Existence of $S$-asymptotically $\omega$-periodic solutions for non-instantaneous impulsive semilinear differential equations and inclusions of fractional order $1 < \alpha < 2$

1.   Introduction

It is known that the action of instantaneous impulses seems not describe some certain dynamics of evolution processes in Pharmacotherapy. For example, in the case of a decompensation, (high or low levels of glucose) one can prescribe some intravenous drugs (insulin). The introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous processes. Thus, we do not expect to use the instantaneous impulses to describe such a process. In fact, the above situation is fallen in a new case of impulsive action, which starts at any arbitrary fixed point and stays active on a finite time interval. To this end, Hernándaz and O'Regan ^[1] introduced the non-instantaneous impulsive differential equations. For recent contributions on non-instantaneous impulsive differential equations and inclusions, we refer the reader to ^{[2,3,4,5,6,7]}.

There are some papers where the nonexistence of non-constant periodic solutions on closed bounded interval for differential equations with fractional order are considered such as ^{[8,9,10,11,12]}. Many authors investigated the existence of $S$-asymptotically $\omega$-periodic solutions for many types of differential equations of fractional order. For example, Maghsoodi et al. ^[13] considered an evolution equation of order$\ \alpha \in (0, 1)$ generated by an evolution system $U(\theta, s)$. Ren et al.^[12] studied semilinear differential equation of order $\alpha \in (0, 1)$ and generated by exponentially stable $C_{0}$-semigroup. Ren et al.^[14] considered semilinear differential equations of order $\alpha \in (1, 2)$ generated by a sectorial operator. Mu et al.^[15] investigated an evolution equation with the Weyl-Liouville fractional derivative of order $\alpha \in (0, 1)$ and generated by $C_{0}$-semigroup. Zhao at al.^[16] demonstrated the existence of an asymptotically almost automorphic mild solution to a semilinear fractional differential equation, and Wang et al. ^[17] studied delay fractional differential equations with an almost sectorial operator of order $\alpha \in (0, 1)$. Moreover, Muslim et al. ^[18] investigated the existence, uniqueness and stability of solutions to second order nonlinear differential equations with non- instantaneous impulses. Very recently, Alsheekhhussain et al. ^[19] proved the existence of $S$-asymptotically $w$-periodic solutions for non-instantaneous impulsive differential equations and inclusions generated by sectorial operators. For more information regarding this subject, we refer the reader to ^{[20,21,22,23,24,25]}.

It is worth noting that the problems considered in all the cited works above, except ^[19], do not contain impulseses effects and the right-hand side is a single-valued function. Moreover, to the best of the authors' knowledge, the literature concerning $S$-asymptotically $w$ -periodic solutions for differential inclusions subject to non-instantaneous impulses and generated by an infinitesimal generator of a cosine family $\{C(\theta):\theta \geq 0\}$ is very new, and this fact is the main aim in the present paper.

When the considered problem contains non-instantaneous impulses, there are two approaches in the literature to prove the existence of the solution. The first one is by keeping the lower limit of the fractional derivative at zero. The second one is by switching it at the impulsive points, which will be considered in the present paper.

Let $\alpha \in (1, 2)$, $E$ be a Banach space, $\mathbb{N}$ be the set of natural numbers, $m$ $\in$ $\mathbb{N}$, $\omega > 0, J = [0, \infty)$,

with $\lim_{i\rightarrow \infty }\theta _{i} = \infty, s_{m+i} = s_{i}+\omega; i\in \{0\}\cup \mathbb{N}$, $\theta _{m+i} = \theta _{i}+\omega$; $\ i\in \mathbb{N},$ and$\ A$ is the infinitesimal generator of cosine family $\{C(\theta):\theta \geq 0\}$. Moreover, let $\Pi :J\times E\rightarrow E$, $g_{i}:[\theta _{i}, s_{i}]\times E\longrightarrow E\; i\in \mathbb{N}$, $x_{0}\in D(A)$ (the domain of $A$ ), and $x_{1}\in E$ a fixed point.

Motivated by the above cited works, we demonstrate the existence and uniqueness of an $S$-asymptotically $\omega$-periodic solution to the following non-instantaneous impulsive semilinear differential equation:

where, $^{c}D_{0, \theta }^{\alpha }x(\theta)$ is the Caputo derivative of the function $x$ at the point $\theta$ with lower limit at $0$ ^[26].

After that, we prove the existence of $S$-asymptotically $\omega$-periodic solutions for the following non-instantaneous impulsive semilinear differential inclusion:

where $F:J\times E\rightarrow 2^{E}-\{\phi \}$ is a multi-valued function.

Unlike the differential equations of integer order, the existence of non-constant periodic solutions for fractional differential equations is not guaranteed. For this reason, the concept of an asymptotically periodic solution is introduced for fractional differential equations. Many researchers uses this approach to investigate the existence of the solution for fractional differential equations. However, up to now, there are no work studying the problem mentioned above. In this paper, we construct sufficient conditions that assure the existence of asymptotically periodic mild solutions for Problems (1.1) and (1.2). Moreover, our results generalize the obtained ones in ^[12], and our method can be used to study the existence of asymptotically periodic mild solutions for the problems considered in ^{[13,15,16,17,20,21,22,23,24,25]}, when these problems contain impulseses effects and the right hand side is a multi-valued function.

Since a multivalued function is a function values are sets, so, our technique to find an asymptotically periodic solution for Problem (2) can be used to extend many recent publications on the same subject in which the right hand side is a single-function see, for example, ^[27,28,29].

In Section 3, we prove the existence and uniqueness of $S$-asymptotically $\omega$-periodic solution for Problem ($1.1)$. Section 4 is devoted to prove the existence of $S$-asymptotically $\omega$-periodic solutions to Problem ($1.2)$. Finally, examples are given to show that the obtained results are applicable.

2.   Preliminaries and notations

Let $J_{0} = [0, \theta _{1}]$, $\ J_{i} = (\theta _{i}, \theta _{i+1}]$, and $i\in \mathbb{N}.$ Because Problem (1.1) contains non-instantaneous impulses effect, we consider the two Banach spaces:

and

where

and $x(\theta _{i}^{+})$ and $x(\theta _{i}^{-})$ are the right and left limits of $x$ at $\theta _{i}$.

Definition 2.1. Let $\omega$ be a positive real number. A function $x\in PC_{b}(J, E)$ is said to be $S$-asymptotically $\omega$-periodic if it satisfies the relation:

Definition 2.2. ^[19] By $SAP_{\omega }PC_{b}(J, E)$, we mean the Banach space of all $S$-asymptotically $\omega$-periodic functions $x\in PC_{b}(J, E)$, where the norm is given by

Definition 2.3. ^[30] A family $\{C(\theta):\theta \in \mathbb{R} \}$, where $C(\theta):D(C(\theta)) = E\rightarrow E$ is a bounded linear operator, is called a strongly cosine family if:

(ⅰ) $C(0) = I$,

(ⅱ) $C(\theta +\tau)+C(\tau -\theta) = 2C(\tau)C(\theta)$ for all $\tau, \theta \in \mathbb{R}$,

(ⅲ) the map $\theta \longmapsto C(\theta)x$ is continuous for each $x\in E$.

If $\{C(\theta):\theta \in \mathbb{R} \}$ is a strongly cosine family, then the strongly continuous sine family associated with it is defined by:

Definition 2.4. The infinitesimal generator of a cosine family $\{C(\theta):\theta \in \mathbb{R} \}$ is an operator $A:D(A)\longmapsto E$ defined by

where $D(A) = \{x\in E:C(t)x$ is twice continuously differentiable of $t\}.$

Lemma 2.1. (^[30], Propositions 2.2 and 2.3]) Let $\{C(t):t\in \mathbb{R} \}$ be a strongly cosine family in $E$ with infinitesimal generator $A$ and

Then, the following statements hold:

1- $D(A)$ is dense in $E$, and $A$ is a closed operator.

2- If $z\in E$, then $S(\theta)z\in Z$.

3- If $z\in Z$, then

(ⅰ) $S(\theta)z\in D(A)$ and $\frac{d^{2}}{d\theta ^{2}}S(\theta)z = AS(\theta)z,$

(ⅱ) $S(\theta)z\in D(A)$ and $\frac{d}{d\theta }C(\theta)z = AS(\theta)z.$

4- If $z\in D(A),$ then

(ⅰ) $C(\theta)z\in D(A$) and $\frac{d^{2}}{d\theta ^{2}}C(\theta)z = AC(\theta)x = C(\theta)Az;$

(ⅱ) $S(\theta)z\in D(A)$ and $AS(\theta)z = S(\theta)Az.$

Definition 2.5. (^[31]) By a mild solution for Problem (1.1), we mean a function $x\in PC(J, E)$ such that

where $q = \frac{\alpha }{2}$, and, for $\vartheta \geq 0$,

and

Remark 2.1. The solution function given by (2.1) satisfies the following properties:

1- $x(0) = C_{q}(0)x_{0} = x_{0}$.

2 $x {\acute{}} (0) = x_{1}$.

3- $x\ $ is continuous on $J_{i}; \ i\in \{0\}\cup \mathbb{N}$.

We will need the following lemma which gives some properties for the operators $C_{q}(\theta), K_{q}(\theta)$ and $P_{q}(\theta)$.

Lemma 2.2. (^[31], Lemma 8). Assume that

$(HA)\ A:D(A)\rightarrow E$ is the infinitesimal generator of strongly continuous cosine family of linear operators $\{C(\theta):\theta \geq 0\}$ which is uniformly bounded by $M$ $> 0$. Then,

(ⅰ) For any fixed $\theta \geq 0, C_{q}(\theta), \, K_{q}(\theta)$ and $P_{q}(\theta)$ are linear bounded operators.

(ⅱ) For $\gamma \in \lbrack 0, 1], \, \int_{0}^{\infty }\theta ^{\gamma }\xi _{\alpha }(\theta)d\theta = \frac{\Gamma (1+\gamma)}{\Gamma (1+\alpha \gamma)}.$

(ⅲ) If $||C_{q}(\theta)||\leq M, \, \theta \geq 0,$ then for any $x\in E, \ ||C_{q}(\theta)x||\ \leq M||x||$, $||K_{q}(\theta)x||\ \leq \theta M||x||$ and $||P_{q}(\theta)x||\ \leq \frac{M}{\Gamma (2q)}||x||\theta ^{q}.$

(ⅳ) $\{C_{q}(\theta), \theta \geq 0\}, \{K_{q}(\theta), \theta \geq 0\}$ and $\{\theta ^{q-1}P_{q}(\theta), \theta \geq 0\}$ are strongly continuous.

3.   Existence and uniqueness of an $S$-asymptotically $\omega$-periodic mild solution for Problem $(1.1).$

We make the following assumptions:

$(HA)^{\ast }A:D(A)\rightarrow E$ satisfies $(HA)$, and the family $\{C(\theta):$ $\theta \geq 0\}$ is exponentially stable. That is, there exist positive numbers $a,$ $M$ such that $||C(\theta)||\leq e^{-a\theta }M$, $\theta \geq 0.$

$(H\Pi)\ \Pi :J\times E\rightarrow E$ is a strongly measurable function, and there are $h_{1}, h_{2}\in C(J, \mathbb{R} ^{+})$ such that $h_{1}$is bounded,

and

$(Hg)$ For any $i\in \mathbb{N}$, $g_{i}:[\theta _{i}, s_{i}]\times E\longrightarrow E\ \ (i\in \mathbb{N})$ such that, for any $x\in E$, the function $\theta \mapsto g_{i}(\theta, x)$ is differentiable at $s_{i}$ , and that:

(ⅰ)

and

(ⅱ) There are $N > 0$ such that

(ⅲ) There is $\mathcal{N} > 0$ such that

(ⅳ) There is $\kappa _{1} > 0$ such that

(ⅴ) There is $\kappa _{2} > 0$ with

The following lemma provides additional properties for the operators $C_{q}(\theta)$ and $P_{q}(\theta)$ when $\{C(\theta):\theta \geq 0\}$ is exponentially stable.

Lemma 3.1. (^[32], Proposition 2.1). If $(HA)^{\ast }$ is verified, then there is $L > 0$ such that

Lemma 3.2. (^[33], Lemma 2.11]) Let $\gamma \in \lbrack 0, 1]$, $0 < a < b$. Then, $|b^{\gamma }-a^{\gamma }|\leq (b-a)^{\gamma }.$

Remark 3.1. In what follows, we mean by $||\ \ \ ||$ the norm in the Banach space $E.$

Theorem 3.1. Under conditions $(HA)^{\ast }, (H\Pi), (Hg_{i})$ and $(H)$, Problem (1.1) has a unique $S$-asymptotically $\omega$-periodic mild solution providedthat the following assumptions are verified:

and

where $h_{1}$ and $h_{2}$ are specified in $(H\Pi)$.

Proof. First, we clarify that if $x\in SAP_{\omega }PC_{b}(J, E)$, then the function $\Phi (x)$ defined by

belongs to $SAP_{\omega }PC_{b}(J, E)$. The proof will be given in the following steps.

Step 1. we will show that $\lim_{\theta \rightarrow \infty }||\Phi (x)(\theta +\omega)-\Phi (x)(\theta)|| = 0$.

Let $\epsilon > 0$. Because $x\in SAP_{\omega }PC(J, E)$, $\lim_{\theta \rightarrow \infty }||x(\theta +\omega)-x(\theta)||_{E} = 0$, and hence there is $\theta _{\epsilon } > \theta _{1}$ such that

Let $\theta > \theta _{\epsilon }$. If $\theta \in (\theta _{i}, s_{i}]$, $i\in \mathbb{N}$, then $\theta +\omega \in (\theta _{i}+\omega, s_{i}+\omega] = (\theta _{i+m}, s_{i+m}]$. So, relations (3.3), (3.5) and (3.14) imply that

Let $\theta \in \lbrack s_{i}, \theta _{i+1}], i\in \mathbb{N}$. Then, $\theta +\omega \in \lbrack s_{i}+\omega, \theta _{i+1}+\omega] = [s_{i+m}, \theta _{i+m+1}]$. By arguing as in (3.16), one obtains

Similarly, by (3.4) and (3.6), we get

Next, notice that

Then,

Note that, from Lemma 3.1, $(\theta +\omega)^{q}-\theta ^{q}\leq \omega ^{q}$. Hence, by taking into account $\tau \in \lbrack -\omega, 0]\Longrightarrow \tau +\omega \in \lbrack 0, \omega]$, it yields from (3.9)

where, $\varkappa = \sup_{s\in \lbrack 0, \omega], ||v||\leq ||x||_{SAP_{\omega }PC_{b}(J, E)}}||\Pi (s, v)||.$

Next, by (3.1), (3.2), (3.9), (3.10) and (3.15), we get

and

where $c_{1} = \sup_{\tau \in \lbrack 0, \theta _{\epsilon }]}||x(\tau +\omega))-x(\tau)||$ and $c_{2} = \sup_{\tau \in \lbrack 0, \theta _{\epsilon }]}h_{1}(\tau)$. Combining (3.19–3.22), one obtains,

Similarly,

Note that $s_{i}\rightarrow \infty$ when$\ \theta \rightarrow \infty$. Therefore, using (3.16)–(3.18), (3.13) and (3.24), we derive $\lim_{\theta \rightarrow \infty }||\Phi (x)(\theta +\omega)-\Phi (x)(\theta)|| = 0$.

Step 2. We show that, for any $x\in SAP_{\omega }PC_{b}(J, E)$, $\Phi (x)$ is bounded.

Let $\theta \in J$.

(ⅰ) Let $\theta \in \lbrack 0, \theta _{1}]$. Then, applying Lemma (1.2) (ⅲ), (3.9) and (3.14), one gets

On the hand, from (3.1), we get

By (3.25) and (3.26), we get

where $B$ is the beta function. Hence, $y$ is bounded on $[0, \theta _{1}]$.

(ⅱ) If $\theta \in (\theta _{i}, s_{i}]$, $i\in \mathbb{N}$, then, by (3.7), it yields

(ⅲ) If $\ \theta \in (s_{i}, \theta _{i+1}]$, then it follows from (3.8) and Lemma (1.2) that

Moreover, as in (3.27),

Similarly, we can derive

As a result of (3.27)–(3.31), we conclude that $y$ is bounded on $J$.

Now, $\Phi (x)$ is continuous on $J_{i}\, i\in \{0\}\cup \mathbb{N}$, and, hence, from Steps 1 and 2, we confirm that $\Phi (x)\in SAP_{\omega }PC_{b}(J, E)$. Thus, $\Phi$ is a function from $SAP_{\omega }PC_{b}(J, E)$ to itself.

Step 3. We show in this step that $\Phi$ is a contraction mapping from $SAP_{\omega }PC_{b}(J, E)$ to $SAP_{\omega }PC_{b}(J, E)$.

To show this, let $x, y\in SAP_{\omega }PC_{b}(J, E)$.We have three cases.

Case 1. $\theta \in \lbrack 0, \theta _{1}]$

Using (3.14), it yields

Using Lemma 1.2, (3.2), (3.4), (3.8) and (3.9), relation (3.32) becomes

Case 2. $\theta \in (\theta _{i}, s_{i}]$, $i\in \mathbb{N}$. Relations (3.5) and (3.14) lead to

where $N = \max_{1\leq i\leq m}\{N_{i}\}$.

Case 3. $\theta \in \lbrack s_{i}, \theta _{i+1}], i\in \mathbb{N}.$ It yields from (3.5), (3.6) and (3.14)

and

where $\ \mathcal{N} = \max_{1\leq i\leq m}\{\mathcal{N}_{i}\}$.

Moreover, similar to (3.33),

and

Due to (3.33)–(3.38), we conclude that

It yields from (3.11) and (3.39) that $\Phi$ is contraction. Applying the Banach fixed point theorem, we have that $\Phi$ has a unique fixed-point which is an $S$-asymptotically$\ \omega$-periodic solution to Problem ($1.1$).

Remark 3.2. If $h_{1}$is bounded on $J$, then relation (3.10) is verified. In fact, suppose that $h_{1}(\tau)\leq \kappa, \forall \tau \in J$. We have

where $B$ is the beta function. Thus, (3.10) is verified.

Remark 3.3. If $\ \lim_{\theta \rightarrow \infty }\int_{0}^{\theta }h_{2}(\tau)d\tau = 0,$ then relation (3.13) is verified. In fact, we have

Corollary 3.1. Suppose that conditions $(HA)$ and$\ (Hg_{i})$ $)$ are satisfied. If $(H\Pi)$ is verified with $h_{1}(\tau)\leq \kappa, \forall \tau \in J$, and $\lim_{\theta \rightarrow \infty }\int_{0}^{\theta }h_{2}(\tau)d\tau = 0$ then, by (3.41) and Theorem (1.1), Problem (1.1) has a unique $S$-asymptotically $\omega$-periodic provided that

Remark 3.4. If there is no impulses effect, then $N = \mathcal{N} = 0$. Hence, relations (3.42) becomes $2L\kappa B(q, q) < 1$.

4.   $S$-asymptotically $\omega$-periodic mild solutions for Problem $(1.2)$

In this section, we demonstrate the existence of $S$-asymptotically $\omega$-periodic mild solutions for 1.2. We denote by $P_{ck}(E)$ the family of non-empty, convex and compact subsets of $E.$

Consider the following assumptions:

$(HF)\, F:J\times E\rightarrow P_{ck}(E)$ is a multi-valued function such that:

(ⅰ) For any $z\in E$, the multi-valued function $\theta \rightarrow$ $F(., z)$ is strongly measurable.

(ⅱ) For any $x\in PC(J, E)$, the set

is not empty.

(ⅲ) There is a measurable bounded, almost everywhere, function $L_{1}:J\rightarrow J$ such that

where $h$ is the Hausdorff distance.

(ⅳ) There is $L_{2}\in C(J, \mathbb{R} ^{+})$ such that

(ⅴ) The function

is bounded almost everywhere on $J.$

We need the following Lemma, which is due to Covitz and Nadler ^[34].

Lemma 4.1. Let $(X, d)$ be a metric space and $G$ be a contraction multi-valuedfunction from $X$ to the family of non-empty closed subsets of $X$. Then, $G$ has a fixed point.

Theorem 4.1. Under conditions $(HA)^{\ast }, (HF), (Hg_{i})$ and $(H)$, Problem (1.2) has an $S$-asymptotically $\omega$-periodic mild solution provided that

and

where $|L_{1}(t)|\leq \lambda _{1}$, $a.e.\, t\in J$.

Proof. Due to $(HF) (ii)$, for any $x\in SAP_{\omega }PC_{b}(J, E),$ the set $S_{F(., x(.))}^{1}$ is not empty. Therefore, for any $x\in SAP_{\omega }PC_{b}(J, E)$, we can define a multi-valued function $R(x)$ as follows: an element $y\in R(x)$ if and only if

where $f\in S_{F(., x(.))}^{1}$. Since the proof is similar to what was shown in Theorem 1.1, we will illustrate only the differences.

Step 1. We show that if $x\in SAP_{\omega }PC_{b}(J, E)$ and $y\in R(x)$, then $\lim_{\theta \rightarrow \infty }||y(\theta +\omega)-y(\theta)|| = 0$.

Let $\epsilon > 0$. Because $x\in SAP_{\omega }PC_{b}(J, E)$, then $\lim_{\theta \rightarrow \infty }||x(\theta +\omega)-x(\theta)|| = 0$ and, hence, there is $\theta _{\epsilon } > \theta _{1}$ such that (3.15) is verified.

Let $y\in R(x)$ and $\theta \in \lbrack s_{i}, \theta _{i+1}]$. According to (4.8), we have

Let $\tau \in \lbrack -\omega, 0]$ be fixed. Since $F(\tau +\omega, 0)$ is compact, there is $v_{\tau +\omega }\in$ $F(\tau +\omega, 0)$ such that

From (4.1), (4.3) and (4.10), we get

Then, by (3.9) and (4.11), it follows that

where $\lambda _{2}$ is a positive number such that $\sigma (\theta)\leq \lambda _{2}, a.e., \theta \in J.$

Next, let $\tau \in \lbrack 0, \theta]$ be fixed. From the fact that$\ F(\tau +\omega, x(\tau))$ is compact, there are $z_{\tau +\omega }, z_{\tau }\in F(\tau, x(\tau +\omega))$ such that $d(f(\tau +\omega), z_{\tau +\omega }) = d(f(\tau +\omega), F(\tau, x(\tau +\omega)))$ and $d(f(\tau), z_{\tau }) = d(f(\tau), F(\tau, x(\tau +\omega))).$ Then, by (4.1) and (4.2), it yields

Moreover, according to (4.1) and (4.3), we get

Then, by (4.13) and (4.14), one obtains

Using (4.4)–(4.6) and (4.15), it yields

Note that $\tau _{i}\rightarrow \infty$ when$\ \theta \rightarrow \infty$. Hence, as above, we derive

Then, due to (3.16)–(3.18), (4.9), (4.12), (4.16) and (4.17), we conclude that

Step 2. In this step, we show that if $x\in SAP_{\omega }PC_{b}(J, E)$ and $y\in R(x)$, then $y$ is bounded.

Let $\theta \in \lbrack 0, \theta _{1}]$. Then, using Lemma 1.2, (3.9) and (4.8), one has

On the hand, from (4.1), we get

By (4.18) and (4.19), it yields

where $B$ is the beta function. Therefore, $y$ is bounded on $[0, \theta _{1}]$. Similarly, one can show that if $\theta \in \lbrack s_{i}, \theta _{i+1}]$, then

and

Then, by (4.20)–(4.22) and by arguing as in (3.28) and (3.29), we deduce that $y$ is bounded on $J$, and our claim in this step is proved.

As a result of Eqs 1.1 and 1.2, $R$ is a multivalued function from $SAP_{\omega }PC_{b}(J, E)$ to the non-empty subsets of $SAP_{\omega }PC_{b}(J, E).$

Next, in order to apply Lemma 3.2 and show that $R$ has a fixed point, we have to show that $R$ is a contraction where its set of values is closed. We do this in two steps.

Step 3. The set of values of $R$ is closed.

Let $x\in SAP_{\omega }PC_{b}(J, E)$ and $(y_{n})_{n\geq 1}$be a sequences in $R(x)$ with $y_{n}\rightarrow y$ in $SAP_{\omega }PC_{b}(J, E)$. Then, there is $f_{n}$ $\in S_{F(., x(.))}^{1}$ such that

We have to show that $y\in R(x)$. By arguing as in (4.19) , one obtains

Now, let $\theta$ be a fixed point in $J$, and $J_{\theta } = [0, \theta]$. From the fact that $\sigma$ and $L_{1}$ are bounded almost everywhere, we can deduce, from (4.24), that the family $\{f_{n}:n\geq 1\}$ is bounded in $L^{2}(J_{\theta }, E)$ and, hence, it is weakly compact in $L^{2}(J_{\theta }, E)$. Thus, it has a subsequence, denoted again by $(f_{n})_{n\geq 1}$, such that $f_{n}\rightarrow f$ weakly in $L^{2}(J_{\theta }, E).$ According to Mazur's lemma, we can find a sequence $(z_{n})_{n\geq 1}$ of convex combinations of $f_{n}$ with $z_{n}\rightarrow f$ strongly in $L^{2}(J_{\theta }, E)$. Then, we can assume, without loss of generality, that $z_{n}(\tau)\rightarrow f\ (\tau), a.e.\, \tau \in J_{\theta }$. Moreover, from (4.24) and Lemma 1.2, we get

Note that the function $\tau \rightarrow (\theta -\tau)^{2q-1}$ belongs to $L^{1}\left([0, \theta], E\right)$. Therefore, by the continuity of $P_{q}(.)$ and applying the Lebesgue dominated convergence theorem, it yields

Thus, from (4.25) and the continuity $P_{q}(.)$ , it follows, by taking the limit as $n\rightarrow \infty$ in (4.23), that

Note that $(HF)(iv)$ leads to$\ f(s)\in F(s, x$ $(s)), a.e.\ s\in J$ and, hence, (4.26) leads to

Then, $y\in R(x)$.

Step 4. We show that $R$ is a contraction.

Let $u_{1}, u_{2}\in SAP_{\omega }PC(J, E)$ and $y_{1}\in R(u_{1})$. Then, there is $f\in S_{F(., u(.))}^{1}$ such that

Consider the multivalued function$\ \Theta :J\rightarrow 2^{E}$ defined by:

We show that the set of values of $\Theta$ is non-empty. Let $\theta \in J$. From (4.1), we get

Thus, from the compactness of $F(\theta, u_{2}(\theta))$, there is $z_{\theta }\in F(\theta, u_{2}(\theta))$ such that

which leads to $\Theta (\theta)\neq \phi, \, \theta \in J$. Moreover, the set $\Lambda (\theta) = \Theta (\theta)\cap F(\theta, u_{2}(\theta)), \, \theta \in J$ is not empty. Because the functions $f_{1}, L_{1}, u_{1}, u_{2}$ are measurable, Proposition 3.4 in ^[35] (Corollary 1.3.1(a) in ^[36]) guarantees that the multivalued map $\theta \rightarrow \Lambda (\theta)$ is measurable. Note that $\Theta (\theta), \, \theta \in J$ is closed. Consequently, the set of values of $\Lambda$ is non-empty and compact and, hence, by Theorem 3.1.1 in ^[37], there exists a measurable selection $f_{2}$ for $\Lambda$ with

Set

Obviously, $y_{2}\in R(u_{1})$. Now, we estimate the value of $||y_{1}-y_{2}||.$ Let $\theta \in \lbrack 0, \theta _{1}]$. Using Lemma 1.2, (3.8), (3.9) and (4.27)–(4.29), we get

Let $\theta \in \lbrack s_{i}, \theta _{i+1}]$, $i\in \mathbb{N}$. As in (4.30), one can show that

and

Combining relations (3.34)–(3.36) and (4.30)–(4.32), it yields

Due to (4.7), relation (4.33) becomes

where $\vartheta = MN+M\omega \mathcal{N}+2L\lambda _{1}B(q, q) < 1$. By interchanging the role of $y_{1}$ and $y_{2}$ in the above discussion and using (4.7) and (4.34), we conclude that $R$ is a contraction.

As a result of Steps 1.1–3.1 and by applying Lemma (3.2), $R$ has a fixed-point which is $S$-asymptotically $\omega$-periodic solution to Problem($1.2$)$.$

Remark 4.1. As in Remark (2.1), if $\ \lim_{\tau \rightarrow \infty }\sigma (\tau) = \lim_{\tau \rightarrow \infty }L_{1}(\tau) = \lim_{\tau \rightarrow \infty }L_{2}(\tau) = 0$, then relations (4.4)–(4.6) are verified.

Remark 4.2. If there is no impulses effect, then $N = \mathcal{N} = 0$ and, hence, relation (4.7) becomes $2L\lambda _{1}B(q, q) < 1.$

5.   Examples

In this section, we give two examples as applications of our results.

Example 5.1. Let $\alpha = \frac{3}{2}, \ q = \frac{3}{4}, \ E = L^{2}[0, \pi]$, $m = 4, \ \omega = 2\pi, \ J = [0, \infty)$, $s_{i} = i\frac{\pi }{2}, \, \ i\in \{0\}\cup \mathbb{N}$, and $\theta _{i} = (2i-1)\frac{\pi }{2};i\in \mathbb{N}$. Observe that $s_{4} = \omega$ and for $i\in \mathbb{N}$, $s_{i+m} = s_{i+4} = (i+4)\frac{\pi }{2} = i\frac{\pi }{2}+2\pi = s_{i}+\omega,$ and $\theta _{i+m} = \theta _{i+4} = (2i+7)\frac{\pi }{4} = (2i-1)\frac{\pi }{4} +2\pi = \theta _{i}+2\pi = \theta _{i}+\omega$.

Consider an operator $A:D(A)\subset E\rightarrow E$ defined as follows: $Av = v^{\prime \prime }$and

Note that the operator $A$ has the representation

where $x_{n}(y) = \sqrt{2}\ \sin ny, n = 1, 2, ...$, is the orthonormal set of eigenfunctions of $A.$ Moreover, $A$ is the infinitesimal generator of a strongly continuous cosine family $C(t)_{t\in \mathbb{R} }$ which is given by

and the associated sine family $S(t)_{t\in \mathbb{R} }$ on $E$ is defined by

It is known that $||C(t)||\leq e^{-\pi ^{2}t}$ and $||S(t)||\leq e^{-\pi ^{2}t}$ for $t\geq 0$ (see ^[38], P.1307). Therefore, the family $\{C(\theta):$ $\theta \geq 0\}$ is exponentially stable and the operator $A$ satisfies $(HA)^{\ast }$ with $M = 1.$ Consider a function $\Pi :J\times E\rightarrow E$ defined by

where $\kappa > 0$. We demonstrate that $\Pi$ satisfies the conditions of Corollary (1.1). Let $u, v\in E = L^{2}[0, \pi]$. One has

Moreover,

Relations (5.3) and (5.4) leads to $(H\Pi)$, where $h_{1}(\theta) = \kappa$ and $h_{2}(\theta) = 0, \, \theta \in J$.

Next, for any $i\in \mathbb{N}$, let $g_{i}:[t_{i}, s_{i}]\times E\rightarrow E$, be defined as:

where $\lambda$ is a positive real number. Then,

Obviously, $g_{i}$ is bounded on bounded subsets. Note that, for any $i\in \mathbb{N},$ any $\theta \in J,$ and any $u, v\in E$, we have

and

In addition,

and

Furthermore,

and

As a result of relations (5.6)–(5.11), $(Hg_{i})$ is satisfied where $N = \mathcal{N} = \lambda$ and $\kappa _{1} = \kappa _{2} = \lambda.$ By applying Corollary (1.1), we conclude that Problem (1.1) has a unique $S-$asymptotically$\ 2\pi -$periodic mild solution provided that

where $A$, $\Pi, g_{i}$ are given by (5.1), (5.2) and (5.5), respectively, and $L$ appears in (3.9). By choosing $\lambda$ and $\kappa$ sufficiently small, we can derive (5.12).

Example 5.2. Assume that $\ A, \alpha, q, \ E$, $m, \ \omega, \ J,$ $s_{i}, \ \theta _{i}, \, i\in \mathbb{N}$ are as in Example (1.1). Let $Z$ be a non-empty convex compact subset of $E, \ L_{1}:J\rightarrow J$ be a measurable bounded almost everywhere function such that $Lim_{\theta \rightarrow \infty }L_{1}(\theta) = 0$ and $F:J\times E\rightarrow P_{ck}(E)$ be a multi-valued function defined by

where $\sigma$ is a constant such that $\ Sup\{\ ||z\ ||\ :\ z\in Z\}\leq \sigma.$ Clearly, for every $u\in E, \, \theta \rightarrow F(\theta, u)$ is strongly measurable and, for any $x\in PC(J, E)$, the function $f(\theta) = \frac{L_{1}(\theta)\ ||x(\theta)||\ \sin \theta \ }{\sigma \ (1+||x(\theta)||)}z_{0}, \,$ $z_{0}\in Z$ is locally integrable, and $f(\theta)\in F(\theta, x(\theta)), \, \theta \in J$. Moreover, using (5.13), for any $u, v\in E\ $and any $\theta \in J,$ we have

and

Then, from (5.14)–(5.16), it follows that assumption $(HF)$ is verified where $L_{2}(\theta) = \sigma (\theta) = 0, \, \ \theta \in J$. Thus, applying Theorem 1.2, Problem (1.2), where $A$, $F, g_{i}$ are given by (5.1), (5.13) and (5.5), respectively, and $L$ appears in (3.9), has an $S-$asymptotically$\ 2\pi -$periodic mild solution provided that

where $\lambda _{1}$ is a positive number such that $|L_{1}(\theta)|\leq$ $\lambda _{1}, a.e.$ for $\theta \in \, J$.

6.   Conclusions

Because, in some works, it was demonstrated that there are no non-stationary periodic solutions of fractional differential equations, studying the existence of $S$-asymptotically $\omega$-periodic solutions for fractional differential equations is necessary and important. Sufficient conditions that assure the existence of $S$-asymptotically $\omega$-periodic solutions for non-instantaneous impulsive semilinear differential equations of order $1 < \alpha < 2$ and generated by the infinitesimal generator of a strongly continuous cosine family of bounded linear operators have been obtained. Also, the case when the single-valued function in the right-hand side is replaced by a multi-valued function is investigated. Examples are given to demonstrate the possibility of applicability of our results. Moreover, our results generalize the obtained one in ^[12] into the case where the order is $1 < \alpha < 2,$ there are non-instantaneous impulse effects, and the right-hand side is a multi-valued function instead of a single-valued-function. Furthermore, our technique can be used to extend many problems that are considered in the literatures such as ^{[13,15,16,17,20,21,22,23,24,25,27,28,29]} to the case where there are non-instantaneous impulse effects and the right-hand side is a multi-valued function instead of a single-valued-function.

Acknowledgments

This research has been funded by the Scientific Research Deanship at University of Ha'il-Saudi Arabia through project number RG-21 101.

Conflict of interest

The authors declare that they have no conflict of interest.