# Existence and uniqueness of C (n) -(ω, c)-periodic mild solutions to some two-term fractional evolution equations

Méthode Koyabanda, Gaston M N'Guérékata

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Méthode Koyabanda, Gaston M N'Guérékata. Existence and uniqueness of C (n) -(ω, c)-periodic mild solutions to some two-term fractional evolution equations. 2025. hal-05269562

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# Existence and uniqueness of $C^{(n)}$- $(\omega, c)$-periodic mild solutions to some two-term fractional evolution equations

Méthode Koyabanda * Gaston M. N'Guérékata †

September 10, 2025

## Abstract

In this paper, we establish some new properties of $C^{(n)}$- $(\omega, c)$-periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following linear fractional differential equations

$$ {}^c D_t^\alpha u(t) = Au(t) + {}^c D_t^{\alpha-1} f(t), \quad 1 < \alpha < 2, \ t \in \mathbb{R}, \quad (1) $$

and the semilinear case

$$ {}^c D_t^\alpha u(t) = Au(t) + {}^c D_t^{\alpha-1} g(t, u(t)), \quad 1 < \alpha < 2, \ t \in \mathbb{R}, \quad (2) $$

where ${}^c D_t^\alpha(\cdot)$ ($1 < \alpha < 2$) stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space $X$, and the functions $f$ and $g$ are $C^{(n)}$- $(\omega, c)$-periodic functions. Our results are obtained using the Leray-Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem.

**Key-words:** Banach space, Leray-Schauder alternative theorem; Arszela-Ascoli theorem, two-term fractional evolution equation; Krasnoselskii's theorem; $C^{(n)}$- $(\omega, c)$-periodic functions; mild solutions.

**MSC (2020):** 35B10; 46E15; 47D06; 47J35; 93D22.

## 1 Introduction.

In 2018, Alvarez et al. [3] introduced the concept of $(\omega, c)$-periodic functions by observing the behavior of any complex values solution $x(t)$ of the so-called

<footnote>*Department of Mathematics and Computer Science, University of Bangui, CAR, email: koyamet@yahoo.fr</footnote>

<footnote>†NEERLab, Department of Mathematics, Morgan State University, Baltimore, MD 21251, USA, email: Gaston.NGuerekata@morgan.edu</footnote>
Matthieu's equation

$$x'' + ax = 2q \cos(2t)x$$

which fulfills the equality $x(t + \omega) = cx(t)$ where $t \in \mathbb{R}, \omega > 0$ and $c$ a non zero complex number.

This theory has rapidly attracted several researchers including Abadias et al. [1], Alvarez et al. [4], Mophou and N'Guérékata [17]. Other contributions can be found in the following papers :[1, 3, 11, 14, 15, 17, 10] and references therein.

In [14], Larrouy and N'Guérékata investigated more interesting properties of $(\omega, c)$-periodic and asymptotically $(\omega, c)$-periodic functions and studied solutions of such types of the following equations:

$${}^c D_t^\alpha u(t) = Au(t) + {}^c D_t^{\alpha-1} f(t, u(t)), \quad 1 < \alpha < 2, \ t \in \mathbb{R}, \quad (3)$$

and the associated semilinear case

$${}^c D_t^\alpha u(t) = Au(t) + {}^c D_t^{\alpha-1} f(t, u(t-h)), \quad 1 < \alpha < 2, \ t, h \in \mathbb{R}_+, \quad (4)$$

where ${}^c D_t^\alpha(\cdot)$ ($1 < \alpha < 2$) stands for the Caputo fractional derivative and $A$ is a linear densely defined operator of sectorial type on a complex Banach space $X$, and the function $f(t, x)$ is $(\omega, c)$-periodic or asymptotically $(\omega, c)$-periodic with respect to the first variable.

Recently, Khallali et al. studied several classes of $(\omega, c)$-almost periodic type functions with one and two variables and applied their results to some evolution equations.

On the other hand, Koyabanda et al. discussed the new concept of $C^{(n)}$- $(\omega, c)$-periodic functions in their paper [12]. This class contains functions which are $(\omega, c)$- periodic as well as their successive derivatives up to the order $n$. The authors proved that under an appropriate norm, this class of functions is a Banach space.

In [21], Siopathis and N'Guérékata investigated more properties of this new concept and applied the results to the following fractional integro-differential equation

$$D_w^\alpha u(t) = Au(t) + \int_{-\infty}^{t} a(t-s)Au(s)ds + f(t) \quad (5)$$

and the semilinear case

$$D_w^\alpha u(t) = Au(t) + \int_{-\infty}^{t} a(t-s)Au(s)ds + f(t, u(t)), \quad (6)$$

where $D_w^\alpha$ stands for the Weyl fractional derivative of order $\alpha > 0$, and $a \in L^1(\mathbb{R}_+)$ is a scalar-valued kernel. Here $A$ is a closed linear operator with domain $D(A)$ dense in a complex Banach $X$.

Motivated by the papers above, among many others, we will discuss in this paper some new properties of $C^{(n)}$- $(\omega, c)$-periodic functions, including an important and useful characterization (Proposition 2.3). Then we study some applications to the two-term fractional equations (1) and (2). We prove the existence and uniqueness of mild solutions of these types using some classical fixed point theorems.
## 2 Preliminaries

In this paper $\mathbb{X} = (\mathbb{X}, \|\cdot\|)$ will denote a complex Banach space and $\mathcal{B}(\mathbb{X})$ the space of all bounded linear operators $\mathbb{X} \to \mathbb{X}$, endowed with the uniform operator topology. First, we recall some definitions and properties about sectorial linear operators and their associated solution operators.

**Definition 2.1** [3] A closed linear operator $A$ is said to be of sectorial type $\tilde{w}$ and angle $\theta$, if there exists $(\theta, M, \tilde{w}) \in (0, \frac{\pi}{2}) \times \mathbb{R}_+^* \times \mathbb{R}$ such that both following assertions hold:

(1) its resolvent exists outside the sector $\tilde{w} + S_\theta := \{\tilde{w} + \lambda, \lambda \in \mathbb{C}, |\arg(-\lambda)| < \theta\}$.

(2) $\|(\lambda - A)^{-1}\| \le \frac{M}{|\lambda - \tilde{w}|}, \lambda \notin \tilde{w} + S_\theta$,

**Definition 2.2** Let $\alpha \in (1, 2)$ and $A$ a closed linear operator with domain $D(A) \subset \mathbb{X}$. The operator $A$ is called a generator of a solution operator if there exists $\tilde{w} \in \mathbb{R}$ and a strongly continuous function $S_\alpha : \mathbb{R}_+ \to B(\mathbb{R})$ such that $\{\lambda^\alpha, \mathrm{Re}(\lambda) > \tilde{w}\} \subseteq \rho(A)$ and $\lambda^{\alpha-1}(\lambda^\alpha - A)^{-1}x = \int_0^\infty e^{-\lambda t} S_\alpha(t) x dt, \mathrm{Re}(\lambda) > \tilde{w}, x \in \mathbb{X}$. In this case $S_\alpha(t)$ is called the solution operator generated by $A$.

**Definition 2.3** ([7]): A family $\{S_\alpha(t)\}_{t \ge 0}$ is said to be uniformly integrable if

$$ \int_0^\infty \|S_\alpha(t)\| dt < \infty $$

We note that if $A$ is a sectorial operator of type $\tilde{w}$ with $\theta \in [0, \pi(1 - \frac{\alpha}{2})]$, then $A$ is the generator of a solution operator given by

$$ S_\alpha(t) := \frac{1}{2i\pi} \int_\xi e^{\lambda t} \lambda^{\alpha-1} (\lambda^\alpha - A)^{-1} $$

where $\xi$ is a suitable path lying outside the sector $\tilde{w} + S_\theta$.

**Lemma 2.1** [8]: Let $A : D(A) \subset \mathbb{X} \to \mathbb{X}$ be a sectorial operator of type $\tilde{w} < 0$ and angle $\theta$ in a complex Banach space $\mathbb{X}$, then there exists $C_{\alpha, \theta} > 0$ depending solely on $\alpha$ and $\theta$ such that:

$$ \|S_\alpha(t)\|_{B(\mathbb{X})} \le \frac{MC_{\alpha,\theta}}{1 + |\tilde{w}|t^\alpha}, \quad t \ge 0 $$

**Definition 2.4** ([2, 20]) The derivative of order $\alpha$ of a function $f : \mathbb{R}_+ \to \mathbb{R}$ in the sense of Caputo is defined as

$$ {}^c D_t^\alpha f(t) = \frac{1}{\Gamma(n-\alpha)} \int_0^t (t-s)^{n-\alpha-1} f^{(n)}(s) ds $$

for $n-1 \le \alpha < n, n \in \mathbb{N}$. If $1 < \alpha < 2$, then

$$ {}^c D_t^\alpha f(t) = \frac{1}{\Gamma(2-\alpha)} \int_0^t (t-s)^{1-\alpha} f^{(2)}(s) ds $$

In the following sections, we will recall some properties of both $C^{(n)} - ((\omega, c))$-periodic functions. But first, we have
**Proposition 2.1** [13] Let $A: D(A) \subset \mathbb{X} \to \mathbb{X}$ be a sectorial operator of type $\tilde{w} < 0$ and angle $\theta$ in a complex Banach space $\mathbb{X}$, then there exists $C_{\alpha, \theta} > 0$ depending solely on $\alpha$ and $\theta$ such that:

$$ \|S_{\alpha}(t)\|_{(\omega,c)} \le \frac{C_{\alpha,\theta}M}{1 + |\tilde{w}|t^{\alpha}}\gamma_1, \quad t \ge 0 $$

where $\gamma_1 = \max\{1, |c|^{-1}\}$

**Proof.** [13]

$$ \begin{aligned} \|S_{\alpha}(t)\|_{(\omega,c)} &= \sup_{t \in [0,w]} \|\hat{c}(-t)S_{\alpha}(t)\| \le \|S_{\alpha}(t)\|_{B(\mathbb{X})} \cdot \sup_{t \in [0,w]} \|\hat{c}(-t)\| \\ &\le \frac{C_{\alpha,\theta}M}{1 + |\tilde{w}|t^{\alpha}} \cdot \sup_{t \in [0,w]} \|\hat{c}(-t)\|, \quad t \ge 0 \\ &\le \frac{C_{\alpha,\theta}M}{1 + |\tilde{w}|t^{\alpha}} \cdot \max\{1, |c|^{-1}\}, \quad t \ge 0 \end{aligned} $$

■

**Definition 2.5** [3] Let $\omega > 0$ and $c$ a non-zero complex number. A function $f \in C(\mathbb{R}, \mathbb{X})$ is said to be $(\omega, c)$-periodic if $f(t + \omega) = cf(t)$, $\forall t \in \mathbb{R}$. In this case $\omega$ is called a $c$-period of the function $f$.

We denote by $P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$ the set of all $(\omega, c)$-periodic functions from $\mathbb{R}$ to $\mathbb{X}$. When $c = 1$, we write $P_{\omega}(\mathbb{R}, \mathbb{X})$ instead of $P_{\omega,1}(\mathbb{R}, \mathbb{X})$ and we say that $f$ is $\omega$-periodic. When $c = -1$ we obtain the class of antiperiodic functions [19]. Using the principal branch of the complex Logarithm, $c^{\frac{t}{\omega}}$ is defined as $c^{\frac{t}{\omega}} := \exp(\frac{t}{\omega}\mathrm{Log}(c)) = c^{\hat{\Lambda}}(t)$ and we will use the notation $c^{\hat{\Lambda}}(t) := c^{\frac{t}{\omega}}$.

In [3], Alvarez et al. gave a useful description of the space $P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$. That is $P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$ is a translation-invariant subspace over $\mathbb{C}$ of $C(\mathbb{R}, \mathbb{X})$. Then for any fixed $h > 0$ and $u \in P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$, we have $u_h(\cdot) := u(\cdot - h) \in P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$.

**Proposition 2.2** ([3]) Let $f \in C(\mathbb{R}, \mathbb{X})$. Then, $f \in P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$ if and only if $f(t) = c^{\frac{t}{\omega}}u(t)$, $u(t) \in P_{\omega}(\mathbb{R}, \mathbb{X})$.

**Theorem 2.1** ([17]) Let $f \in P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$ and $A \in \mathcal{B}(\mathbb{X})$. Then $Af \in P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$.

We state and prove this basic property which follows naturally from the $(\omega, c)$-periodicity definition. It will be very useful in the sequel.

**Lemma 2.2** [13] Let $f \in P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$. Then $f \in P_{-\omega, c^{-1}}(\mathbb{R}, \mathbb{X})$.

**Proof.** Let $f \in P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$. According to Proposition (2.1), $f(t) = c^{\frac{t}{\omega}}u(t)$ with $u(t) \in P_{\omega}(\mathbb{R}, \mathbb{X})$. Then, we have $f(t - \omega) = c^{\frac{t-\omega}{\omega}}u(t - \omega) = c^{-1}c^{\frac{t}{\omega}}u(t) = c^{-1}f(t)$. So $f \in P_{-\omega, c^{-1}}(\mathbb{R}, \mathbb{X})$.

■
**Theorem 2.2 ([3])** $P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$ is a Banach space with the norm

$$ \|f\|_{\omega,c} := \sup_{t \in [0, \omega]} \|c^{\wedge}(-t)f(t)\| $$

We note that if $f \in P_{(\omega,c)}(\mathbb{R}, \mathbb{X})$, then $\|f\|_{\omega,c} < \infty$ and we say that $f$ is $c$-bounded. The use of $\|f\|_{\omega,c}$ instead of $\|f\|_{\infty}$ will allow us to handle the $(\omega, c)$-periodicity properties of $f$ (see [3] for more details).

In the following paragraph, we will recall some properties of $C^{(n)}-(\omega, c)$-periodic functions.

Let $f: \mathbb{R} \to \mathbb{X}$. Denote by $C^{(n)}(\mathbb{R}, \mathbb{X})$ (briefly $C^{(n)}(\mathbb{X})$) the space of all functions $\mathbb{R} \to \mathbb{X}$ which have a continuous $n^{th}$ derivative on $\mathbb{R}$. Let $C_b^{(n)}(\mathbb{R}, \mathbb{X})$ (briefly $C_b^{(n)}(\mathbb{X})$) be the subspace of $C^{(n)}(\mathbb{R}, \mathbb{X})$ consisting of such functions satisfying

$$ \sup_{t \in \mathbb{R}} \sum_{i=0}^{n} \|f^{(i)}(t)\| < +\infty, $$

where $f^{(i)}$ denotes the $i^{th}$ derivative of $f$ and $f^{(0)} = f$. Clearly $C^{(n)}(\mathbb{X})$ turns out to be a Banach space with the norm

$$ \|f\|_n = \sup_{t \in \mathbb{R}} \sum_{i=0}^{n} \|f^{(i)}(t)\| $$

(cf. for instance see [12]).

**Definition 2.6** [12] A function $f \in C^{(n)}(\mathbb{X})$ is said to be a $C^n-(\omega, c)$-periodic if $f, f', f^{(2)}, \dots, f^{(n)}$ are $(\omega, c)$-periodic.

The set of all $C^{(n)}-(\omega, c)$-periodic functions will be denoted by $P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathbb{X})$, (briefly $P_{(\omega,c)}^{(n)}(\mathbb{X})$).

$P_{(\omega,c)}^{(0)}(\mathbb{X}) = P_{(\omega,c)}(\mathbb{X})$, is the classical space of all $(\omega, c)$-periodic functions.

We now state and prove this important result.

**Proposition 2.3** Let $f \in C(\mathbb{R}, \mathbb{X})$. Then

$$ f \in P_{(\omega,c)}^{(n)}(\mathbb{X}) \iff \exists u \in P_{\omega}^{(n)}(\mathbb{X}) \setminus f^{(n)}(t) = c^{\wedge}(t) \sum_{k=0}^{n} \binom{k}{n} \left(\frac{\ln(c)}{w}\right)^{(k)} u^{(n-k)}(t). $$
Proof.

$$f(t) = c^{\wedge}(t)u(t) = c^{\frac{t}{w}}u(t) = u(t)e^{\frac{t}{w} \ln(c)}$$

$$f'(t) = u'(t)e^{\frac{t}{w} \ln(c)} + \frac{\ln(c)}{w} e^{\frac{t}{w} \ln(c)} u(t) = \left[ u'(t) + \frac{\ln(c)}{w} u(t) \right] e^{\frac{t}{w} \ln(c)}$$

$$f''(t) = \left[ u''(t) + \frac{\ln(c)}{w} u'(t) \right] e^{\frac{t}{w} \ln(c)} + \frac{\ln(c)}{w} \left[ u'(t) + \frac{\ln(c)}{w} u(t) \right] e^{\frac{t}{w} \ln(c)} \\ = \left[ u''(t) + \frac{\ln(c)}{w} u'(t) + \frac{\ln(c)}{w} u'(t) + \left( \frac{\ln(c)}{w} \right)^2 u(t) \right] e^{\frac{t}{w} \ln(c)}$$

$$f''(t) = \left[ u''(t) + 2 \frac{\ln(c)}{w} u'(t) + \left( \frac{\ln(c)}{w} \right)^2 u(t) \right] e^{\frac{t}{w} \ln(c)}$$

$$f'''(t) = \left[ u'''(t) + 2 \frac{\ln(c)}{w} u''(t) + \left( \frac{\ln(c)}{w} \right)^2 u'(t) \right] e^{\frac{t}{w} \ln(c)} \\ + \frac{\ln(c)}{w} \left[ u''(t) + 2 \frac{\ln(c)}{w} u'(t) + \left( \frac{\ln(c)}{w} \right)^2 u(t) \right] e^{\frac{t}{w} \ln(c)} \\ = \left[ u^3(t) + 2 \frac{\ln(c)}{w} u''(t) + \left( \frac{\ln(c)}{w} \right)^2 u'(t) + \frac{\ln(c)}{w} u''(t) \right] e^{\frac{t}{w} \ln(c)} \\ + \left[ 2 \left( \frac{\ln(c)}{w} \right)^2 u'(t) + \left( \frac{\ln(c)}{w} \right)^3 u(t) \right] e^{\frac{t}{w} \ln(c)}$$

$$f^{(3)}(t) = \left[ u^3(t) + 3 \frac{\ln(c)}{w} u''(t) + 3 \left( \frac{\ln(c)}{w} \right)^2 u'(t) + \left( \frac{\ln(c)}{w} \right)^3 u(t) \right] e^{\frac{t}{w} \ln(c)}$$

step by step we have

$$f^{(n)}(t) = c^{\wedge}(t) \sum_{k=0}^{n} \binom{k}{n} \left(\frac{\ln(c)}{w}\right)^{(k)} u^{(n-k)}(t)$$

therefore

$$f \in P_{(\omega,c)}^{(n)}(\mathbb{X}) \iff \exists u \in P_{\omega}^{(n)}(\mathbb{X}) \setminus f^{(n)}(t) = c^{\wedge}(t) \sum_{k=0}^{n} \binom{k}{n} \left(\frac{\ln(c)}{w}\right)^{(k)} u^{(n-k)}(t).$$

$$\forall k \le n, \exists v_k \in P_{\omega}^{(n)}(\mathbb{X}) \setminus f^{(n)}(t) = c^{\wedge}(t)v_k \text{ then } f \in P_{(\omega,c)}^{(n)}(\mathbb{X}) \blacksquare$$

**Proposition 2.4** [12] The set $P_{(\omega,c)}^{(n)}(\mathbb{X})$ equipped with the $\|\cdot\|_{(\omega,c)}^n$ norm below, turns out to be a Banach space.

$$\|f\|_{(\omega,c)}^n = \sup_{t \in \mathbb{R}} \sum_{i=0}^{n} \|f^{(i)}(t)\|_{(\omega,c)}$$
**Proof.** see [12] ■

**Theorem 2.3** [12] Let $f \in P_{(\omega, c_1)}^{(n)}(\mathbb{C})$ and $g \in P_{(\omega, c_2)}^{(n)}(\mathbb{X})$ then $(fg) \in P_{(\omega, c_1, c_2)}^{(n)}(\mathbb{X})$.

**Proof.**

$$ 
\begin{aligned}
(fg)(t + \omega) &= f(t + \omega) \cdot g(t + \omega) \\
&= c_1 f(t) \cdot c_2 g(t) = c_1 \cdot c_2 f(t) g(t) \\
(fg)(t + \omega) &= c_1 \cdot c_2 (fg)(t)
\end{aligned}
$$

Therefore $(fg)$ is $(\omega, c_1 \cdot c_2)$-periodic

It is easy to show that

$(fg)'(t + \omega) = c_1 \cdot c_2 (f'g + fg')(t) = c_1 \cdot c_2 (fg)'(t)$, then $(fg)'$ is also $(\omega, c_1 \cdot c_2)$-periodic

suppose that $(fg)^{(n)}$ is $(\omega, c_1 \cdot c_2)$-periodic and let prove that $(fg)^{(n+1)}$ is also $(\omega, c_1 \cdot c_2)$-periodic.

Set $h = (fg)^{(n)}$ then $h' = (fg)^{(n+1)}$, if $h = (fg)^{(n)}$ is $(\omega, c_1 \cdot c_2)$ then $h' = (fg)^{(n+1)}$ is also $(\omega, c_1 \cdot c_2)$-periodic

$$ 
\begin{aligned}
h'(t) &= \lim_{\psi \to 0} \frac{h(t + \psi) - h(t)}{\psi} \\
h'(t + \omega) &= \lim_{\psi \to 0} \frac{h(t + \omega + \psi) - h(t + \omega)}{\psi} \\
&= \lim_{\psi \to 0} \frac{c_1 \cdot c_2 h(t + \psi) - c_1 \cdot c_2 h(t)}{\psi} \\
&= c_1 \cdot c_2 \lim_{\psi \to 0} \frac{h(t + \psi) - h(t)}{\psi} \\
h'(t + \omega) &= c_1 \cdot c_2 h'(t)
\end{aligned}
$$

$h' = (fg)^{(n+1)}$ is $(\omega, c_1 \cdot c_2)$-periodic

Therefore $(fg) \in P_{(\omega, c_1 \cdot c_2)}^{(n)}(\mathbb{X})$. The proof is complete.

<figure><img src="image_1.png" /></figure>

In particular cases:

(a) For $c_1 = c_2 = c$, we obtain $(fg) \in P_{(\omega, c^2)}^{(n)}(\mathbb{X})$.

(b) for $g = f$ we have the same result

Let $f$ be a continuous and bounded function. We recall that the Nemytskii operator of superposition is defined by:

$$ N_F(\varphi)(t) = F(t, \varphi(t)) $$

we have the following result:
**Theorem 2.4** [12] Let $f \in C^{(n)}(\mathbb{R} \times \mathbb{X})$, then the following assertions are equivalent:

$$ (i) \quad \forall \varphi \in P_{(\omega,c)}^{(n)}(\mathbb{X}), \quad N_f(\varphi) \in P_{(\omega,c)}^{(n)}(\mathbb{X}) $$

$$ (ii) \quad \forall (t, x) \in \mathbb{R} \times \mathbb{X}, \quad f^{(n)}(t + w, cx) = c f^{(n)}(t, x). $$

before establishing the proof of this theorem, we will prove the following very useful lemma

**Lemma 2.3** [12] $\forall (t, x) \in \mathbb{R} \times \mathbb{X}, \exists \varphi \in P_{(\omega,c)}^{(n)}(\mathbb{X}) : \varphi(t) = x$

**Proof.** Let $\psi : \mathbb{R} \to \mathbb{X}$ the function define by: $\psi(t) = c^{\frac{t}{w}}$

we have $\psi \in P_{(\omega,c)}^{(n)}(\mathbb{X})$ and $\psi(0) = 1$

set $\varphi(s) = \psi(s-t)x = c^{\frac{s-t}{w}} \cdot x = xe^{\frac{s-t}{w}Ln(c)}$, where $Ln(\cdot)$ stands for the principal value of the complex Logarithm.

$\varphi \in P_{(\omega,c)}^{(n)}(\mathbb{X})$ and $\varphi(t) = x$

Indeed $\varphi'(s) = \frac{Ln(c)}{w^2} \psi(s-t)x$

$$ \varphi''(s) = \left(\frac{Ln(c)}{w}\right)^2 \psi(s-t)x $$

step by step we have,

$$ \varphi^{(n)}(s) = \left(\frac{Ln(c)}{w}\right)^n \psi(s-t)x $$

in addition,

$$ \varphi^{(n)}(s+w) = \left(\frac{Ln(c)}{w}\right)^n \psi(s+w-t)x = \left(\frac{Ln(c)}{w}\right)^n c\psi(s-t)x = c\varphi^{(n)}(s) $$

we have just shown that $\varphi \in P_{(\omega,c)}^{(n)}(\mathbb{X})$ ■

Let us now prove Theorem 2.4

**Proof.** $\forall n = 1, 2, ...$

(i) $\Rightarrow$ (ii) Suppose that $N_f(\varphi)$ is $C^{(n)} - (\omega, c)$ -periodic.

So, $N_f^{(n)}(\varphi)$ is $(\omega, c)$ -periodic.

We have

$$ N_f^{(n)}(\varphi)(t+w) = cN_f^{(n)}(\varphi)(t) $$

$$ f^{(n)}(t+w, \varphi(t+w)) = c f^{(n)}(t, \varphi(t)) $$

$$ f^{(n)}(t+w, c\varphi(t)) = c f^{(n)}(t, \varphi(t)) $$

if we set $x = \varphi(t)$ then we get

$$ f^{(n)}(t+w, cx) = c f^{(n)}(t, x) $$

let us now prove (ii) $\Rightarrow$ (i) Suppose that

$$ \forall (t, x) \in \mathbb{R} \times \mathbb{X}, f^{(n)}(t+w, cx) = c f^{(n)}(t, x) $$

by the lemma above $\forall (t, x) \in \mathbb{R} \times \mathbb{X}, \exists \varphi \in P_{(\omega,c)}^{(n)}(\mathbb{X})$ such that $x = \varphi(t)$ then
$$
\begin{align*}
f^{(n)}(t + w, c\varphi(t)) &= c f^{(n)}(t, \varphi(t)) \\
f^{(n)}(t + w, \varphi(t + w)) &= c f^{(n)}(t, \varphi(t)) \\
N_f^{(n)}(\varphi)(t + w) &= c N_f^{(n)}(\varphi)(t)
\end{align*}
$$

Then $N_f^{(n)}(\varphi) \in P_{(\omega,c)}(\mathbb{X})$ and $N_f(\varphi)$ is $C^{(n)} - (\omega,c)$-periodic.
From where $(ii) \Rightarrow (i)$ ■

**Proposition 2.5** Let $f \in P_{(\omega,c)}^{(n)}(\mathbb{X})$. Then $f \in P_{(-w,c^{-1})}^{(n)}(\mathbb{X})$.

**Proof.** Let $f \in P_{(\omega,c)}^{(n)}(\mathbb{X})$. According to proposition(2.3), $f^{(n)}(t) = c^{\wedge}(t)v_k(t)$ with $v_k(t) \in P_{(\omega,c)}^{(n)}(\mathbb{X})$.

Then, we have $f^{(n)}(t-w) = c^{\frac{t-w}{w}} v_k(t-w) = c^{-1}c^{\wedge}(t)v_k(t) = c^{-1}f^{(n)}(t)$.

So $f \in P_{(-w,c^{-1})}^{(n)}(\mathbb{X})$ ■

3 Applications to fractional differential equa-
tions

**3.1 Linear case**

In this subsection, we mainly deal with the existence and uniqueness result of
$C^{(n)} - (\omega, c)$-periodic mild solutions to the following linear fractional differential
equation:

$$
{}^c D_t^\alpha u(t) = Au(t) + c D_t^{\alpha-1} f(t), \quad 1 < \alpha < n, \, t \in \mathbb{R}. \quad (7)
$$

**Definition 3.1** Assume that A is of sectorial type $\tilde{w} < 0$ and angle $\theta \in [0, \pi(1 - \frac{\alpha}{2}))$ which generates an operator solution $S_\alpha(t)$ which is uniformly integrable. A continuous function $u : \mathbb{R} \to \mathbb{X}$ is called a mild solution to Equation (7) on $\mathbb{R}$, if the function $s \mapsto S_\alpha(t-s)f^{(n)}(s)$ is integrable on $[-\infty, t)$ for each $t \in \mathbb{R}$ and

$$
u(t) := \int_{-\infty}^{t} S_{\alpha}(t-s)f(s)ds \quad \text{for any } t \in \mathbb{R}.
$$

**Theorem 3.1** Let {$S_\alpha(t)$}$_{t \ge 0} \subset B(\mathbb{X})$ be a uniformly integrable and strongly continuous family.

If $f \in P_{(\omega,c)}^{(n)}(\mathbb{X})$, then the function $(S_\alpha \diamond f)$ given by

$$
(S_{\alpha} \star f)(t) = \int_{-\infty}^{t} S_{\alpha}(t-s)f(s)ds
$$

is also in $P_{(\omega,c)}^{(n)}(\mathbb{X})$
**Proof.** It is obvious that:

$$ (S_{\alpha} \star f)^{(n)}(t) = \int_{-\infty}^{t} S_{\alpha}(t-s)f^{(n)}(s)ds $$

So, we have by applying the theorem (2.1)

$$ \begin{aligned} (S_{\alpha} \star f)^{(n)}(t+w) &= \int_{-\infty}^{t+w} S_{\alpha}(t+w-s)f^{(n)}(s)ds = \int_{-\infty}^{t} S_{\alpha}(t-s)f^{(n)}(w+s)ds \\ &= \int_{-\infty}^{t} S_{\alpha}(t-s)c^{\frac{w+s}{w}}v_{k}(w+s)ds = \int_{-\infty}^{t} S_{\alpha}(t-s)cc^{\wedge}(s)v_{k}(s)ds \\ &= \int_{-\infty}^{t} cS_{\alpha}(t-s)f^{(n)}(s)ds = c(S_{\alpha} \star f)^{(n)}(t) \end{aligned} $$

The proof is complete.

The following result is straightforward from the above.

**Theorem 3.2** Let {$S_{\alpha}(t)}_{t \ge 0} \subset B(\mathbb{X})$ be a uniformly integrable and strongly continuous family generated by a sectorial A of type $\tilde{w} < 0$ and angle $\theta \in [0, \pi(1 - \frac{\alpha}{2})]$. If f is $C^{(n)} - (\omega, c)$-periodic, then the the Equation (7) has a unique mild solution u which is $C^{(n)} - (\omega, c)$-periodic also.

## 3.2 Semilinear case

In this subsection, we study the existence and uniqueness result of $C^{(n)} - (\omega, c)$-periodic mild solutions to the following fractional semilinear fractional differential equation:

$$ {}^{c}D_{t}^{\alpha}u(t) = Au(t) + {}^{c}D_{t}^{\alpha-1}f(t, u(t)), \quad 1 < \alpha < 2, \ t \in \mathbb{R}. \quad (8) $$

**Definition 3.2** Assume that A is of sectorial type $\tilde{w} < 0$ and angle $\theta \in [0, \pi(1 - \frac{\alpha}{2}))$ which generates a solution operator uniformly integrable $S_{\alpha}(t)$. A continuous function $u: \mathbb{R} \to \mathbb{X}$ is called a mild solution to Equation (8) on $\mathbb{R}$, if the function $s \to S_{\alpha}(t-s)f^{(n)}(s, u(s))$ is integrable on $[-\infty, t)$ for each $t \in \mathbb{R}$ and

$$ u(t) := \int_{-\infty}^{t} S_{\alpha}(t-s)f(s, u(s))ds \quad \text{for any } t \in \mathbb{R} $$

Because of the estimate on $\|S_{\alpha}(t)\|_{B(\mathbb{X})}$, we can make the following assumptions:

(HA): The operator A is sectorial of type $\tilde{w} < 0$ and angle $\theta \in [0, \pi(1 - \frac{\alpha}{2}))$.

(HB): Let $(S_{\alpha}(t))_{t \ge 0} \subset B(\mathbb{X})$ be a strongly continuous family of linear operators generated by A.

(HC): f is $C^{(n)} - (\omega, c)$-periodic such that for all $(t, u) \in \mathbb{R} \times \mathbb{X}$, $f^{(n)}(t+w, cu) = cf^{(n)}(t, u)$ and there exist $\mu \in (0, 1]$ such that we have

$$ \|f^{(n)}(t, u) - f^{(n)}(t, v)\|_{(\omega, c)}^{n} \le \mu\|u - v\|_{(\omega, c)}^{n} \quad \forall u, v \in \mathbb{X}, $$
(HCA): f is $C^{(n)} - (\omega, c)$-periodic such that for all $(t, u) \in \mathbb{R} \times \mathbb{X}$,
$f^{(n)}(t+w, cu) = cf^{(n)}(t, u)$, and there exist $M \in (0, 1]$ such that we have

$$
\|f^{(n)}(t, u_h(t))\|_{(\omega, c)}^n \le M \|u_h(t)\|_{(\omega, c)}^n \quad \forall u \in \mathbb{X}, \quad \forall t, h \in \mathbb{R}_+
$$

With all of this in mind, we can have the following results.

**Theorem 3.3** Under previous assumptions, if we assume that (HA)-(HC)
hold, then there exist a unique $C^{(n)} - (\omega, c)$-periodic mild solution $u$ to Equa-
tion (8), provided that there is a constant $\delta_{\alpha,\theta,\mu} \in (0,1)$ such that
$\delta_{\alpha,\theta} \ge \frac{C_{\alpha,\theta} K \gamma_1 \mu |\tilde{w}|^{-\frac{1}{\alpha}} \pi}{\alpha \sin(\frac{\pi}{\alpha})}$
where $\gamma_p = \max\{1, |c|^{-p}\}, \forall p \in (0, \infty)$.

**Proof.** consider the operator $\Psi : P_{(\omega,c)}^{(n)}(\mathbb{X}) \to P_{(\omega,c)}^{(n)}(\mathbb{X})$ such that

$$
\Psi u(t) := \int_{-\infty}^{t} S_{\alpha}(t-s)f(s, u(s))ds, \quad t \ge 0,
$$

then $\Psi$ is well defined. In fact, let $u \in P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathbb{X})$ using Theorem 2.4, $s \to f(s, u(s))$ belong to $P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathbb{X})$. Then by Theorem 3.1 with $f \in P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathbb{X})$, we have $\Psi u \in P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathbb{X})$. Thus we infer that $\Psi$ maps $P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathbb{X})$ into itself.
For $u, v \in P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathbb{X})$, we get

$$
\begin{align*}
\|\Psi u(t) - \Psi v(t)\|_{(\omega,c)}^n &= \left\| \int_{-\infty}^{t} S_{\alpha}(t-s) [f^{(i)}(s, u(s)) - f^{(i)}(s, v(s))] ds \right\|_{(\omega,c)}^n \\
&\le \sum_{i=0}^{n} \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \|c|^{\alpha} (-t) S_{\alpha}(t-s) [f^{(i)}(s, u(s)) - f^{(i)}(s, v(s))] \|ds \right\} \\
&\le \sum_{i=0}^{n} \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \|c|^{\alpha} (-t) S_{\alpha}(t-s) \|\mu\|u(s) - v(s)\|_{(\omega,c)}^n ds \right\} \\
&\le \mu \sum_{i=0}^{n} \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \|S_{\alpha}(t-s)\|_{(\omega,c)} ds \right\} \|u-v\|_{(\omega,c)}^n \\
&\le C_{\alpha,\theta} K \gamma_1 \mu \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \frac{1}{1+|\tilde{w}|(t-s)^\alpha} ds \right\} \|u-v\|_{(\omega,c)}^n \\
&\le C_{\alpha,\theta} K \gamma_1 \mu \sup_{t \in [0,w]} \left\{ \int_{0}^{\infty} \frac{1}{1+|\tilde{w}|s^\alpha} ds \right\} \|u-v\|_{(\omega,c)}^n \\
&\le C_{\alpha,\theta} K \gamma_1 \mu \sup_{t \in [0,w]} \left\{ \frac{|\tilde{w}|^{-\frac{1}{\alpha}} \pi}{\alpha \sin(\frac{\pi}{\alpha})} \right\} \|u-v\|_{(\omega,c)}^n \\
&\le \frac{C_{\alpha,\theta} K \gamma_1 \mu |\tilde{w}|^{-\frac{1}{\alpha}} \pi}{\alpha \sin(\frac{\pi}{\alpha})} \|u-v\|_{(\omega,c)}^n \\
&\le \delta_{\alpha,\theta,\mu} \|u-v\|_{(\omega,c)}^n
\end{align*}
$$
Finally, $\Psi$ is a contraction. So by using the Banach fixed point theorem, there exists a unique $u \in P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathcal{X})$ mild solution to Equation(8) such that $\Psi u = u$
The proof is complete. ■

We need the following theorems for the sequel:

**Theorem 3.4 (Arzela-Ascoli):** Let $\mathcal{X}$ be a compact metric space and $\mathcal{Y}$ be a metric space. Then $A \subset C(\mathcal{X}, \mathcal{Y})$ is relatively compact if and only if both conditions are satisfied

(1) $A(x) := \{f(x), f \in A\}$ is relatively compact in $\mathcal{Y}$ for all $x \in \mathcal{X}$,

(2) A is equicontinuous

**Theorem 3.5 (Schauder):** Let E be a Banach space, C a nonempty closed convex set of E and T : C $\rightarrow$ C continuous. If T(C) is relatively compact, then T has at least one fixed point.

Let us assume that:

(HD): $f^{(n)}(t, u)$ is of Carathedory, that is, for any $t \in \mathbb{R}$, $f^{(n)}(t, u)$ is continuous with respect to $u \in P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathcal{X})$ and for any $u \in P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathcal{X})$, $f^{(n)}(t, u)$ is strongly measurable with respect to $t \in \mathbb{R}$. Now, we state and prove this additional existence theorem:

**Theorem 3.6 : (HA), (HB), (HCa) and (D) hold. Then Equation (8) has at least on $C^{(n)} - (\omega, c)$-periodic mild solution.**

**Proof.** Let $\xi = \frac{|\tilde{w}|^{-\frac{1}{\alpha}}\pi}{\alpha \sin(\frac{\pi}{\alpha})}$. We recall that $\sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \frac{1}{1+|\tilde{w}|(t-s)^{\alpha}} ds \right\} \le \xi$
Now choose

$$r \ge 2C_{\alpha,\theta} K \gamma_1 \xi M \|u\|_{(\omega,c)}^n$$

and consider $B_r = \{u \in P_{(\omega,c)}^{(n)}(\mathbb{R}, \mathcal{X}), \|u\|_{(\omega,c)}^n \le r\}$. Define the operator $\Lambda$ on $B_r$ by

$$(\Lambda u)(t) = \int_{-\infty}^{t} S_{\alpha}(t-s)f(s, u(s))ds$$

Let us observe that if $u \in B_r$. Indeed, we have
$$
\begin{aligned}
\|\Lambda u(t)\|_{(\omega,c)}^n &= \left\| \int_{-\infty}^{t} S_{\alpha}(t-s) f^{(i)}(s, u(s)) ds \right\|_{(\omega,c)}^n \\
&\le \sum_{i=0}^{n} \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \|c|^{\alpha} (-t) S_{\alpha}(t-s) f^{(i)}(s, u(s)) \| ds \right\} \\
&\le \sum_{i=0}^{n} \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \|c|^{\alpha} (-t) S_{\alpha}(t-s) \| \cdot M \|u(s)\|_{(\omega,c)}^n ds \right\} \\
&\le M \sum_{i=0}^{n} \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \|S_{\alpha}(t-s)\|_{(\omega,c)} ds \right\} \|u\|_{(\omega,c)}^n \\
&\le C_{\alpha,\theta} K \gamma_1 M \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \frac{1}{1 + |\tilde{w}|(t-s)^{\alpha}} ds \right\} \|u\|_{(\omega,c)}^n \\
&\le C_{\alpha,\theta} K \gamma_1 \xi M \|u\|_{(\omega,c)}^n \\
&\le r
\end{aligned}
$$

Now let us prove that $\Lambda$ is continuous and relatively compact. we show the continuity first.

Let $(u_n)_{n \in \mathbb{N}}$ be a sequence in $B_r$ such that $(u_n) \to u$ in $B_r$.

According to (HD) $f$ is continuous on $\mathbb{R} \times \mathbb{X}$. It comes that $f^{(n)}(s, u_n(s)) \to f^{(n)}(s, u(s))$ whenever $n \to \infty$ that is for $\epsilon > 0$, there exist $N \in \mathbb{N}$ such that for all $n \ge N$, $\|f^{(n)}(s, u_n(s)) - f^{(n)}(s, u(s))\| < \epsilon$. Then, for any $t \in \mathbb{R}$, we have

$$
\begin{aligned}
\|(\Lambda u_n)(t) - (\Lambda u_n)(t)\|_{(\omega,c)}^n &\le \sum_{i=0}^{n} \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \|S_{\alpha}(t-s)\|_{B(\mathbb{X})} \|c|^{\alpha}(-s) \right\} \\
&\quad \times \left\{ f^{(i)}(s, u_n(s)) - f^{(i)}(s, u(s)) \|ds\| \right\} \\
&\le C_{\alpha,\theta} K \gamma_1 \epsilon \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \frac{1}{1 + |\tilde{w}|(t-s)^{\alpha}} ds \right\} \\
&\le (C_{\alpha,\theta} K \gamma_1 \xi) \epsilon
\end{aligned}
$$

Now, we show the compactness of $\Lambda$, using Ascoli's theorem. First, we prove $(\Lambda u)(t) : u \in B_r$ is uniformly bounded.

$$
\begin{aligned}
\|(\Lambda u)(t)\|_{(\omega,c)}^n &\le \sum_{i=0}^{n} \sup_{t \in [0,w]} \left\{ \int_{-\infty}^{t} \|S_{\alpha}(t-s)\|_{B(\mathbb{X})} \|c|^{\alpha}(-s) f^{(i)}(s, u(s)) ds \right\} \\
&\le C_{\alpha,\theta} K \xi M \|u\|_{(\omega,c)}^n
\end{aligned}
$$

So, $\|(\Lambda u)(t)\|_{(\omega,c)}^n < \infty$ and it is proved. Now, let us prove that $\Lambda(B_r)$ is equicontinuous.
Let $t \in (-\infty, b], b \in \mathbb{R}$. For $t_1, t_2 \in (-\infty, b]$ such that $t_2 < t_1$, we have

$$
\begin{align*}
\|(\Lambda u)(t_1) - (\Lambda u)(t_2)\|_{(\omega,c)}^n &= \left\| \int_{-\infty}^{t_1} S_{\alpha}(t-s)[f^{(i)}(s, u(s))ds \right. \\
&\qquad \left. - \int_{-\infty}^{t_2} S_{\alpha}(t_2-s)[f^{(i)}(s, u(s))ds\right\|_{(\omega,c)}^n \\
&= \left\| \int_{-\infty}^{t_1} S_{\alpha}(t_1-s)[f^{(i)}(s, u(s))ds \right. \\
&\qquad \left. + \int_{t_1}^{t_2} S_{\alpha}(t_1-s)[f^{(i)}(s, u(s))ds \right. \\
&\qquad \left. - \int_{-\infty}^{t_1} S_{\alpha}(t-s)[f^{(i)}(s, u(s))\right\|_{(\omega,c)}^n \\
&\le M \int_{-\infty}^{t_2} \|S_{\alpha}(t_1-s) - S_{\alpha}(t_2-s)\|_{B(\mathbb{X})} \|u\|_{(\omega,c)}^n ds \\
&\quad + M \int_{t_2}^{t_1} \|S_{\alpha}(t_1-s)\|_{B(\mathbb{X})} \|u\|_{(\omega,c)}^n ds
\end{align*}
$$

Let $J_1 := \int_{-\infty}^{t_2} \|S_\alpha(t_1 - s) - S_\alpha(t_2 - s)\|_{B(\mathbb{X})} \|u\|_{(\omega,c)}^n ds$ and

$J_2 := \int_{t_2}^{t_1} \|S_\alpha(t_1 - s)\|_{B(\mathbb{X})} \|u\|_{(\omega,c)}^n ds$. Actually, both $J_1$ and $J_2$ tend to 0 independently of $u \in B_r$ when $t_2 \to t_1$.

Therefore the continuity of the function $t \to \|S_\alpha(t)\|_{B(\mathbb{X})}$ for $t \in (-\infty, b]$ allows us to conclude that $\lim_{t_2 \to t_1} J_1 = 0$. In the other hand, we have

$$
J_2 = \int_{t_2}^{t_1} \|S_{\alpha}(t_1 - s)\|_{B(\mathbb{X})} \|u\|_{(\omega,c)}^{n} ds \le (C_{\alpha,\theta} K \xi \|u\|_{(\omega,c)}^{n}) |t_1 - t_2|
$$

And consequently, it comes that $\lim_{t_2 \to t_1} J_2 = 0$

So, using Theorem 3.4 we have proved that $\Lambda(B_r)$ is relatively compact for $t \in (-\infty, b]$ for all $b \in \mathbb{R}$. Then $\Lambda u : u \in (B_r)$ is a family of equicontinuous functions. Then Schauder's theorem (see Theorem 3.5 allows us to conclude that Equation (8) has at least one $C^{(n)-(\omega,c)}$-periodic mild solution. $\blacksquare$

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