Asymptotic Periodicity of Bounded Mild Solutions for Evolution Equations with Non-Densely Defined and Fractional Derivative

Abstract

In the present article, we establish conditions for the asymptotic periodicity of bounded mild solutions in two distinct cases of evolution equations. The first class involves non-densely defined operators, while the second class incorporates densely defined operators with fractional derivatives that generate a semigroup of contractions. Our method integrates the theory of spectral properties of uniformly bounded continuous functions defined on the positive real semi-axis. Additionally, we apply extrapolation theory to evolution equations with non-densely defined operators. To illustrate our main results, we provide a concrete example.

1. Introduction

One of the main issues facing the theory of evolution equations is the study of solution periodicity. Although in the recent literature there are many different kinds of fractional models [1].Based on the various methods that can be used in this vast field, we will concentrate on the problem given in [2] (Theorem 4.4). Indeed, Fourier’s law ensures the relationship of proportionality between the flux and the temperature gradient, more previously, if on the one hand the function $u=u(t,z)$ represents the temperature. However, the function $\varphi =\varphi (t,z)$ represents the flux, then we can write.

$${\varphi }_l(t,z)=-u_z(t,z).$$

(1)

Here, l denotes the local character. However, Equation (1) was exposed by various authors, for example, Pipkin Gurtin in 1968 [3] for memory fluxes. More recently, non-local fluxes, as proposed by Atanackovi have been considered, as discussed in [4]. This paper will focus on an evolution-type problem for the non-local flux law introduced in [5], which is described by the following equation

$${\varphi }_{nl}(t,z)=-{\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\int }_0^z{u}_z(t,\zeta ){(z-\zeta )}^{-\gamma }d\zeta ,$$

(2)

where $\Gamma$ is the gamma function given by $\Gamma \left(z\right)={\int }_0^{\infty }{t}^{z-1}exp(-t)dt,$ for all $z>0$. This equation ensures that the flux at any position z and time t is an infinitesimal sum of the local flux at every position between the extreme points of the slab 0 and z. Equation (2) can be written in terms of fractional derivatives.

$${\varphi }_{nl}(t,z)=-{𝒟}^{\gamma }u(t,z).$$

(3)

Recall that the balance equation is derived by the principle of thermodynamics

$${\displaystyle \frac{\partial u}{\partial t}}(t,z)=-{\displaystyle \frac{\partial \varphi }{\partial z}}(t,z)-G\left(t\right).$$

(4)

By inserting the non-local flux (3) into Equation (4), we rediscover the fractional diffusion equation in space as follows

$${\displaystyle \frac{\partial u}{\partial t}}(t,z)-{\displaystyle \frac{\partial }{\partial z}}{𝒟}^{\gamma }u(t,z)+G\left(t\right)u(t,z)=0,$$

(5)

which serves as the governing equation in the liquid phase, provided that $G\left(t\right)u(t,z)$ represents fluid friction at time-position $(t,z)$. If we set $A={\displaystyle \frac{\partial }{\partial z}}{𝒟}^{\gamma }$, Equation (5) becomes

$${\displaystyle \frac{\partial u}{\partial t}}(t,z)=(A+G\left(t\right))u(t,z),$$

(6)

Periodic solutions for various significant types of densely defined evolution equations have been established through classical methodologies. These include the application of the fixed point method (the reader can see [6,7,8,9,10,11]), the utilization of spectral theory of functions [12,13], or the ergodic approach. There are instances where we must address operators that are not densely defined, as mentioned in [14]. Numerous results concerning the uniqueness and existence of periodic solutions for evolution problems with non-dense domains have been derived [1,8,15,16]. In particular, in [17], M. Jazar and K. Ezzinbi introduced a new criterion based on the approach of Massera, which proves to be more expansive than the established exponential dichotomy. This criterion is utilized to establish the almost periodic and periodic existence of solutions for specific evolution problems. These type of equations have the form of that in (6) with initial conditions $u(0,z)=u_0$ given as follows

$$\left\{\begin{array}{c}{\displaystyle \frac{d}{dt}}u\left(t\right)=(A+G\left(t\right))u\left(t\right)+f\left(t\right),\mathrm{for}t\ge 0\hfill \\ \hfill \\ u\left(0\right)=u_0,\hfill \end{array}\right.$$

(7)

with $A:\mathcal{D}\left(A\right)\subset E⟶E$ linear operator that is a nondensely defined and E is a Banach space with A verifies the condition of Hille-Yosida

H₁. 

There exist ${\omega }_0\in \mathbb{R}$ and $M_0\ge 1$ such that $\left({\omega }_0,+\infty \right)\subset \rho \left(A\right)$ and

$$\begin{array}{c}\hfill \left|\mathcal{R}{(\xi ,A)}^n\right|\le {\displaystyle \frac{M_0}{{\left(\xi -{\omega }_0\right)}^n}},forn\in \mathbb{N}and\xi >{\omega }_0,\end{array}$$

where $\mathcal{R}(\xi ,A)={(\xi -A)}^{-1}$ and $\rho \left(A\right)$ is the resolvent set of A that is the set of all $\xi \in \mathbb{C}$ such that the operator $\xi -A$ has an inverse.

To make the problem (7) fairly feasible in relation to phenomena in physics related to a phase change in an infinite slab due to heat transfer, we take A${{\displaystyle \frac{\partial }{\partial z}}}^c\mathrm{=}{𝒟}^{\gamma }$ and $G\left(t\right)$ is a bounded linear operator on E for all $t\ge 0$, $f:{\mathbb{R}}^{+}\to E$ is a continuous bounded function that is almost periodic or 1-periodic.

In a recent publication [18], Luong et al. examined the case where the operator in (7) is densely defined and $A\left(t\right):=A+G\left(t\right)$ is the infinitesimal generator of ${\left\{{\mathcal{U}}_{t,s}\right\}}_{t\ge s\ge 0}$, the 1-periodic strongly continuous evolution process across the entire space E. Additionally, f considered as an asymptotically 1-periodic function, characterized by being bounded, continuous and satisfying $\underset{t\to \infty }{lim}(f\left(t\right)-f(t+1))=0$. (see, e.g., [19] and related references therein). We note that $u(·)$ is said to be an asymptotic solution to the problem (7) if there exists a continuous function $ϵ(·)$ with $\underset{t\to \infty }{lim}ϵ\left(t\right)=0$ and

$$\begin{array}{c}\hfill u^{\prime }\left(t\right)=(A+G\left(t\right))u\left(t\right)+ϵ\left(t\right)+f\left(t\right),\mathrm{for}\mathrm{all}t\ge 0.\end{array}$$

By applying the spectral theory of uniformly continuous functions on the half-line and the associated evolution semigroups in several spectral function spaces, Luong et al. proposed a new criterion for the unique existence of bounded solutions that are asymptotically 1-periodic on the half-line. Consequently, establishing the existence of asymptotically 1-periodic solutions is simplified to finding $u(·)$ with $\sigma \left(u\right(·\left)\right)\subset \left\{1\right\}$. The pursuit of such asymptotic solutions $u(·)$ satisfying the above condition can be accomplished by employing the evolution semigroup related with the equation $u^{\prime }\left(t\right)=A\left(t\right)u\left(t\right)$ within suitable function spaces. If the operator A lacks a dense definition, the linear component $A+G\left(t\right)$ fails to induce a strongly continuous evolution process throughout the entirety of the space E, thereby undermining the assurances provided in [18]. Additionally, the inhomogeneous term $f(·)$ spans the entire space E, whereas the values of the mild solution $u(·)$ are precisely within $E_0=\overline{\mathcal{D}\left(A\right)}$. To surmount these challenges, in this work we initially employ the extrapolation spaces theory to represent the mild solution of (7) in terms of an evolutionary process ${\left\{{\mathcal{U}}_{t,s}\right\}}_{t\ge s\ge 0}$ defined on a closed subspace E (for further elaboration, refer to [15] and the associated literature). Consequently, leveraging the periodicity and boundedness of some process and the circular spectrum of functions, we specify hypothesis under which the unique bounded solution of (7) becomes asymptotically periodic, a scenario suitable for densely defined non-autonomous linear operators.

The structure of this paper is outlined as follows: After this introduction, Section 2 offers a brief overview of the key notations, definitions, and properties of circular spectra of functions on the half-line. The main result of the paper is detailed in Section 3, which is divided into three subsections that address the asymptotic periodicity of solutions to non-autonomous evolution problems represented by (7), a special case with A densely defined operator but not $A\left(t\right)$ and finally an example to ensure the results.

2. Preliminaries

In this paper, the space $\mathcal{L}\left(E\right)$ consists of all linear and bounded operators on E, where E is a Banach space equipped with the sup norm. And ${\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E\right)$ the set of all continuous and bounded functions defined from ${\mathbb{R}}^{+}$ to E with the uniform convergence norm, and $C_0({\mathbb{R}}^{+},E)$ is the space of all functions belonging to ${\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E\right)$ and vanishing at infinity, and $\Gamma$ is the unit circle. Let $\mathcal{D}={\displaystyle \frac{d}{dt}}$ be the operator of differentiation in ${\mathcal{B}}_{uc}({\mathbb{R}}^{+},E)$, such that

$$\mathcal{D}\left(𝒟\right)=\left\{f\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0):f^{\prime }\mathrm{exists}\mathrm{and}{f}^{\prime }\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)\right\}.$$

be its domain. The semigroup of translation $\left(S{\left(t\right)}_{t\ge 0}\right)$ in the space ${\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$ is strongly continuous with $𝒟$ as its infinitesimal generator. To be convinced, you can see [20]. Since the subspace $C_0({\mathbb{R}}^{+},E)$ is closed in ${\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$ and is invariant under the translation semigroup $\left(S{\left(t\right)}_{t\ge 0}\right)$, so we can introduce the relation ≡ in ${\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$ by

$$h_1\equiv h_2\mathrm{equivalent}\mathrm{to}{h}_1-h_2\in C_0({\mathbb{R}}^{+},E_0).$$

It is an equivalence relation and let $\mathbb{E}:={\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)/\equiv$ is a Banach space, we denote also its norm by $\parallel .\parallel$ if there is no ambiguity. The class of an element f in ${\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E\right)$ will be denoted by $\tilde{f}$. Now let us define an operator $\widetilde{𝒟}$ in $\mathbb{E}$ by

$$\mathcal{D}\left(\widetilde{𝒟}\right)=\left\{\tilde{f}\in \mathbb{E}:\mathrm{exists}u\in \tilde{f},u\in \mathcal{D}\left(𝒟\right)\right\},$$

and for all $\tilde{f}\in \mathcal{D}\left(\widetilde{𝒟}\right)$, we define

$$\widetilde{𝒟}\tilde{f}=\widetilde{𝒟u},u\in \tilde{f}.$$

Lemma 1.

$\widetilde{𝒟}$ is single valued linear operator which is well defined in $\mathbb{E}$.

Proof. 

The linearity of $\widetilde{𝒟}$ is clear. We demonstrate that the operator $\widetilde{𝒟}$ is well-defined and single-valued. Let $\tilde{f}\in \mathcal{D}\left(\widetilde{𝒟}\right)$, we will show that $\widetilde{𝒟}\tilde{f}$ does not depend on the choice of the representative of the class $\tilde{f}$. Suppose that u and v are two representatives of $\tilde{f}$ such that $u,v\in \mathcal{D}\left(𝒟\right)$, then $\widetilde{𝒟u}=\widetilde{𝒟v}$, which means that $𝒟(u-v)\in C_0({\mathbb{R}}^{+},E)$. If we set $h:=u-v$, we have $u,v\in \tilde{f}$ then $h\in C_0({\mathbb{R}}^{+},E)$ and $h\in \mathcal{D}\left(𝒟\right)$.

Therefore,

$$\underset{t\to 0^{+}}{lim}{\displaystyle \frac{S\left(t\right)h-h}{t}}=𝒟h.$$

(8)

Remark that $𝒟h=𝒟(u-v)$. Which proves that $\widetilde{𝒟}$ is a well defined single valued operator. □

For any given $f\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E)$, we consider the complex function in $\lambda$ defined as

$$\widehat{f}\left(\lambda \right)={(\lambda -\widetilde{𝒟})}^{-1}\tilde{f}.$$

(9)

Definition 1.

Let $f\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E)$. The set of all points ${\xi }_0\in \mathbb{R}$ such that $\widehat{f}\left(\lambda \right)$ has no analytic extension to any neighborhood of $i{\xi }_0$ is defined to be the spectrum of f, denoted by $\sigma \left(f\right)$.

Several notions and results discussed here can be found in [18]. If S is the 1-translation operator, then it induces an operator on the quotient space that will be denoted by $\overline{S}$ and the above operator is an isometry, so $\sigma \left(\overline{S}\right)\subset \Gamma$. For each $u\in {\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)$ we consider the function $\mathcal{S}u\left(\lambda \right)$ in $\lambda \in \mathbb{C}\setminus \Gamma$ given by

$$\begin{array}{c}\hfill \mathcal{S}u\left(\lambda \right)={(\lambda -\overline{S})}^{-1}\overline{u},\lambda \in \mathbb{C}\setminus \Gamma .\end{array}$$

Definition 2.

Let $u\in {\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)$ The spectrum of a function of u (often called a circular spectrum) is determined as the set of all ${\xi }_0\in \Gamma$ for which $\mathcal{S}u\left(\lambda \right)$ has no analytic extension into any neighborhood of ${\xi }_0$ in the complex plane. It is noted by $\sigma \left(u\right)$.

The subsequent lemma provides justification for the introduction of these spectrum concepts.

Lemma 2.

([18]). Let $u\in {\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)$ be given. Then, for every $u\in {\mathcal{B}}_c\left({\mathbb{R}}^{+},E_0\right)$,

$$\begin{array}{c}\hfill \sigma \left(\mathcal{Q}u\right)\subset \sigma \left(u\right),\end{array}$$

where $\mathcal{Q}$ is an operator in ${\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)$ which commutes with S and leaves $C_0\left({\mathbb{R}}^{+},E_0\right)$ invariant.

Extrapolation space and Mild solutions. With the notation in the introduction, it is widely recognized that the restriction $A_0$ of A to $E_0$ generates a $C_0$-semigroup denoted by ${\left(T_0\left(t\right)\right)}_{t\ge 0}$ satisfying $\parallel T_0\left(t\right)\parallel \le M{e}^{\omega t}$, for all $t\ge 0$. In addition, for $\lambda$ an element of $\rho \left(A_0\right)$, $\mathcal{R}(\lambda ,A_0)$ is the restriction of $\mathcal{R}(\lambda ,A)$ to $E_0$. And we introduce the norm

$$\begin{array}{c}\hfill {\parallel z\parallel }_{-1}=\parallel \mathcal{R}(\mu ,A_0)z\parallel \end{array}$$

defined on $E_0$, where $\mu \in \rho \left(A\right)$ is fixed. The extrapolation space of $E_0$ with respect to A is completion $E_{-1}$ of $E_0$ with respect to ${\parallel .\parallel }_{-1}$, and the extrapolated semigroup ${\left(T_{-1}\left(t\right)\right)}_{t\ge 0}$ serves as the unique continuous extension of the operators $T_0\left(t\right),t\ge 0$, on $E_{-1}$. This strongly continuous semigroup has an infinitesimal generator $A_{-1}$ that represents the unique linear continuous extension of $A_0$ onto $\mathcal{L}(E_0,E_{-1})$. Consequently, $A_0$ and A constitute the components of $A_{-1}$ within $E_0$ and E, respectively. With this background, we proceed to provide the definition of a mild solution of (7).

Definition 3.

Let $u_0\in E_0$. An element $u\in C({\mathbb{R}}^{+},E_0)$ that satisfies the following equation

$$u\left(t\right)=T_0(t-s)u\left(s\right)+{\int }_s^t(f\left(\tau \right)+T_{-1}(t-\tau )\left(B\left(\tau \right)u\left(\tau \right)\right)d\tau ,forallt\ge s\ge 0.$$

(10)

is said to be a mild solution of (7).

Consider the problem

$$\left\{\begin{array}{c}{\displaystyle \frac{du}{dt}}=(A+G\left(t\right))u\left(t\right),t\ge 0\hfill \\ \hfill \\ u\left(0\right)=u_0\in E_0.\hfill \end{array}\right.$$

(11)

In the sequel, we need to the following hypothesis

H₂. 

$t\mapsto G\left(t\right)u$ is measurable for each $u\in E$,

H₃. 

$B(·)$ is 1-periodic operator.

Theorem 1.

([15]). Let $f\in L_{\mathrm{loc}}^1({\mathbb{R}}^{+},E)$ and $u\in E_0$. Then, there is at most one mild solution $u(·)\in C\left({\mathbb{R}}^{+},E_0\right)$ of (7) such that

$$\begin{array}{c}\hfill u\left(t\right)={\mathcal{U}}_{t,s}u\left(s\right)+\underset{\xi \to \infty }{lim}{\int }_s^t{\mathcal{U}}_{t,h}\xi \mathcal{R}(\xi ,A)f\left(h\right)dhfort\ge s\ge 0.\end{array}$$

Moreover, ${\displaystyle \underset{\xi \to \infty }{lim}{\int }_s^t{\mathcal{U}}_{t,h}\xi \mathcal{R}(\xi ,A)f\left(h\right)dh\in E_0}$ exists uniformly for $t\ge s$ in any compact sets of ${\mathbb{R}}^{+}$.

3. Main Results

3.1. Spectral Characterization of Asymptotic Periodic Functions

We impose a period that is equal to 1 and is not restrictive, but is used merely for the convenience of the reader. All results can be readily extended for any period.

Definition 4.

In the context of ${\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E\right)$, a function f is said to be asymptotically 1-periodic if it verifies the condition

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}(f(t+1)-f\left(t\right))=0.\end{array}$$

Remark 1.

The authors in [21] explored the notion of asymptotic periodicity of functions as defined in our Definition 4. However, an incompatibility arises when attempting to demonstrate the equivalence of this concept with the widely adopted definition of asymptotic 1-periodicity found in the literature, such that f is asymptotic periodic if

$$f\left(\tau \right)=\eta \left(\tau \right)+ϵ\left(\tau \right),$$

(12)

where ϵ is continuous with $\underset{t\to \infty }{lim}ϵ\left(t\right)=0$ and η is a continuous 1-periodic.

A simpler counterexample for asymptotic periodicity of Definition 4 is given by $f\left(t\right)=cos\sqrt{t},t\in {\mathbb{R}}^{+}$, we have, ${\displaystyle \underset{t\to \infty }{lim}(f(t+1)-f\left(t\right))=0.}$ If we take $f\left(t\right)=\eta \left(t\right)+ϵ\left(t\right)$, where $\eta$ is continuous 1-periodic and $\underset{t\to \infty }{lim}ϵ\left(t\right)=0$ then, we have

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}f\left(n\right)=\underset{n\to \infty }{lim}cos\sqrt{n}=\eta \left(1\right)\in \mathbb{R}.\end{array}$$

If we take $n=k^2$ in the last equation, it yields

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}cosk=\eta \left(1\right)\in \mathbb{R}.\end{array}$$

A simple argument of the standard analysis shows that this is impossible. Indeed, if this equality holds, then $lim\underset{k\to \infty }{lim}(cos(k+2)-cos\left(k\right))=0.$ Consequently,

$$\begin{array}{c}\hfill \underset{k\to \infty }{lim}2cos\left(1\right)cos(k+1)=0.\end{array}$$

Hence, we obtain that $\underset{k\to \infty }{lim}cosk=0$. Using a similar argument, we can also prove that $\underset{k\to \infty }{lim}sink=0$. Therefore, $1={lim}_{k\to \infty }\left({sin}^{2}k+{cos}^{2}k\right)=0$. This contradiction demonstrates that $\underset{k\to \infty }{lim}cosk$ does not exist, implying that f cannot be represented in the form given by (12).

Proposition 1. 

([18]). The following statements

(i)

Let $p\in \mathbb{R}$ and $u\in {\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)$. Then, a necessary and sufficient condition for ${lim}_{t\to \infty }\left(u(t+1)-e^{ip}u\left(t\right)\right)=0,$ is $\sigma \left(u\right)\subset \left\{e^{ip}\right\}$;

(ii)

Let $u\in {\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)$. Then, $\sigma \left(u\right)=\varnothing$ if and only if $u\in C_0\left({\mathbb{R}}^{+},E_0\right)$;

hold.

Lemma 3.

([18]). For every $t\in {\mathbb{R}}^{+}$, assume that $\mathcal{Q}\left(t\right)$ is a linear operator in $E_0$ that it bounded and satisfied

1. 

The function ${\mathbb{R}}^{+}\times E_0\ni (t,u)\mapsto \mathcal{Q}\left(t\right)u\in E_0$ is continuous,

2. 

$\mathcal{Q}(t+1)=\mathcal{Q}\left(t\right)$ for all $t\in {\mathbb{R}}^{+}$,

3. 

${sup}_{0\le t\le 1}\parallel \mathcal{Q}\left(t\right)\parallel <\infty$.

Then, for each $u(·)\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$ we have

$$\sigma \left(\mathcal{Q}u\right(·\left)\right)\subset \sigma \left(u\right(·\left)\right),$$

where $\mathcal{Q}$ denotes the operator in ${\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$ defined as

$$\begin{array}{c}\hfill \left[\mathcal{Q}u(·)\right]\left(t\right)=\mathcal{Q}\left(t\right)u\left(t\right),t\in {\mathbb{R}}^{+}.\end{array}$$

3.2. Asymptotic Periodic Solution

Definition 5.

A function $u(·)\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$ is said to be an asymptotic mild solution of (7) if there exists a function $ϵ(·)\in C_0({\mathbb{R}}^{+},E)$ such that

$$u\left(t\right)={\mathcal{U}}_{t,s}u\left(s\right)+\underset{\xi \to \infty }{lim}{\int }_s^t{\mathcal{U}}_{t,h}\xi \mathcal{R}(\xi ,A)[f\left(h\right)+ϵ\left(h\right)]dh,$$

(13)

for all $t\ge s\ge 0$ and ${\mathcal{U}}_{t,s}$ is 1-periodic strongly continuous evolution process ${\left\{{\mathcal{U}}_{t,s}\right\}}_{t\ge s\ge 0}$ that satisfies the usual properties of the evolution process.

Now, for T is an operator in a Banach space $E_0$, we denote ${\sigma }_{\Gamma }\left(T\right)=\sigma \left(T\right)\cap \Gamma$. We also recall the following well known result on the spectrum of the operators

$$L\left(t\right)={\mathcal{U}}_{t+1,t}$$

(14)

for each $t\ge 0$. When $t=1$ we denote $L=L\left(1\right)$. In particular, $L={\mathcal{U}}_{1,0}$ if ${\left\{{\mathcal{U}}_{t,s}\right\}}_{t\ge s\ge 0}$ is a 1-periodic process. Let us denote by $\mathcal{P}$ the operator of multiplication $u\mapsto \mathcal{P}u$ defined as

$$\mathcal{P}u\left(t\right)=L\left(t\right)u\left(t\right).$$

(15)

The unique existence of asymptotic mild solution of (7) is implied from Theorem 1, by in fact that $f\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E)\subset L_{loc}^1{\mathcal{B}}_{uc}({\mathbb{R}}^{+},E)$. Now we prove the relation between the spectral of asymptotic mild solution u with spectral of L and f.

Lemma 4.

Let $u(·)\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$ be an asymptotic mild solution of (7) and $f\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E)$. Then,

$$\sigma \left(u\right)\subset {\sigma }_{\Gamma }\left(L\right)\cup \sigma \left(f\right).$$

(16)

Proof. 

By the definition of asymptotic mild solutions there is a function $ϵ(·)\in C_0\left({\mathbb{R}}^{+},E\right)$ such that, for each $t\in {\mathbb{R}}^{+}$

$$u(t+1)={\mathcal{U}}_{t+1,t}u\left(t\right)+\underset{\xi \to \infty }{lim}{\int }_t^{t+1}{\mathcal{U}}_{t+1,h}\xi \mathcal{R}(\xi ,A)(f\left(h\right)+ϵ\left(h\right))dh.$$

(17)

For $\xi >\omega$ we set $f_{\xi }=\xi R(\xi ,A)f$. Note that $\sigma \left(f_{\xi }\right)\subset \sigma \left(f\right)$ and $f_{\xi }\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$.

Let us denote

$$\begin{array}{c}\hfill {\mathcal{F}}_{\xi }\left(t\right)={\int }_t^{t+1}{\mathcal{U}}_{t+1,h}{f}_{\xi }\left(h\right)dh.\end{array}$$

Remark that the operator that maps $f_{\xi }$ to ${\mathcal{F}}_{\xi }$ commutes with S, and it is a linear operator which is bounded from ${\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)$ into itself, hence by Lemma 3,

$$\begin{array}{c}\hfill \sigma \left(F_{\xi }\right)\subset \sigma \left(f_{\xi }\right).\end{array}$$

Moreover, ${\mathcal{F}}_{\xi }\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+},E_0)$ and

$$\begin{array}{c}\hfill {\mathcal{F}}_{\xi }\left(t\right)=\underset{\xi \to \infty }{lim}{\int }_t^{t+1}{\mathcal{U}}_{t+1,h}{f}_{\xi }\left(h\right)dh\in E_0.\end{array}$$

which proves that

$$\begin{array}{c}\hfill \sigma \left(\mathcal{F}\right)\subset \sigma \left({\mathcal{F}}_{\xi }\right)\subset \sigma \left(f_{\xi }\right)\subset \sigma \left(f\right).\end{array}$$

Also, if we denote

$$\begin{array}{c}\hfill \epsilon \left(t\right)=\underset{\xi \to \infty }{lim}{\int }_t^{t+1}{\mathcal{U}}_{t+1,h}\xi \mathcal{R}(\xi ,A)ϵ\left(h\right)dh,\end{array}$$

then $\epsilon (·)\in C_0({\mathbb{R}}^{+},E)$. Hence, for the function

$$\begin{array}{cc}\hfill \underset{\xi \to \infty }{lim}{\int }_t^{t+1}{\mathcal{U}}_{t+1,h}\xi \mathcal{R}(\xi ,A)(f\left(h\right)+ϵ\left(h\right))dh& =\mathcal{F}\left(t\right)+\epsilon \left(t\right)\hfill \\ \hfill & =w\left(t\right),\hfill \end{array}$$

we have $\sigma \left(w\right)=\sigma \left(\mathcal{F}\right)\subset \sigma \left(f\right).$ Therefore, ref. (17) gives

$$\begin{array}{c}\hfill \overline{S}\overline{u}=\overline{\mathcal{P}}\overline{u}+\overline{\mathcal{F}}.\end{array}$$

Let $0\ne {\lambda }_0\notin {\sigma }_{\Gamma }\left(L\right)\cup \sigma \left(f\right)$ and let V be a small enough open neighborhood of ${\lambda }_0$ which satisfies

$$\begin{array}{c}\hfill V\cap \left({\sigma }_{\Gamma }\left(L\right)\cup \sigma \left(f\right)\right)=\varnothing .\end{array}$$

By the identity

$$\begin{array}{c}\hfill \mathcal{R}(\lambda ,\overline{S})\overline{S}\overline{u}=\lambda \mathcal{R}(\lambda ,\overline{S})\overline{u}-\overline{u},\mathrm{for}\lambda \in V:\left|\lambda \right|\ne 1,\end{array}$$

then we can write,

$$\begin{array}{cc}\hfill \mathcal{R}(\lambda ,\overline{S})(\overline{\mathcal{P}}\overline{u}+\overline{\mathcal{F}})& =\mathcal{R}(\lambda ,\overline{S})\overline{S}\overline{u}\hfill \\ \hfill & =\lambda \mathcal{R}(\lambda ,\overline{S})\overline{u}-\overline{u}.\hfill \end{array}$$

And since, $\mathcal{R}(\lambda ,\overline{S})\overline{\mathcal{P}}\overline{u}=\overline{\mathcal{P}}\mathcal{R}(\lambda ,\overline{S})\overline{u}$, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \overline{u}+\mathcal{R}(\lambda ,\overline{S})\overline{\mathcal{F}}& =\lambda \mathcal{R}(\lambda ,\overline{S})\overline{u}-\overline{\mathcal{P}}\mathcal{R}(\lambda ,\overline{S})\overline{u}\hfill \\ \hfill & =(\lambda -\overline{\mathcal{P}})\mathcal{R}(\lambda ,\overline{S})\overline{u}.\hfill \end{array}\end{array}$$

Since $\lambda \in V$ the operator $\lambda -\overline{\mathcal{P}}$ is invertible and its inverse is given by $\mathcal{R}(\lambda ,\overline{\mathcal{P}})$. Therefore, for all $\lambda \in V$ such that $\left|\lambda \right|\ne 1$, we have

$$\begin{array}{c}\hfill \mathcal{R}(\lambda ,\overline{S})\overline{u}=\mathcal{R}(\lambda ,\overline{\mathcal{P}})(\overline{u}+\mathcal{R}(\lambda ,\overline{S})\overline{\mathcal{F}}).\end{array}$$

Since $\mathcal{R}(\lambda ,\overline{\mathcal{P}})\overline{u}$ is analytic in V and $\mathcal{R}(\lambda ,\overline{S})\overline{\mathcal{F}}$ is analytic in a neighborhood of ${\lambda }_0$, then the function $\mathcal{R}(\lambda ,\overline{S})\overline{u}$ is analytic as complex function in a neighborhood of ${\lambda }_0$. That is ${\lambda }_0\notin \sigma \left(u\right)$. This proves (16). □

Theorem 2.

Suppose $({\mathbf{H}}_{\mathbf{1}}$–${\mathbf{H}}_{\mathbf{3}})$ are fulfilled. If $\sigma \left(L\right)\subset \left\{1\right\}$ and $u\in {\mathcal{B}}_{uc}\left({\mathbb{R}}^{+},E_0\right)$ is an asymptotic mild solution of (7), and if additionally $f\in \mathcal{B}uc({\mathbb{R}}^{+},E)$ in (7) is asymptotic 1-periodic, then $u(·)$ is asymptotic 1-periodic, which means

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}(u(t+1)-u\left(t\right))=0.\end{array}$$

Proof. 

If we assume that f is asymptotic 1-periodic, then

$$\begin{array}{c}\hfill \sigma \left(f\right)\subset \left\{1\right\}.\end{array}$$

By Lemma 4,

$$\begin{array}{c}\hfill \sigma \left(u\right)\subset \sigma \left(P\right)\cup \sigma \left(f\right)\subset \left\{1\right\}.\end{array}$$

Therefore, based on Proposition 1, we deduce that $u(·)$ is asymptotic 1-periodic.

□

3.3. The Operator ${{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }$ as an Infinitesimal Generator of a $C_0$-Semigroup

In this section, we will focus on a special case that consists of the particular operator A, which appears in (7) as being ${{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }$ all this to meet the requirements resulting from physical phenomena as we mentioned in the introduction We will recall some preliminaries of the fractional derivatives and some related Sobolev spaces

Definition 6.

Let $\gamma >0$. For $f\in L^1(0,1)$, we introduce Riemann–Liouville (RL) fractional integral

$${\mathcal{J}}^{\gamma }f\left(z\right){\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{(z-\tau )}^{\gamma -1}f\left(\tau \right)d\tau .$$

If f is regular enough the (RL) fractional derivative can be defined as follows

$${\partial }^{\gamma }f\left(z\right)={\displaystyle \frac{\partial }{\partial z}}{\mathcal{J}}^{1-\gamma }f\left(z\right)={\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\displaystyle \frac{\partial }{\partial z}}{\int }_0^z{(z-\tau )}^{\gamma -1}f\left(\tau \right)d\tau ,$$

and the Caputo derivative is given by

$${}^c𝒟^{\gamma }f\left(z\right)={\displaystyle \frac{\partial }{\partial z}}\left({\mathcal{J}}^{1-\gamma }[f\left(z\right)-f\left(0\right)]\right)={\displaystyle \frac{1}{\Gamma (1-\gamma )}}{\displaystyle \frac{\partial }{\partial z}}{\int }_0^z{(z-\tau )}^{\gamma -1}[f\left(\tau \right)-f\left(0\right)]d\tau .$$

Let us define the following functional spaces

$${}_{0}H^{\gamma }(0,1)=\left\{\begin{array}{c}{H}^{\gamma }(0,1)for\gamma \in (0,{\displaystyle \frac{1}{2}})\hfill \\ \hfill \\ \{u\in H^{{\displaystyle \frac{1}{2}}}(0,1):{\int }_0^1{\displaystyle \frac{{\left|u\left(z\right)\right|}^2}{z}}dx<\infty \},for\gamma ={\displaystyle \frac{1}{2}}\hfill \\ \hfill \\ \{u\in H^{\gamma }(0,1):u\left(1\right)=0\},for\gamma \in ({\displaystyle \frac{1}{2}},1)\hfill \end{array}\right.$$

where $H^{\gamma }(0,1)$ denote the fractional Sobolev space of order γ. The corresponding are given by ${\parallel u\parallel }_{{}_{0}H^{\gamma }(0,1)}={\parallel u\parallel }_{H^{\gamma }(0,1)}$ for $\gamma \ne {\displaystyle \frac{1}{2}}$ and

$${\parallel u\parallel }_{{}_{0}H^{{\displaystyle \frac{1}{2}}}(0,1)}={\left({\parallel u\parallel }_{H^{{\displaystyle \frac{1}{2}}}(0,1)}^2+{\int }_0^1{\displaystyle \frac{{\left|u\left(z\right)\right|}^2}{1-z}}dx\right)}^{{\displaystyle \frac{1}{2}}}.$$

Proposition 2.

([22]). $For\gamma \in [0,1]$ the operators ${\mathcal{J}}^{\gamma }:{\mathcal{L}}^2(0,1)\to {}_{0}H(0,1)$ and ${\partial }^{\gamma }:{}_{0}H(0,1)\to {\mathcal{L}}^2(0,1)$ are isomorphism and the following inequalities hold

1. 

$c_{\gamma }^{-1}{\parallel u\parallel }_{{}_{0}H(0,1)}\le \parallel {\partial }^{\gamma }{u\parallel }_{{\mathcal{L}}^2(0,1)}\le c_{\gamma }{\parallel u\parallel }_{{}_{0}H(0,1)},foru\in {}_{0}H^{\gamma }(0,1)$;

2. 

$c_{\gamma }^{-1}\parallel {\mathcal{J}}^{\gamma }{u\parallel }_{{}_{0}H(0,1)}\le {\parallel f\parallel }_{{\mathcal{L}}^2(0,1)}\le c_{\gamma }{\parallel {\mathcal{J}}^{\gamma }f\parallel }_{{}_{0}H(0,1)},forf\in {\mathcal{L}}^2(0,1)$,

where $c_{\gamma }$ denotes a positive constant.

For $\gamma \in (0,{\displaystyle \frac{1}{2}})$, the operator ${\mathcal{J}}^{\gamma }$ can be extended to a linear operator that is bounded from $H^{-\gamma }(0,1):={\left({}_{0}H^{\gamma }(0,1)\right)}^{\prime }$ to ${\mathcal{L}}^2(0,1)$. Indeed, by the Fubini’s theorem for $u,v\in {\mathcal{L}}^2(0,1)$, we can obtain $({\mathcal{J}}^{\gamma },v)=(u,{\mathcal{J}}_1^{\gamma })$, where ${\mathcal{J}}_1^{\gamma }v\left(z\right)={\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_z^{\beta }{(z-\tau )}^{\gamma -1}f\left(\tau \right)d\tau$, $\beta >0$. Then, by Proposition 2 we may estimate,

$$|({\mathcal{J}}^{\gamma },v)|\le \parallel {\mathcal{J}}_1^{\gamma }{v\parallel }_{H^{\gamma }(0,1)}{\parallel u\parallel }_{{\left(H^{\gamma }(0,1)\right)}^{\prime }}\le c_{\gamma }{\parallel v\parallel }_{{\mathcal{L}}^2(0,1)}{\parallel u|}_{H^{-\gamma }(0,1)},$$

since $\gamma <{\displaystyle \frac{1}{2}})$, we have ${\left({}_{0}H^{\gamma }(0,1)\right)}^{\prime }=\left(H^{\gamma }(0,1)\right)$, the last inequality finishes the result. we can prove with the same argument that for $\gamma \in (0,{\displaystyle \frac{1}{2}})$ the operator ${\partial }^{\gamma }$ can be extended to a linear operator, which is bounded from ${\mathcal{L}}^2(0,1)$ to $H^{-\gamma }(0,1)$.

Firstly, we will characterize the domain of the operator ${{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }$ in ${\mathcal{L}}^2(0,1)$.

Lemma 5.

Operator ${{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }:{𝒟}_{\gamma }\subseteq {\mathcal{L}}^2(0,1)\to {\mathcal{L}}^2(0,1)$ generates a $C_0$-semigroup of contractions.

Proof. 

Firstly, we see that ${{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }$ is densely defined. In order to satisfy assumptions of LumerCPhillips theorem [20], [Ch.1, Theorem 4.3] we have to show that $Ran(I-{{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma })={\mathcal{L}}^2(0,1)$ and $-{\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma }$ is accretive. Let $u\in {𝒟}_{\gamma }$, we use [22], (Proposition 6.5) and integration by part, we get

$$\begin{array}{cc}\hfill \mathrm{Re}\left({{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }u,u\right)=& -\mathrm{Re}{\int }_0^1\left({\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma }u\right)\left(z\right).\overline{u\left(z\right)}dx\hfill \\ \hfill =& {\int }_0^1{}^c𝒟^{\gamma }\mathrm{Re}u\left(z\right).{\displaystyle \frac{\partial }{\partial z}}\mathrm{Re}u\left(z\right)dx+{\int }_0^1{}^c𝒟^{\gamma }\Im u\left(z\right).{\displaystyle \frac{\partial }{\partial z}}\Im u\left(z\right)dx\hfill \\ \hfill =& {\int }_0^1{}^c𝒟^{\gamma }\mathrm{Re}u\left(z\right).{\partial }^{1-\gamma }\mathrm{Re}u\left(z\right)dx+{\int }_0^1{}^c𝒟^{\gamma }\Im u\left(z\right).{\partial }^{1-\gamma }\Im u\left(z\right)dx,\hfill \end{array}$$

since $u_z\in {}_{0}H_{\gamma }(0,1)$, then ${}^c𝒟^{\gamma }={\mathcal{J}}^{1-\gamma }{u}_z\in {}_{0}H_{\gamma }(0,1)$. we can apply [22] (Proposition 6.5) with two functions ${}^c𝒟^{\gamma }\mathrm{Re}u$ and ${}^c𝒟^{\gamma }\Im u$ we have

$$\begin{array}{ccc}\hfill \mathrm{Re}\left(-{\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma }u,u\right)& \ge & c_{\gamma }{\parallel }^c{𝒟}^{\gamma }{u\parallel }_{H^{{\displaystyle \frac{1-\gamma }{2}}}(0,1)}^2\ge c_{\gamma }{\parallel {\partial }^{{\displaystyle \frac{1-\gamma }{2}}}{}^c𝒟^{\gamma }u\parallel }_{{\mathcal{L}}^2(0,1)}^2\hfill \\ & \ge & c_{\gamma }{{\parallel }^c{𝒟}^{{\displaystyle \frac{1+\gamma }{2}}}u\parallel }_{{\mathcal{L}}^2(0,1)}^2\hfill \end{array}$$

where $c_{\gamma }$ is a constant dependent on $\gamma$. Now, we have to prove that

$$\begin{array}{c}\hfill Ran(I-{\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma })={\mathcal{L}}^2(0,1).\end{array}$$

For every $\lambda \in \mathbb{C}$ belonging to a well-chosen sector $S_{\gamma }$, the equality

$$Ran(\lambda I-{\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma })={\mathcal{L}}^2(0,1)$$

holds. Indeed, we fixe $h\in {\mathcal{L}}^2(0,1)$ and $\lambda$, we have to show that there exists $u\in {𝒟}_{\gamma }$ such that

$$(\lambda I-{\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma })u=h.$$

(18)

Firstly, we solve this equation with arbitrary condition $u\left(0\right)=u_0\in \mathbb{C}$. We note that if we search u in $\{g\in H^{1+\gamma }(0,1):{\displaystyle \frac{\partial }{\partial z}}g\in {}_{0}H(0,1)\}$, then Equation (18) is equivalent to

$$u_0=\lambda {\mathcal{J}}^{1+\gamma }u-{\mathcal{J}}^{1+\gamma }h=u.$$

(19)

Applying ${\mathcal{J}}^{\gamma }$ to both sides of (19), we obtain

$$\lambda {\mathcal{J}}^{\gamma }u-{\mathcal{J}}^{\gamma }h=u_z.$$

Integrating the above equation we obtain (19). If we assume that $u\in {\mathcal{L}}^2(0,1)$ is a solution of (19), then by Proposition 2$u\in \left\{g\in H^{\gamma +1}(0,1):{\displaystyle \frac{\partial }{\partial z}}\in {}_{0}H^{\gamma }(0,1)\right\}$ and a sufficient condition to have (18) is to apply the operator ${\partial }^{\gamma }{\displaystyle \frac{\partial }{\partial z}}$ to (19). Hence, we are going to solve (19), we apply to it the operator ${\mathcal{J}}^{\gamma +1}$ it yields

$$u_0+\lambda \left({\mathcal{J}}^{\gamma +1}{u}_0\right)\left(z\right)-\left({\mathcal{J}}^{\gamma +1}h\right)\left(z\right)+{\lambda }^2\left({\mathcal{J}}^{2(\gamma +1)}u\right)\left(z\right)-\lambda \left({\mathcal{J}}^{2(\gamma +1)}h\right)\left(z\right)=u\left(z\right).$$

By iterating this operator n times, we arrive to the following equation

$$u_0\sum _{k=0}^n\left({\lambda }^k{\mathcal{J}}^{k(\gamma +1)}1\right)\left(z\right)-\sum _{k=0}^n{\lambda }^k\left({\mathcal{J}}^{(k+1)(\gamma +1)}h\right)\left(z\right)+{\lambda }^{n+1}\left({\mathcal{J}}^{(n+1)(\gamma +1)}u\right)\left(z\right)=u\left(z\right).$$

(20)

We have

$$\left|{\lambda }^n\left({\mathcal{J}}^{n(\gamma +1)}u\right)\left(z\right)\right|\le {\parallel u\parallel }_{L^{\infty }(0,1)}{\displaystyle \frac{{\left|\lambda \right|}^n{z}^{(\gamma +1)n}}{\Gamma \left(\right(\gamma +1)n+1)}}\le {\displaystyle \frac{{\parallel u\parallel }_{L^{\infty }(0,1)}{\left|\lambda \right|}^n}{\Gamma \left(\right(\gamma +1)n+1)}},$$

since ${\displaystyle \frac{{\parallel u\parallel }_{L^{\infty }(0,1)}{\left|\lambda \right|}^n}{\Gamma \left(\right(\gamma +1)n+1)}}\to 0$ as $n\to \infty$ for each $\lambda \in \mathbb{C}$ uniformly with respect to $z\in [0,1]$. Hence, if we passing to the limit with respect to n in (20), we will have

$$u_0\sum _0^{\infty }\left({\lambda }^k{\mathcal{J}}^{k(\gamma +1)}1\right)\left(z\right)-\sum _0^{\infty }{\lambda }^k\left({\mathcal{J}}^{(k+1)(\gamma +1)}h\right)\left(z\right)=u\left(z\right).$$

(21)

By comparison criterion of the series, we can show that both series in (21) are uniformly convergent with respect to z to the sum

$$\sum _{k=0}^{\infty }\left({\mathcal{J}}^{(\gamma +1)k}1\right)\left(z\right)=E_{\gamma +1}\left(\lambda z^{\gamma +1}\right),$$

(22)

with $E_{\gamma }$ denote the Mittag–Leffler function of order $\gamma$. For the sum of the second series, we have by the definition of the fractional integral operator

$$\sum _{k=0}^{\infty }{\lambda }^k\left({\mathcal{J}}^{(\gamma +1)(k+1)}h\right)\left(z\right)=\sum _{k=0}^{\infty }{\lambda }^k{\int }_0^{z}h\left(\tau \right){\displaystyle \frac{{(z-\tau )}^{(\gamma +1)k+\gamma }}{\Gamma \left(\right(\gamma +1)k+\gamma +1)}}d\tau ,$$

in the previous equality we need to invert the order of summation and integration. Let us write

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }{\lambda }^k{\int }_0^{z}h\left(\tau \right){\displaystyle \frac{{(z-\tau )}^{(\gamma +1)k+\gamma }}{\Gamma \left(\right(\gamma +1)k+\gamma +1)}}d\tau =& \underset{n\to \infty }{lim}\sum _{k=0}^n{\lambda }^k{\int }_0^{z}h\left(\tau \right){\displaystyle \frac{{(z-\tau )}^{(\gamma +1)k+\gamma }}{\Gamma \left(\right(\gamma +1)k+\gamma +1)}}d\tau \hfill \\ \hfill =& \underset{n\to \infty }{lim}{\int }_0^{z}h\left(\tau \right)\sum _{k=0}^n{\lambda }^k{\displaystyle \frac{{(z-\tau )}^{(\gamma +1)k+\gamma }}{\Gamma \left(\right(\gamma +1)k+\gamma +1)}}d\tau .\hfill \end{array}$$

Also, we have the following major ration

$$\begin{array}{cc}\hfill \left|h\left(\tau \right)\sum _{k=0}^n{\lambda }^k{\displaystyle \frac{{(z-\tau )}^{(\gamma +1)k+\gamma }}{\Gamma \left(\right(\gamma +1)k+\gamma +1)}}\right|\le & \left|h\left(\tau \right)\right|\sum _{k=0}^{\infty }{\displaystyle \frac{{\left|\lambda \right|}^k}{\Gamma \left(\right(\gamma +1)k+\gamma +1)}}\hfill \\ \hfill =& \left|h\left(\tau \right)\right|E_{\gamma +1}\left(\right|\lambda \left|\right),\hfill \end{array}$$

since $h\in {\mathcal{L}}^2(0,1)$, the function $\tau \to \left|h\left(\tau \right)\right|E_{\gamma +1}\left(\right|\lambda \left|\right)$ is integrable. Then, by the well known dominated convergence theorem of Lebesgue, we obtain

$$\begin{array}{cc}\hfill \sum _{k=0}^{\infty }{\lambda }^k\left({\mathcal{J}}^{(\gamma +1)(k+1)}h\right)\left(z\right)=& h\ast z^{\gamma }{E}_{\gamma +1}\left(\lambda z^{\gamma +1}\right).\hfill \end{array}$$

(23)

By using the results (22) and (23) into (21), obtain that the solution u of (18) with condition $u\left(0\right)=u_0\in \mathbb{C}$ is given as follows

$$u\left(z\right)=u_0{E}_{\gamma +1}\left(\lambda z^{\gamma +1}\right)-h\ast z^{\gamma }{E}_{\gamma +1}\left(\lambda z^{\gamma }\right).$$

(24)

Then, we will choose $u_0$ which ensures the fulfillment of the zero condition at the opposite end of the interval. For this, we teak $z=1$ in (23) to have

$$u\left(1\right)=u_0{E}_{\gamma +1}\left(\lambda \right)-(h\ast z^{\gamma }{E}_{\gamma +1}\left(\lambda z^{\gamma +1}\right))\left(1\right)=0.$$

(25)

Hence,

$$u_0={\left(E_{\gamma +1}\left(\lambda \right)\right)}^{-1}(h\ast z^{\gamma }{E}_{\gamma +1}\left(\lambda z^{\gamma +1}\right))\left(1\right).$$

Since $E_{\gamma +1}>0$ for $\gamma$ belonging to the sector $v_{\gamma }$, then $u_0$ is well defined, now placing $u_0$ into (23) we obtain that

$$u\left(z\right)={\left(E_{\gamma +1}\left(\lambda \right)\right)}^{-1}(h\ast z^{\gamma }{E}_{\gamma +1}\left(\lambda z^{\gamma +1}\right))\left(1\right)E_{\gamma +1}(\lambda z^{\gamma }+1)-h\ast z^{\gamma }{E}_{\gamma +1}\left(\lambda z^{\gamma +1}\right).$$

solves (19). Then, we have proven that

$$Ran(\lambda I-{\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma })={\mathcal{L}}^2(0,1),$$

for every $\lambda$ in a sector $v_{\gamma }$, in particular we have proven that $Ran(I-{\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma })={\mathcal{L}}^2(0,1)$. This and the fact that $-{\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma }$ is accretive enables the application of Lumer-Phillips theorem, Which ends the proof of Lemma 5.

□

Now, we consider the following problem

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial }{\partial t}}u(t,z)={{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }u(t,z)+G\left(t\right)u(t,z)+f\left(t\right),\mathrm{for}t\ge 0\hfill \\ \hfill \\ u(0,z)=u_0.\hfill \end{array}\right.$$

(26)

Definition 7.

A function $u\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+}\times \mathbb{R},E)$ (the space of uniformly bounded and continuous on t uniformly every compact of $\mathbb{R}$ ) is said to be an asymptotic mild solution of (26) if there exists a function $ϵ\in C_0({\mathbb{R}}^{+}\times \mathbb{R},E)$, such that

$$u(t,z)={\mathcal{U}}_{t,s}u(s,z)+\underset{\xi \to \infty }{lim}{\int }_s^t{\mathcal{U}}_{t,h}\xi \mathcal{R}(\xi ,A)[f\left(h\right)+ϵ(h,z)]dh,$$

(27)

for all $t\ge s\ge 0$, $z\in K$, compact of $\mathbb{R}$ and ${\left\{{\mathcal{U}}_{t,s}\right\}}_{t\ge s\ge 0}$ denote the unique 1-periodic strongly continuous evolution process associate to the operators ${{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }$ and $B(.)$ that satisfies all properties of evolution process, where we keep the same notation if there is no ambiguity. Note that both spaces ${\mathcal{B}}_{uc}({\mathbb{R}}^{+}\times \mathbb{R},E)$ and $C_0({\mathbb{R}}^{+}\times \mathbb{R},E)$ have the same properties like the spaces ${\mathcal{B}}_{uc}({\mathbb{R}}^{+},E)$ and $C_0({\mathbb{R}}^{+},E)$, respectively.

Remark 2.

Since the operator $A={{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }$ is an infinitesimal generator of a $C_0$-semigroup, then in particular it verifies the Hille-Yosida condition $\left({\mathbf{H}}_{\mathbf{1}}\right)$.

Theorem 3.

Assume that $\left({\mathbf{H}}_{\mathbf{2}}\right)$ and $\left({\mathbf{H}}_{\mathbf{3}}\right)$ are satisfied. Let ${\sigma }_{\Gamma }\left(L\right)\subset \left\{1\right\}$ and $u\in {\mathcal{B}}_{uc}\left({\mathbb{R}}^{+}\times \mathbb{R},E\right)$ be an asymptotic mild solution of (26) Furthermore, let $f\in {\mathcal{B}}_{uc}({\mathbb{R}}^{+}\times \mathbb{R},E)$ be asymptotic 1-periodic. Then, $u(·,z)$ is asymptotic 1-periodic, i.e.,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim}(u(t+1,z)-u(t,z))=0,\mathit{for}\mathit{all}z\mathit{in}\mathrm{a}\mathit{compact}\mathit{of}\mathbb{R}.\end{array}$$

Proof. 

Since, by the result in Lemma 5 the operator ${{\displaystyle \frac{\partial }{\partial z}}}^c{𝒟}^{\gamma }$ satisfies the Hille-Yosida condition $\left({\mathbf{H}}_{\mathbf{1}}\right)$, so all hypothesis of Theorem 5 hold, then the rest of the proof is a consequence of that of our first theorem. □

Example 1.

Consider the linear non-homogeneous parabolic partial differential equation and let us study the existence of asymptotic periodic mild solutions.

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u}{\partial t}}(t,\xi )={\displaystyle \frac{{\partial }^{2}u}{\partial {\xi }^2}}(t,\xi )-b\left(t\right)u(t,\xi )+c\left(t\right)g\left(\xi \right),\xi \in [0,\pi ],\end{array}$$

(28)

$$\begin{array}{c}\hfill u(t,0)=u(t,\pi )=0,t>0,\end{array}$$

(29)

where $b(.)$ is a scalar continuous, 1-periodic function, $c(.)$ uniformly continuous 1-periodic function, $g\in \mathbb{X}:=L^2[0,\pi ]$ and the operator define by $A:𝒟\left(A\right)\subset \mathbb{X}\to \mathbb{X}$ with

$$\begin{array}{c}\hfill 𝒟\left(A\right)=\left\{v\in \mathbb{X}:v,\mathrm{are}\mathrm{absolutely}\mathrm{continuous},v^{\prime \prime }\in \mathbb{X},v\left(0\right)=v\left(\pi \right)=0\right\}\end{array}$$

and $A={\displaystyle \frac{{\partial }^2}{\partial {\zeta }^2}}$ be the operator defined as the second partial derivative

To make our result manageable, we consider, in particular, the following nondensely defined nonautonomous PDE

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}}(t,\zeta )={\displaystyle \frac{{\partial }^2}{\partial {\zeta }^2}}u(t,\xi )-\nu \left(t\right)u(t,\zeta )+g\left(\zeta \right)·cos\sqrt{t},\mathrm{for}t\in {\mathbb{R}}_{+}\mathrm{and}\zeta \in [0,\pi ],\hfill \\ \hfill \\ u(t,0)=u(t,\pi )=0,\mathrm{for}t\in {\mathbb{R}}_{+},\hfill \end{array}\right.$$

(30)

We have $(0,\infty )\subset \rho \left(A\right)$ and the following estimate

$$\begin{array}{c}\hfill \parallel \mathcal{R}(\lambda ,A)\parallel \le {\displaystyle \frac{1}{\lambda }},\forall \lambda >0,\end{array}$$

Hence, $\left({\mathbf{H}}_1\right)$ is satisfied. Since A is an infinitesimal generator of a strongly continuous semigroup ${\left(T_0\left(t\right)\right)}_{t\ge 0}$ on E, which is exponentially, and verifies

$$\begin{array}{c}\hfill ∥T_0\left(t\right)∥\le e^{-t},\forall t\ge 0.\end{array}$$

Additionally, as mentioned in [23] (p. 414), the eigenvalues of A on $i\mathbb{R}$ are found from the solutions of equations

$$\lambda -1=-n^2,n=1,2,\dots .$$

Clearly, there exists a single root at $\lambda =0$ situated on the imaginary axis, thus $\sigma \left(A\right)\cap i\mathbb{R}=0$. As this semigroup is compact, applying the spectral mapping theorem yields that $\sigma \left(T_0\left(1\right)\right)=e^{\sigma \left(A\right)}=\left\{1\right\}$.

For every $t\ge 0$, we consider the operator $G\left(t\right)$ defined on E by $G\left(t\right)=-\nu \left(t\right)I$. Since $\nu (.)\in L_{loc}^1\left({\mathbb{R}}_{+}\right),t\mapsto G\left(t\right)z$ is strongly measurable. Hence, $\left({\mathbf{H}}_2\right)$ is verified. We can verify that $B(·)$ is 1-periodic so $\left({\mathbf{H}}_3\right)$ is fulfilled. We find by Definition (7) that $A+G\left(t\right)$ is an infinitesimal generator of a unique 1-periodic strongly continuous evolution process ${\left\{{\mathcal{U}}_{t,s}\right\}}_{t\ge s\ge 0}$ on E given by

$$\begin{array}{c}\hfill {\mathcal{U}}_{t,s}=exp\left(-{\int }_s^t\nu \left(\tau \right)d\tau \right)T_0(t-s).\end{array}$$

The spectrum of the monodromy operator $L={\mathcal{U}}_{1,0}=exp\left({\displaystyle -{\int }_0^1\nu \left(\tau \right)d\tau }\right)T_0\left(1\right)$ is reduced to one. Moreover, under the assumption that $g\in E$, the function $f\left(t\right)=cos\sqrt{t}·g(·)$ becomes an asymptotic 1-periodic function with values in E.

Hence, utilizing Theorem 2, we deduce that every asymptotic solution to (30) is asymptotic 1-periodic.

4. Conclusions

We studied the problem (7) in two important cases: the first is when the A is not a densely defined operator, where we showed that if f is 1-periodic and under a condition about its spectrum inasmuch as a bounded, uniformly continuous, then the mild solution is 1-periodic asymptotic. In the second case is when the operator $A={\displaystyle \frac{\partial }{\partial z}}{}^c𝒟^{\gamma }$ in this case A generates a $C_0$-semigroup of contraction; even better, this semigroup is analytical, and also in this case we have shown that the mild solution of (7) is 1-periodic asymptotic.