$ is uniformly
continuous and bounded, and and $a>0$ is given. We define the space
${\mathbb{X}}:=L^{2}(\Omega)$ and $A:{\mathbb{X}}\to{\mathbb{X}}$ by the
formula

(3.19) $\begin{cases}\hskip 14.22636ptAy=ay^{\prime\prime},\cr
D(A)=H^{1}_{0}(\Omega)\cap H^{2}(\Omega).\end{cases}$

Problem 3.18 can be rewritten in the form of an evolution equation

(3.20) $D^{\alpha}_{t}u(t)=Au(t)+f(t),\ t\in\mathbb{R}^{+},u(0)=u_{0},$

where $u(t)\in{\mathbb{X}}$, $A$ is defined as above. As is well known the
spectrum of $A$ consists of eigenvalues that satisfy this equation

$\left\\{\begin{array}[]{l}\dfrac{d^{2}u}{dx^{2}}=\frac{1}{a}\lambda u\\\
u(0)=u(\pi)=0\end{array}\quad\text{ in }\Omega.\right.$

This implies

$\lambda_{n}=-an^{2},u_{n}=C\sin(nx),n\in\mathbb{N}$

Furthermore, $A$ generates a compact semigroup $S(t)$ with

$\|S(t)\|\leq M\cdot e^{-at},\forall t\geq 0$

According to the subordination principle, $A$ generates the subordinated
resolvent $S_{\alpha}(t),\alpha\in(0;1)$ such that
$\lim_{t\rightarrow\infty}\left\|S_{\alpha}(t)\right\|=0.$ By definition,

$\Sigma_{i}(A,\alpha)=\\{\lambda\in i\mathbb{R}:\
\left(\lambda^{\alpha}-A\right)^{-1}\ \mbox{does not exists as a bounded
operator}\\}.$

As

$\lambda^{\alpha}\in\sigma(A)\Longleftrightarrow\lambda^{\alpha}=-an^{2},n=1,2,..$

we have

$\begin{gathered}\lambda^{\alpha}=|\lambda|^{\alpha}\cdot
e^{i\alpha\arg\lambda}\\\
\lambda^{\alpha}=-an^{2}=an^{2}e^{i\pi},n=1,2,..\end{gathered}$

this is equivalent to

$\left\\{\begin{array}[]{ l }{|\lambda|^{\alpha}=an^{2},n=1,2,..}\\\
{\operatorname{arg}\lambda=\frac{\pi}{\alpha}}\end{array}\Rightarrow\left\\{\begin{array}[]{l}|\lambda|=a^{\frac{1}{\alpha}}n^{\frac{2}{\alpha}},n=1,2,..\\\
\arg\lambda=\frac{\pi}{\alpha}\end{array}\right.\right.$

If the choices are $\alpha=\frac{2}{3},a=\sqrt[3]{\pi^{2}}$, we get:

$\lambda_{n}=n^{3}\pi e^{i\cdot\frac{3\pi}{2}}=-in^{3}\pi,n=1,2,..$

Therefore

$\Sigma_{i}(A,2/3)=\left\\{-in^{3}\pi,n=1,2,..\right\\}$

and

$e^{\Sigma_{i}(A,2/3)}=\\{1\\}\cup\\{-1\\}.$

If $f(\cdot,t)$ is asymptotic $1$-periodic in $t$, then $e^{i\cdot
sp(f)}=\\{1\\}$. Therefore, by Theorem 3.16, there is an asymptotic 1-periodic
solution $u$ of (3.20) if and only if there exists an asymptotic mild solution
to Eq.(3.20). Similar conclusions can be made if $f$ is asymptotic $1$-anti-
periodic. This is an analog of Massera Theorem for the fractional differential
equation (3.20).

######  Example 3.18.

We consider Problem 3.18 again with different values of $\alpha$ to illustrate
a Katznelson-Tzafriri type result. We will assume that $\alpha$ is a number
that satisfies

$\frac{\pi}{\alpha}=\frac{\pi}{2}+k\pi,$

for a certain $k=1,2,...$ For instance,

$\alpha=\frac{2}{2k+1},$

where $k$ is a positive integer. Then

$\lambda_{n}=\left((2\pi)^{\frac{2}{2k+1}}\cdot
n^{2}\right)^{\frac{2k+1}{2}}\cdot e^{i\cdot{\frac{2k+1}{2}\pi}}$
$\lambda_{n}=\pm i2n^{(2k+1)}\pi=2mi\pi,$

where $m\in\mathbb{N}^{*}$.

Therefore, assuming that $f(\cdot)$ is asymptotic $1$-periodic, by Theorem
3.9, if $u\in BCU(\mathbb{R}^{+},\mathbb{X})$ is an asymptotic mild solution
of Eq. (3.20), $u$ must be asymptotic $1$-periodic.

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