Existence and stability analysis of solutions for periodic conformable differential systems with non-instantaneous impulses

1.   Introduction

In the natural sciences and human social activities, impulsive and periodic phenomena are both widespread and of profound significance. Impulse systems, a prevalent mathematical model, are used to depict sudden change phenomena in practical problems across various fields such as economics, population dynamics systems, physics, machinery, and biotechnology. Not only can they fully consider the impact of perturbation or disturbance on the system state, but they can also accurately reflect the characteristics of the system itself. With the deepening of research, researchers have conducted extensive studies on the impulsive effect, such as ^{[1,2,3,4,5,6,7,8,9]}.

Periodic theory is an attractive subject in the qualitative theory of differential equations. Applications in fields such as physics, mathematical biology, control theory, and other technical sciences are of great significance. There are many papers concerning the periodic systems, and they have yielded a variety of valuable outcomes, such as ^[10,11]. Further, in biomathematics, periodic phenomena and impulsive phenomena often occur simultaneously within one system. Hence, many scholars focus on the periodic differential systems with impulsive effects, such as ^{[12,13,14,15,16,17]}. As an extension and expansion of integer order calculus, a large number of scholars have conducted research on different types of derivatives and obtained many excellent results, such as ^[18,19]. Subsequently, following the introduction of the concept of conformable derivatives ^[20], many results have been published about conformable derivatives, such as ^{[21,22,23,24,25,26,27,28,29,30,31,32]}. Previous research has focused on the existence, uniqueness, and stability of solutions to non-instantaneous impulsive differential equations, as well as the existence and asymptotic stability of periodic solutions. However, the relationship between the solutions and the periodic solutions of periodic conformable differential systems with non-instantaneous impulses has not been investigated. Such research can help us establish a more complete theoretical system and further deepen our understanding of related issues. In this paper, we mainly analyze the existence and stability of solutions for the systems given as below:

and

and

in which $\mathcal{B}\in\mathbb{R}^{n\times n}$ and $\mathcal{C}_{l}\in\mathbb{R}^{n\times n}$, $a:\bigcup\limits_{l = 0}^{\infty}(\varsigma_{l}, \iota_{l+1}]\to \mathbb{R}^{n}$ and $\mathcal{A}:\bigcup\limits_{l = 0}^{\infty}(\varsigma_{l}, \iota_{l+1}]\times\mathbb{R}^{n}\to \mathbb{R}^{n}$. $\{\iota_{l}\}_{l\in\mathbb{N}_{0}}$ and $\{\varsigma_{l}\}_{l\in\mathbb{N}_{0}}$ satisfy $\mathcal{Q} = \iota_0 = \varsigma_0 < \iota_1\leq\varsigma_1 < \cdots < \iota_{l}\leq\varsigma_{l} < \iota_{l+1}$ and $\mathcal{B}(\mathcal{I}+\mathcal{C}_{l}) = (\mathcal{I}+\mathcal{C}_{l})\mathcal{B}$ for $l\in \mathbb{N}.$ What's more, $\mathcal{I}$ denotes the unit matrix.

We propose the following assumptions:

$(G_{1})$ There are $q\in\mathbb{N}$ and $\delta\in \mathbb{R}_+$ satisfy $\iota_{l+q} = \iota_{l}+\delta, \; l\in\mathbb{N}$ and $\varsigma_{l+q} = \varsigma_{l}+\delta, \; l\in\mathbb{N}_0$.

$(G_{2})$ For each $l\in \mathbb{N}$, $\mathcal{C}_{l+q} = \mathcal{C}_{l}$ and $b_{l+q} = b_{l}$.

Among the 6 sections of this paper, section 2 introduces the conformable Cauchy matrix $\Xi(\cdot, \cdot)$ and discusses its properties. Section 3 discusses the results concerning the periodicity and stability of the solution for (1.1). Section 4 gives the expression of the solution and sufficient conditions for the existence of a periodic solution for (1.2). Section 5 studies the periodic solution of (1.3). Last, examples are given to verify our results.

2.   Preliminaries

Set $\mathbb{Q} = [\mathcal{Q}, +\infty)$ and $PC(\mathbb{Q}, \mathbb{R}^{n}): = \{\gamma: \mathbb{Q}\rightarrow \mathbb{R}^{n}: \gamma\in C\big((\iota_{l}, \iota_{l+1}], \mathbb{R}^{n}\big), \; l\in\mathbb{N}_0,$ there exists $\gamma(\iota_{l}^{-})$ and $\gamma(\iota_{l}^{+}), \; l\in\mathbb{N}$ with $\gamma(\iota_{l}^{-}) = \gamma(\iota_{l})\}$, where $C\big((\iota_{l}, \iota_{l+1}], \mathbb{R}^{n}\big)$ denotes the space of all continuous functions from $(\iota_{l}, \iota_{l+1}]$ into $\mathbb{R}^{n}$ with $\|\gamma\| = \sup\limits_{\iota\in \mathbb{Q}}\|\gamma(\iota)\|$. One denotes $a = (a_{1}, \cdots, a_{n})^{\top}\in \mathbb{R}^{n}$ with $\|a\| = \max\limits_{1\leq i\leq n}|a_{i}|$ and $\alpha\in\mathbb{R}^{n\times n}$ with $\|\alpha\| = \max\limits_{1\leq i\leq n}\sum\limits_{j = 1}^{n}|\alpha_{ij}|$.

To start, one presents the relevant concepts.

Definition 2.1. (see [20, Definition 2.1]) The conformable derivative of a function $\gamma:\mathbb{Q}\rightarrow \mathbb{R}$ is

Remark 2.2. If $\gamma\in C^{1}(\mathbb{Q}, \mathbb{R})$, then $\mathfrak{D}_{\kappa}^{\mathcal{Q}}\gamma(\iota) = (\iota-\mathcal{Q})^{1-\kappa}\gamma'(\iota)$.

Definition 2.3. (see [20, Notation]) The conformable integral of a function $\gamma:\mathbb{Q}\rightarrow \mathbb{R}$ is

if $\mathcal{Q} = 0$, then we write $d_{\kappa}(\varsigma, \mathcal{Q})$ as $d_{\kappa}(\varsigma)$.

In order to derive the solution for (1.1), we present this definition.

Definition 2.4. Denote the conformable Cauchy matrix $\Xi(\cdot, \cdot)$ as

where $\phi(\mathcal{Q}, \iota)$ denotes the number of $\iota_{l}$ existing in $(\mathcal{Q}, \iota)$ and $z_{+}: = \max\{0, z\}$ for $z\in\mathbb{R}$. If $\phi(\mathcal{Q}, \iota) = \phi(\mathcal{Q}, \varsigma)$, then $\prod\limits_{l = \phi(\mathcal{Q}, \varsigma)+1}^{\phi(\mathcal{Q}, \iota)}(\mathcal{I}+\mathcal{C}_{l}) = \mathcal{I}, \; \sum\limits_{l = \phi(\mathcal{Q}, \varsigma)}^{\phi(\mathcal{Q}, \iota)-1} = 0$.

Theorem 2.5. The solution $\gamma(\iota, \varsigma; \gamma_{\varsigma})\in PC(\mathbb{Q}, \mathbb{R}^{n})$ of (1.1) with $\gamma(\varsigma) = \gamma_{\varsigma}$ is

Particularly,

Proof.

There are many conditions to consider.

Condition 1: $\phi(\mathcal{Q}, \iota) = \phi(\mathcal{Q}, \varsigma)$.

(ⅰ) Set any $\iota, \varsigma\in (\varsigma_{l}, \iota_{l+1}]$, for $\iota\in[\varsigma_{0}, \iota_{1}]$, by Remark 2.2, we obtain

and when $\iota\in (\iota_{1}, \varsigma_{1}]$,

For $\iota\in(\varsigma_{1}, \iota_{2}]$,

and

So

Then for a positive integer $l$, we suppose that the following equalities hold.

When $\iota\in(\varsigma_{l}, \iota_{l+1}]$,

and when $\iota\in(\iota_{l+1}, \varsigma_{l+1}]$,

Thus, for $\iota\in(\varsigma_{l+1}, \iota_{l+2}]$,

and

In conclusion,

(ⅱ) Set any $\iota, \varsigma\in (\iota_{l}, \varsigma_{l}]$, by

we know $\gamma(\iota) = \gamma(\varsigma)$.

(ⅲ) Set any $\varsigma\in (\iota_{\phi(\mathcal{Q}, \varsigma)}, \varsigma_{\phi(\mathcal{Q}, \varsigma)}]$ and any $\iota\in (\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$,

then

Condition 2: $\phi(\mathcal{Q}, \iota) = \phi(\mathcal{Q}, \varsigma)+1$.

(ⅰ) Set any $\varsigma\in(\varsigma_{\phi(\mathcal{Q}, \varsigma)}, \iota_{\phi(\mathcal{Q}, \varsigma)+1}]$ and any $\iota\in (\iota_{\phi(\mathcal{Q}, \iota)}, \varsigma_{\phi(\mathcal{Q}, \iota)}]$,

(ⅱ) Set any $\varsigma\in (\varsigma_{\phi(\mathcal{Q}, \varsigma)}, \iota_{\phi(\mathcal{Q}, \varsigma)+1}]$ and any $\iota\in (\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$,

(ⅲ) Set any $\varsigma\in (\iota_{\phi(\mathcal{Q}, \varsigma)}, \varsigma_{\phi(\mathcal{Q}, \varsigma)}]$ and any $\iota\in (\iota_{\phi(\mathcal{Q}, \iota)}, \varsigma_{\phi(\mathcal{Q}, \iota)}]$,

(ⅳ) Set any $\varsigma\in (\iota_{\phi(\mathcal{Q}, \varsigma)}, \varsigma_{\phi(\mathcal{Q}, \varsigma)}]$ and any $\iota\in (\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$,

Condition 3: general $\phi(\mathcal{Q}, \iota)$ and $\phi(\mathcal{Q}, \varsigma)$.

(ⅰ) Set any $\varsigma\in(\varsigma_{\phi(\mathcal{Q}, \varsigma)}, \iota_{\phi(\mathcal{Q}, \varsigma)+1}]$ and any $\iota\in (\iota_{\phi(\mathcal{Q}, \iota)}, \varsigma_{\phi(\mathcal{Q}, \iota)}]$,

(ⅱ) Set any $\varsigma\in (\varsigma_{\phi(\mathcal{Q}, \varsigma)}, \iota_{\phi(\mathcal{Q}, \varsigma)+1}]$ and any $\iota\in (\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$,

(ⅲ) Set any $\varsigma\in (\iota_{\phi(\mathcal{Q}, \varsigma)}, \varsigma_{\phi(\mathcal{Q}, \varsigma)}]$ and any $\iota\in (\iota_{\phi(\mathcal{Q}, \iota)}, \varsigma_{\phi(\mathcal{Q}, \iota)}]$,

(ⅳ) Set any $\varsigma\in (\iota_{\phi(\mathcal{Q}, \varsigma)}, \varsigma_{\phi(\mathcal{Q}, \varsigma)}]$ and any $\iota\in (\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$,

To sum up,

Let

then (2.2) can be written as

Particularly, when $\varsigma = \mathcal{Q}$,

and

□

Following this, we introduce the definitions and lemma that are used in this work.

Definition 2.6. If $\gamma(\iota) = \gamma(\iota+\delta), \; \iota\geq \mathcal{Q}$, then $\gamma(\cdot)$ is $\delta$-periodic.

Definition 2.7. If there are constants $L\geq1$ and $v > 0$ satisfying

then system (1.1) is exponentially stable.

Lemma 2.8. (see ^[33]) Let $\mathcal{B}\in\mathbb{R}^{n\times n}$ and $\varphi(\mathcal{B}) = \max\{\Re\ell|\ell\in\sigma(\mathcal{B})\}$. Then for any $\theta > 0$, there is $L_{\theta}\geq1$ such that

for any $\iota\geq0$. Here $\sigma(\mathcal{B})$ is the spectrum of $\mathcal{B}$.

Lemma 2.9. (Gronwall inequality, see ^[34]) Set $x(\cdot), \; f(\cdot)$ as the nonnegative continuous function on $[t_{0}, \infty)$. If

then

Next, we present the properties of $\Xi(\iota, \varsigma)$.

For the following results, we assume

and let

Theorem 2.10. When $\mathcal{Q}\leq \varsigma\leq \iota$, there are

or

Proof. For any $\theta > 0$, with $\varsigma\in(\varsigma_{\phi(\mathcal{Q}, \varsigma)}, \; \iota_{\phi(\mathcal{Q}, \varsigma)+1}], \; \iota\in(\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$, by taking the norm of (2.1), we can get

With $\varsigma\in(\varsigma_{\phi(\mathcal{Q}, \varsigma)}, \iota_{\phi(\mathcal{Q}, \varsigma)+1}], \; \iota\in(\iota_{\phi(\mathcal{Q}, \iota)}, \varsigma_{\phi(\mathcal{Q}, \iota)}]$, there is

□

Theorem 2.11. Suppose that $(G_{1})$ and $(G_{2})$ hold, we have

Proof. By the form of (2.1), we have

□

Theorem 2.12. If $(G_{1})$ and $(G_{2})$ hold, we can obtain

Proof. By using (2.1), there is

□

Theorem 2.13. Let $(G_{1})$ and $(G_{2})$ be satisfied, for $N\in\mathbb{N}$, then

Proof. Theorems 2.11 and 2.12 obtain

□

3.   Linear homogeneous problem

The focus of this section is on the linear homogeneous problem.

Theorem 3.1. Let $(G_{1})$ and $(G_{2})$ be satisfied; one of the results follows:

(i) (1.1) has the unique trivial $\delta-$periodic solution iff $rank(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) = n$.

(ii) (1.1) has at least one nontrivial $\delta-$periodic solution iff $rank(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) < n$.

Proof. Sufficiency: (ⅰ) If $rank(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) = n$, then $(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q}))\gamma = 0$ only has the zero solution, thus the solution of (1.1) are trivial.

(ⅱ) If $rank(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) < n$, then there exist nonzero solutions of $(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q}))\gamma = 0$ and (1.1) has nontrivial $\delta-$periodic solutions.

Next, we prove the necessity via the method of proof by contradiction:

(ⅰ) If $rank(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) < n$, then (1.1) has nontrivial $\delta-$periodic solutions, which is contradictory to the given condition. Thus, $rank(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) = n$.

(ⅱ) If $rank(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) = n$, then (1.1) has the unique trivial $\delta-$periodic solution, which contradicts the fact that (1.1) has at least one nontrivial $\delta-$periodic solution. Thus, $rank(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) < n$. □

Before the discussion about the stability, we introduce this relationship.

Theorem 3.2. Let $(G_{1})$ be satisfied; one has

Proof. Using $(G_{1})$, with $\varsigma\in[\mathcal{M}\delta, (\mathcal{M}+1)\delta]$ and $\iota\in[\mathcal{N}\delta, (\mathcal{N}+1)\delta]$ where $\mathcal{M}\leq \mathcal{N}$, there are

and

Hence,

It obviously holds that $\iota-\varsigma\to\infty$ iff $\mathcal{N}-\mathcal{M}\to\infty$. So,

and

□

Next, we consider the stability of (1.1).

Theorem 3.3. Suppose that $(G_{1})$ and $(G_{2})$ are satisfied. If $\ln\varrho+(\varphi(\mathcal{B})+\theta)\lambda < 0$, then system (1.1) is exponentially stable.

Proof. Theorem 3.2 implies that for an arbitrarily small $\varepsilon > 0$, one has

and

For $\iota\in(\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$, with any $\varepsilon\in(0, \frac{q}{\delta})$, Theorem 2.10 implies

in which $L = L_{\theta}e^{|\varphi(\mathcal{B})+\theta|\lambda }\geq1$ and $v = (\varepsilon-\frac{q}{\delta})(\ln\varrho+(\varphi(\mathcal{B})+\theta)\lambda) > 0$.

For $\iota\in(\iota_{\phi(\mathcal{Q}, \iota)}, \varsigma_{\phi(\mathcal{Q}, \iota)}]$, one has

in which $L = L_{\theta}$ and $v = (\varepsilon-\frac{q}{\delta})(\ln\varrho+(\varphi(\mathcal{B})+\theta)\lambda) > 0$.

Hence, (1.1) is exponentially stable.

□

Theorem 3.4. Suppose that $(G_{1})$ and $(G_{2})$ are satisfied. If there is a $\zeta > 0$ such that

where

then system (1.1) is exponentially stable.

Proof.

It is clear that

Combining (3.2) with (3.3), we obtain

and

So

Equation (3.1) implies

With (3.4)–(3.6), we obtain

Finally, we know

in which $L = L_{\theta}e^{\overline{\lambda}|\zeta+\varphi(\mathcal{B})+\theta|}\geq1$ and $v = \zeta\lambda_{1}(\frac{q}{\delta}-\varepsilon) > 0$.

Hence, (1.1) is exponentially stable. □

4.   Linear nonhomogeneous problem

This section considers the linear nonhomogeneous problem.

We present the following condition: $(G_{3})$ $a(\iota) = a(\iota+\delta)$ for $\iota\in\bigcup\limits_{l = 0}^{\infty}(\varsigma_{l}, \iota_{l+1}]$.

Following this, we present the solution of (1.2).

Theorem 4.1. The solution of (1.2) with $\gamma(\mathcal{Q}) = \gamma_{\mathcal{Q}}$ is

Proof. For $\iota\in[\varsigma_{0}, \iota_{1}]$,

If (4.1) holds for $\iota\in(\varsigma_{\phi(\mathcal{Q}, \iota)-1}, \iota_{\phi(\mathcal{Q}, \iota)}]$, then

and for $\iota\in(\iota_{\phi(\mathcal{Q}, \iota)}, \varsigma_{\phi(\mathcal{Q}, \iota)}]$,

Next, for $\iota\in(\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$,

With the mathematical induction method, we obtain

□

Theorem 4.2. Suppose that $(G_{1})$, $(G_{2})$, and $(G_{3})$ hold. If the solution of (1.2) is bounded, then it is $\delta-$periodic.

Proof. Since the solution of (1.2) is bounded, one set $\widetilde{\gamma}(\mathcal{Q}+n\delta)$ is bounded. Using Theorems 2.11 and 2.12, one obtains

where

Hence,

With the proof by contradiction, one supposes that $\widetilde{\gamma}(\iota)$ is not the $\delta-$periodic solution of (1.2). So there is not a $\gamma_{\mathcal{Q}}\in\mathbb{R}^{n}$ such that

Fredholm alternative Theorem implies that there is a $\mathcal{Z}\in\mathbb{R}^{n}$ satisfying

Since $(\mathcal{I}-\Xi^{\top}(\mathcal{Q}+\delta, \mathcal{Q}))\mathcal{Z} = 0$, with any $n\in\mathbb{N}$, we have $[\Xi^{n}(\mathcal{Q}+\delta, \mathcal{Q})]^{\top}\mathcal{Z} = \mathcal{Z}$. Also,

which is contradictory to the boundedness of $\widetilde{\gamma}(\iota)$. So $\widetilde{\gamma}(\iota)$ is a $\delta-$periodic solution of (1.2).

□

We analyze the existence of periodic solutions for (1.2) in two different situations.

Theorem 4.3. Suppose $(G_{1})$ and $(G_{2})$ hold. If $\det(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q}))\neq0$, (1.2) has a $\delta-$periodic solution with

Next, if $\det(\mathcal{I}-\Xi(\mathcal{Q}+\delta, \mathcal{Q})) = 0$, we present

Theorem 4.4. Let $(G_{1})$, $(G_{2})$, and $(G_{3})$ be satisfied. (1.2) has a $\delta-$periodic solution iff $< \beta_{\mathcal{Q}}, \Gamma_{q} > = 0$, where $\beta_{\mathcal{Q}}$ is the initial value of (4.2).

Proof. (1.2) has a $\delta-$periodic solution iff there exists $\gamma_{\mathcal{Q}}$ such that

Then,

□

5.   Nonlinear problem

This section studies the $\delta-$periodic solution of (1.3).

One presents the conditions:

$(G_{4})$ For $\gamma\in\mathbb{R}^{n}$ and $\iota\in\bigcup\limits_{l = 0}^{\infty}(\varsigma_{l}, \iota_{l+1}]$, $\mathcal{A}(\iota+\delta, \gamma) = \mathcal{A}(\iota, \gamma)$.

$(G_{5})$ For $\gamma\in\mathbb{R}^{n}$ and $\iota\in\bigcup\limits_{l = 0}^{\infty}(\varsigma_{l}, \iota_{l+1}]$, there is a $\overline{\mathcal{A}} > 0$ such that $\|\mathcal{A}(\iota, \gamma)\|\leq\overline{\mathcal{A}}$.

One studies

and the solution is

We set this mapping

Equation (5.1) implies

and (5.2) implies

where $\overline{b} = \max\limits_{l\in \mathbb{N}}\|b_{l}\|$.

Then we construct this operator

and set $\chi_{l} = e^{|\varphi(\mathcal{B})+\theta|\frac{(\iota_{l}-\varsigma_{l-1})^{\kappa}}{\kappa}}$, $\varpi = \varrho L_{\theta}$.

Next, one presents the norm estimation of $\mathcal{G}$.

Theorem 5.1. If $(G_5)$ holds, there is

Proof. For $l = 1$, there is

For $l = 2$, there is

For $l = 3$, there is

Suppose that (5.3) holds for $l = q-1$; when $l = q$,

□

Following this, we derive the equivalence between the $\delta-$periodic solution for (1.3) and the fixed point of $\mathcal{G}$.

Theorem 5.2. If $(G_{1})$, $(G_{2})$, and $(G_{4})$ hold, (1.3) has a $\delta-$periodic solution iff $\mathcal{G}$ has a fixed point.

Proof. Sufficiency.

According to the definition of $\mathcal{G}$, one has

For $\iota = \widetilde{\iota}+N\delta$, Theorems 2.11–2.13 derive

and

then $\gamma(\iota+\delta) = \gamma(\iota)$.

Necessity.

If $\gamma(\iota)$ is a $\delta-$periodic solution of (1.3), then $\mathcal{G}(\gamma_{\mathcal{Q}}) = \gamma_{\mathcal{Q}}$ and $\gamma_{\mathcal{Q}}$ is a fixed point of $\mathcal{G}$. □

Next, we present this condition:

$(G_{6})$ For $\gamma\in\mathbb{R}^{n}$ and $\iota\in\bigcup\limits_{l = 0}^{\infty}(\varsigma_{l}, \iota_{l+1}]$, there is a $L_{\mathcal{A}} > 0$ such that $\|\mathcal{A}(\iota, \gamma)-\mathcal{A}(\iota, \overline{\gamma})\|\leq L_{\mathcal{A}}\|\gamma-\overline{\gamma}\|$.

Theorem 5.3. Let $(G_{1})$, $(G_{2})$, $(G_{4})$, and $(G_{6})$ be satisfied. If

then (1.3) has a unique $\delta-$periodic solution.

Proof. Set $\gamma(\iota)$ and $\overline{\gamma}(\iota)$ respectively as the solutions of (1.3) with initial values $\gamma_{\mathcal{Q}}$ and $\overline{\gamma}_{\mathcal{Q}}$.

By calculation, for $\iota\in[\varsigma_{0}, \iota_1]$, there is

then

Using Lemma 2.9, one has

and

Next, (5.2) implies

For $\iota\in(\varsigma_{1}, \iota_2]$, there is

then

Using Lemma 2.9, one has

and

Next, (5.2) implies

According to the above calculation, there is

For $\iota\in(\varsigma_{q-1}, \iota_q]$, there is

then

Using Lemma 2.9, one has

and

Next, (5.2) implies

Hence,

Since

$\mathcal{G}$ is a contraction mapping. Then, $\mathcal{G}$ has a unique fixed point such that $\mathcal{G}\gamma_{\mathcal{Q}} = \gamma_{\mathcal{Q}}$. □

Theorem 5.4. Let $(G_{1})$, $(G_{2})$, $(G_{4})$, and $(G_{5})$ be satisfied. If

the Eq (1.3) has at least one $\delta-$periodic solution and $\| \gamma_{\mathcal{Q}}\|\leq\psi: = \frac{\widetilde{\rho}}{1-\rho},$ where

Proof. $\| \gamma_{\mathcal{Q}}\|\leq\psi$ and (5.3) imply

Then $\mathcal{G}:\overline{B(0, \psi)}\to\overline{B(0, \psi)}$. Brouwers fixed-point theorem implies that $\mathcal{G}$ has fixed points, and Theorem 5.2 obtains that (1.3) has at least one $\delta-$periodic solution. □

6.   Examples

Example 6.1 Consider (1.1). Let $\varsigma_{0} = \frac{1}{2}, \; \kappa = \frac{1}{2}, \; \varsigma_{l} = l+\frac{1}{2}, \; \iota_{l} = l, \; q = 1, \; \delta = 1, \; l\in\mathbb{N},$ and

And we can obtain

so $\|\mathcal{I}+\mathcal{C}_{l}\| = e^{2}$, $\ln\varrho = 2$ and

Then, we set $\varepsilon = \theta = 0.5$ and obtain

in which $L = e^{4.5\sqrt{2}} > 1$ and $v = 0.5(4.5\sqrt{2}-2) > 0$.

Thus, (1.1) is exponentially stable with $L = e^{4.5\sqrt{2}}$ and $v = 0.5(4.5\sqrt{2}-2)$.

Further,

Then (1.1) only has the trivial $1-$periodic solution.

Example 6.2 Consider (1.2) and $\kappa = \frac{1}{2}, \varsigma_{0} = 0, \varsigma_{l} = l, \iota_{l} = l-\frac{1}{2}, \delta = 1, q = 1, l\in N$,

Set

It is easily obtain

Next,

By

and

so

Thus, for $t\in(\varsigma_{\phi(\mathcal{Q}, \iota)}, \iota_{\phi(\mathcal{Q}, \iota)+1}]$,

Then

so there is a $1$-periodic solution. The component of the solution is in Figure 1. Further,

and Theorem 4.2 is verified.

Example 6.3 Consider (1.3) and $\kappa = \frac{1}{2}, \varsigma_{0} = 0, \varsigma_{l} = l, \iota_{l} = l-\frac{1}{2}, \delta = 1, q = 1, l\in N$,

Set

Next, $\overline{\mathcal{A}} = \frac{1}{2}$, $\chi_{1} = e^{|\varphi(\mathcal{B})+\theta|\frac{(\iota_{1}-\varsigma_{0})^{\kappa}}{\kappa}} = e^{\sqrt{2}}$, $\varpi = \varrho L_{\theta} = \frac{1}{10},$ $\rho: = \varpi^{q}\prod\limits_{l = 1}^{q}\chi _{l} = \frac{e^{\sqrt{2}}}{10} < 1$ and

Thus, (1.3) has at least one $1-$periodic solution and $1 = \|\gamma_{Q}\| < \psi = 1.96$.

7.   Conclusions

In this paper, we investigate the existence and stability of solutions for periodic conformable systems with non-instantaneous impulses. A key focus is the introduction and analysis of the conformable Cauchy matrix and its properties. We conduct separate studies on linear homogeneous, linear nonhomogeneous, and nonlinear systems. For the linear nonhomogeneous system, we employ the constant variation method to derive the solution expression. Moreover, we discuss the existence of periodic solutions for the linear nonhomogeneous system under two conditions. Regarding the nonlinear system, the existence and uniqueness of periodic solutions are transformed into the existence and uniqueness of fixed points of a corresponding operator. This transformation enables us to utilize related fixed-point theory to analyze the existence of periodic solutions.

Use of Generative-AI tools declaration

The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author is grateful to the referees for their careful reading of the manuscript and valuable comments. The author thanks the help from the editor too.

Conflict of interest

The author declares that he has no conflict of interest.