Asymptotic stability of solutions of impulsive multi-delay differential equations

Abstract

In this paper, we consider the asymptotic stability of solutions to impulsive multi-delayed differential equations with linear parts defined by pairwise permutable matrices. First, we introduce the concept for an impulsive multi-delayed Cauchy matrix and then use it to obtain the representation of solutions to linear impulsive Cauchy problems via the variation of constants principle. Next, we give a norm estimate of the impulsive multi-delayed Cauchy matrix and establish sufficient conditions to guarantee that the trivial solutions are asymptotically stable when the nonlinear terms satisfy appropriate conditions. Finally, two numerical examples are given to illustrate the effectiveness of the results.

Keywords

Impulsive differential equations multiple delays impulsive multi-delayed Cauchy matrix solution asymptotic stability

Introduction

Khusainov and Shuklin (2003) constructed a delayed matrix exponential and gave the representation of the solution to a linear autonomous time-delay system with permutable matrices. Following the work of Diblík and Khusainov (2006a), Khusainov and Shuklin (2003) used this idea to represent the solution of discrete delayed systems by introducing a discrete matrix delayed exponential. There are many new results in the literature on the theory of ordinary differential or difference equations with single or multiply delays, such as controllability, exponential stability, boundary-value problems (Boichuk et al., 2010; Diblík and Khusainov, 2006b; Diblík and Morávková, 2013; Diblík et al., 2016; Khusainov et al., 2008; Medveď and Pospíšil, 2012; Medveď et al., 2011; Pospíšil, 2012) and some relative controllability problems (Diblík et al., 2008, 2010, 2014; Khusainov and Shuklin, 2005).

Medveď and Pospíšil (2012) gave the representation of solutions of Cauchy initial value problems for the following linear multi-delay differential equations

\begin{matrix} {\begin{matrix} u' (t) = A_{1} u (t) + \sum_{m = 1}^{n} B_{m} u (t - ϱ_{m}), A_{1}, B_{m} \in R^{N \times N} \\ t \geq 0, ϱ_{m} > 0, m = 1, \dots, n \\ u (t) = ψ (t), - ϱ \leq t \leq 0, ϱ : = max {ϱ_{1}, \dots, ϱ_{n}} \end{matrix} \end{matrix}

(1)

with the linear parts $A_{1} B_{m} = B_{m} A_{1}$ and $ψ \in C_{ϱ}^{1} : = C^{1} ([- ϱ, 0], R^{N})$ . In regards to equation (1), a concept of multi-delayed matrix exponential $E_{ϱ_{1}, \dots, ϱ_{j}}^{B_{1}, \dots, B_{j} \cdot}$ (Medveď and Pospíšil, 2012: Corollary 9) is introduced, which has the form

\begin{matrix} E_{ϱ_{1}, \dots, ϱ_{j}}^{B_{1}, \dots, B_{j} t} = \\ \begin{matrix} {\begin{matrix} Θ, t < - ϱ_{j} \\ X_{j - 1} (t + ϱ_{j}), - ϱ_{j} \leq t < 0 \\ X_{j - 1} (t + ϱ_{j}) + B_{j} \int_{0}^{t} X_{j - 1} (t - s_{1}) X_{j - 1} (s_{1}) d s_{1} \\ + \dots + B_{j}^{k} \int_{(k - 1) ϱ_{j}}^{t} \int_{(k - 1) ϱ_{j}}^{s_{1}} \dots \int_{(k - 1) ϱ_{j}}^{s_{k - 1}} X_{j - 1} (t - s_{1}) \\ \times Π_{i = 1}^{k - 1} X_{j - 1} (s_{i} - s_{i + 1}) X_{j - 1} (s_{k} - (k - 1) ϱ_{j}) d s_{k} \dots d s_{1} \\ (k - 1) ϱ_{j} \leq t < k ϱ_{j}, k = 1, 2, \dots \end{matrix} \end{matrix} \end{matrix}

(2)

where $X_{j - 1} (t) = E_{ϱ_{1}, \dots, ϱ_{j - 1}}^{B_{1}, \dots, B_{j - 1} (t - ϱ_{j - 1})}$ , $j = 2, \dots, n$ and $Θ$ is the zero matrix.

Many processes and phenomena are characterized by rapid changes in state; the duration of these changes is relatively short compared with the overall duration of the whole process. For the theory of impulsive differential equations, we refer the reader to the monographs of Bainov and Simeonov (1993) and Samoilenko et al. (1995) and for applications to the works of Benchohra et al. (2006) and Ivanov and Slyn’ko (2009) and the references therein. You and Wang (2016) introduced the notation of an impulsive delayed matrix function and used the variation of constants method to obtain the representation of solutions to linear impulsive delay differential equations. However, there are only a few papers in the literature seeking the explicit solution of linear impulsive multi-delay differential equations with permutable matrices. Because of the double impact from the impulsive perturbation and the multi-time delay it is a challenge to obtain the presentation of the solution. The topic of control of impulsive multi-delayed differential equations has attracted some attention. In the literature, related control and estimation issues are considered under a model-driven framework, such as an adaptive neural-network-based approach for fault-tolerant control of nonlinear time-varying delay systems with unmodelled dynamics (Yin et al., 2017) and adaptive fuzzy control of strict-feedback nonlinear time-delay systems with unmodelled dynamics (Yin et al., 2016). We note that impulsive multi-delayed differential equations can describe a model of living organism dynamics involving delayed birthrates and delayed logistic terms under impulsive perturbation.

In this paper, we consider the representation of solutions of Cauchy initial value problems and asymptotic stability for the following linear impulsive multi-delay differential equations

\begin{matrix} {\begin{matrix} u' (t) = A_{1} u (t) + \sum_{m = 1}^{n} B_{m} u (t - ϱ_{m}), t \geq 0 \\ t \neq τ_{i}, ϱ_{m} > 0, m = 1, \dots, n \\ Δ u (τ_{i}) = C_{i} u (τ_{i}), i = 1, 2, \dots \\ u (t) = ψ (t), - ϱ \leq t \leq 0 \end{matrix} \end{matrix}

(3)

where $A_{1}, B_{m}, C_{i} \in R^{N \times N}$ are constant matrices, $A B_{m} = B_{m} A$ , $B_{j} B_{m} = B_{m} B_{j}$ , $A_{1} C_{i} = C_{i} A_{1}$ and $B_{m} C_{i} = C_{i} B_{m}$ for each $m, j = 1, 2, \dots, n$ , $i = 1, 2, \dots$ , $ψ \in C_{ϱ}^{1}$ , $u (t) \in R^{N}$ , and the time sequences ${τ_{i}}_{i = 1}^{\infty}$ satisfy $0 = τ_{0} < τ_{1} < \dots < τ_{i} < \dots$ , the impulsive conditions $Δ u (τ_{i}) : = u (τ_{i}^{+}) - u (τ_{i}^{-})$ , $u (τ_{i}^{+}) = lim_{ϵ \to 0^{+}} u (τ_{i} + ϵ)$ and $u (τ_{i}^{-}) = u (τ_{i})$ represent the right and left limits of $u (t)$ at $t = τ_{i}$ and $lim_{i \to + \infty} τ_{i} = \infty$ .

We note that the assumptions $A_{1} B_{m} = B_{m} A_{1}$ , $B_{j} B_{m} = B_{m} B_{j}$ , $A_{1} C_{i} = C_{i} A_{1}$ and $B_{m} C_{i} = C_{i} B_{m}$ are not too strong for many matrices, e.g. the normal matrix or the diagonal matrix. For more details on finding commutable matrices, we refer the reader to Arnold (1988: Section 30, Lemma 4) or Chow et al. (1994: Corollary 9.8, p. 152).

Motivated by Medveď and Pospíšil (2012) and You and Wang (2016), we introduce an impulsive multi-delayed Cauchy matrix $Y (\cdot, \cdot)$ defined in equation (6) and a generalization of the multi-delayed matrix exponential defined in equation (2) for equation (1), and seek a possible representation of the solutions for equation (3) and the corresponding nonhomogeneous problem

{\begin{matrix} u' (t) = A_{1} u (t) + \sum_{m = 1}^{n} B_{m} u (t - ϱ_{m}) + f (t), t \geq 0 \\ t \neq τ_{i}, ϱ_{m} > 0, m = 1, \dots, n \\ Δ u (τ_{i}) = C_{i} u (τ_{i}), i = 1, 2, \dots \\ u (t) = ψ (t), - ϱ \leq t \leq 0 \end{matrix}

(4)

using the new impulsive multi-delayed Cauchy matrix via the variation of constants principle.

We study local asymptotic stability of solutions to equation (3) and local asymptotic stability of solutions to impulsive multi-delay differential equations with nonlinear terms of the form

{\begin{matrix} u' (t) = A_{1} u (t) + \sum_{m = 1}^{n} B_{m} u (t - ϱ_{m}) + F (u (t), u (t - ϱ_{1}), \dots, u (t - ϱ_{n})), t \geq 0, t \neq τ_{i} \\ Δ u (τ_{i}) = C_{i} u (τ_{i}), i = 1, 2, \dots \\ u (t) = ψ (t), - ϱ \leq t \leq 0 \end{matrix}

(5)

where

F \in C (\underset{n + 1}{\underset{︸}{R^{N} \times \dots \times R^{N}}}, R^{N})

The main innovation is to derive the impulsive multi-delayed Cauchy matrix for equation (3) and give its norm estimate. Based on the impulsive multi-delayed Cauchy matrix via the variation of constants principle, we obtain an explicit formula for solutions to equations (3) and (4). Deriving properties of $Y (t, s)$ defined in equation (6), and putting mild conditions on impulsive terms, we present sufficient conditions to guarantee that the trivial solution to equation (3) is locally asymptotically stable. We also show that the trivial solution to equation (5) is locally asymptotically stable by using a modified integral Grönwall inequality with delay.

The main contributions are as follows:

We give the explicit solution of impulsive multi-delayed Cauchy problems with linear parts defined by pairwise permutable matrices, using a new concept for the impulsive multi-delayed Cauchy matrix and the variation of constants principle.

Based on the presentation of solutions and a norm estimate of the impulsive multi-delayed Cauchy matrix, we apply the multi-delayed Grönwall inequality to establish some asymptotic stability results for trivial solutions.

In the next section, we recall some notation and properties for multi-delayed exponential matrices. Then we introduce the impulsive multi-delayed Cauchy matrix and its norm estimate, and verify that it is the fundamental matrix for linear impulsive multi-delay differential equations. After this, we give explicit formulae for solutions to linear impulsive homogeneous or nonhomogeneous multi-delay differential equations via an impulsive multi-delayed Cauchy matrix associated with the variation of constants approach. Next, sufficient conditions ensuring local asymptotic stability of solutions are presented. Two examples are given to illustrate our theoretical results in the final section.

Preliminaries

For $z \in R^{N}$ and $Z \in R^{N \times N}$ , we introduce the vector norm $∥ z ∥ = max_{1 \leq i \leq N} | z_{i} |$ and the matrix norm

∥ Z ∥ = max_{1 \leq i \leq N} \sum_{j = 1}^{N} | c_{ij} |

respectively, where $z_{i}$ and $c_{ij}$ are the elements of the vector $z$ and the matrix $Z$ . Let $L (R^{N})$ be the space of bounded linear operators in $R^{N}$ . Denote by $C (R^{+}, R^{N})$ the Banach space of vector-value bounded continuous functions from $R^{+} \to R^{N}$ endowed with the norm $∥ z ∥_{C} = sup_{t \in R^{+}} ∥ z (t) ∥$ . We introduce a space $C^{1} (R^{+}, R^{N}) = {z \in C (R^{+}, R^{N}) : z' \in C (R^{+}, R^{N})}$ . Denote $PC (R^{+}, R^{N}) : = {z : R^{+} \to R^{N} : z \in C ((τ_{i}, τ_{i + 1}], R^{N})$ ; there exist $z (τ_{i}^{-})$ and $z (τ_{i}^{+})$ with $z (τ_{i}^{-}) = z (τ_{i})$ for any $i = 1, 2, \dots}$ and $P C^{1} (R^{+}, R^{N}) : = {z : R^{+} \to R^{N} : z' \in PC (R^{+}, R^{N})}$ .

We need the following basic definitions and lemmas.

Definition 1. A function $u \in C^{1} ([- ϱ, 0], R^{N}) \cup P C^{1} (R^{+}, R^{N})$ is called the solution of equation (3) (or equation (5)), if $u$ satisfies $u (t) = ψ (t)$ for $- ϱ \leq t \leq 0$ and the first and second equations in equation (3) (or equation (5)).

Definition 2. The trivial solution of equation (3) is called locally asymptotically stable, if there exists $δ > 0$ such that $∥ ψ ∥_{1} : = max_{[- ϱ, 0]} ∥ ψ (t) ∥ + max_{[- ϱ, 0]} ∥ ψ' (t) ∥ < δ$ , $lim_{t \to \infty} ∥ u (t) ∥ = 0$ holds, where the solution $u$ is generated by $ψ$ . If $δ$ can be arbitrary, then $u$ is globally asymptotically stable.

It is well known that local and global asymptotic stabilities are coincident for equation (3).

Now we recall a Grönwall inequality with time multi-delays.

Lemma 1. (Bainov and Simeonov, 1992: Theorem 14.8). Let $J = [t_{0}, \infty)$ ; $g \in C (J, R^{+})$ is nondecreasing, $b, c_{i} \in C (J, R^{+})$ , $φ \in C ([t_{0}, t_{0} + ϱ], R^{+})$ . If $u \in C (J, R^{+})$ and

\begin{matrix} {\begin{matrix} u (t) \leq g (t) + \int_{t_{0}}^{t} b (s) u (s) d s + \sum_{i = 1}^{n} \int_{t_{0}}^{t} c_{i} (s) u (s - ϱ_{i}) d s, t \geq t_{0} (i) \\ u (t) = φ (t), t_{0} \leq t \leq t_{0} + ϱ (ii) \end{matrix} \end{matrix}

where $ϱ_{i} > 0, ϱ = max {ϱ_{i} : i = 1, 2, \dots, n}$ , then

u (t) \leq [g (t) + \sum_{i = 1}^{n} \int_{[t_{0}, t] ⋂ E_{i}} c_{i} (s) φ (s - ϱ_{i}) d s] \times e^{(\int_{t_{0}}^{t} b (s) d s + \sum_{i = 1}^{n} \int_{[t_{0}, t] \ E_{i}} c_{i} (s) d s)}

for $t \geq t_{0}$ , where $E_{i} = [t_{0}, t_{0} + ϱ_{i}]$ .

To deal with our next problem, we improve the result in Lemma 1.

Remark 1. In Lemma 1, if we replace the original condition $(ii)$ with $u (t) = ϕ (t), t_{0} - ϱ \leq t \leq t_{0}, ϕ \in C ([t_{0} - ϱ, t_{0}], R^{+}),$ then

u (t) \leq [g (t) + \sum_{i = 1}^{n} \int_{[t_{0}, t] ⋂ E_{i}} c_{i} (s) ϕ (s - ϱ_{i}) d s] \times e^{(\int_{t_{0}}^{t} b (s) d s + \sum_{i = 1}^{n} \int_{[t_{0}, t] \ E_{i}} c_{i} (s) d s)}

Proof. Note that

\begin{array}{l} u (t) \leq g (t) + \int_{t_{0}}^{t} b (s) u (s) d s + \sum_{i = 1}^{n} \int_{t_{0}}^{t} c_{i} (s) u (s - ϱ_{i}) d s \\ \leq g (t) + \sum_{i = 1}^{n} \int_{[t_{0}, t] \cap E_{i}} c_{i} (s) ϕ (s - ϱ_{i}) d s \\ + \int_{t_{0}}^{t} b (s) u (s) d s + \sum_{i = 1}^{n} \int_{[t_{0}, t] \ E_{i}} c_{i} (s) u (s - ϱ_{i}) d s \end{array}

From the proof of Bainov and Simeonov (1992: Theorem 14.8), we get the result.

Impulsive multi-delayed Cauchy matrix and its property

We now introduce a concept of impulsive multi-delay matrix function, an extension of the multi-delay matrix function for linear multi-delay differential equations, which helps us to seek explicit formulae of solutions to impulsive multi-delay differential equations.

Using the multi-delayed matrix exponential (equation (2)), we define $Y (\cdot, \cdot) : R \times R \to R^{N \times N}$ and

Y (t, s) = e^{A_{1} (t - s)} X (t, s + ϱ), t > s

(6)

where

X (t, s) = E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - s)} + \sum_{s - ϱ < τ_{j} \leq t} C_{j} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ - τ_{j})} X (ϱ_{j}, s)

and ${\tilde{B}}_{i} = e^{- A_{1} ϱ_{i}} B_{i}, i = 1, \dots, n .$

Here, we call $Y (\cdot, \cdot)$ the impulsive multi-delayed Cauchy matrix associated with equation (3). Note that $Y (t, t) = I$ and $Y (t, s) = Θ, t < s,$ where $I$ is an identity matrix.

Lemma 2. The impulsive multi-delayed Cauchy matrix $Y (\cdot, \cdot)$ is the fundamental matrix of equation (3).

Proof.

Step 1. Having differentiated $Y (t, s)$ for $t \in (τ_{i}, τ_{i + 1}]$ and $t > s$ , and using Medveď and Pospíšil (2012: Theorem 8), we obtain

\begin{matrix} \frac{d}{d t} Y (t, s) = A_{1} e^{A_{1} (t - s)} X (t, s + ϱ) \\ + e^{A_{1} (t - s)} {{\tilde{B}}_{1} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{1} - ϱ - s)} \\ + \dots + {\tilde{B}}_{n} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{n} - ϱ - s)} \\ + {\tilde{B}}_{1} \sum_{s < τ_{j} \leq t} C_{j} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{1} - ϱ - τ_{j})} X (τ_{j}, s + ϱ) \\ + \dots \\ + {\tilde{B}}_{n} \sum_{s < τ_{j} \leq t} C_{j} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{n} - ϱ - t_{j})} X (ϱ_{j}, s + ϱ)} \\ = A_{1} Y (t, s) + B_{1} e^{A_{1} (t - ϱ_{1} - s)} {E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{1} - ϱ - s)} \\ + \sum_{s < τ_{j} \leq t - ϱ_{1}} C_{j} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{1} - ϱ - τ_{j})} X (τ_{j}, s + ϱ)} \\ + \dots \\ + B_{n} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{n} - ϱ - s)} \\ + \sum_{s < τ_{j} \leq t - ϱ_{n}} C_{j} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{n} - ϱ - τ_{j})} X (τ_{j}, s + ϱ)} \\ = A_{1} Y (t, s) + B_{1} Y (t - ϱ_{1}, s) + \dots + B_{n} Y (t - ϱ_{n}, s) \end{matrix}

Step 2. Let $i : = i (0, t)$ be the number of impulsive points that belong to $(0, t)$ . Note that $E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} t^{+}} = E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} t^{-}}$ for all $t > - ϱ$ (see You and Wang, 2017: Lemma 2.4), and then

\begin{matrix} Y (τ_{i}^{+}, s) = e^{A_{1} (τ_{i}^{-} - s)} {E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (τ_{i}^{-} - ϱ - s)} \\ + \sum_{s < τ_{j} \leq τ_{i}^{-}} C_{j} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (τ_{i}^{-} - ϱ - τ_{j})} X (τ_{j}, s + ϱ) \\ + C_{i} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (τ_{i}^{+} - ϱ - τ_{i})} X (ϱ_{i}, s + ϱ)} \\ = e^{A_{1} (τ_{i}^{-} - s)} X (τ_{i}^{-}, s + ϱ) + C_{i} e^{A_{1} (τ_{i}^{-} - s)} X (τ_{i}, s + ϱ) \\ = Y (τ_{i}^{-}, s) + C_{i} Y (τ_{i}, s) \end{matrix}

This ends the proof.

Now we introduce the following condition.

$(H_{0})$ . Let $σ (A_{1}) = {p_{1}, p_{2}, \dots, p_{N}}$ be the eigenvalues of $A_{1}$ with $Re p_{1} \leq Re p_{2} \leq \dots \leq Re p_{N} \leq - k < 0, k > 0,$ i.e. there exist $K, k > 0$ such that

∥ e^{A_{1} t} ∥ \leq K e^{- kt} for all t \geq 0

(7)

To estimate $Y (\cdot, \cdot)$ , we consider the norm estimate on ${\tilde{B}}_{i} = e^{- A_{1} ϱ_{i}} B_{i}, i = 1, 2, \dots, n$ . For given ${\tilde{B}}_{i}$ there exists $α_{i} \in R^{+}$ , such that $∥ {\tilde{B}}_{i} ∥ \leq α_{i} e^{α_{i} ϱ_{i}}$ . From Ortega and Rheinboldt (1970: Chapter 2) and $(H_{0})$ , we have $∥ {\tilde{B}}_{i} ∥ = ∥ e^{- A_{1} ϱ_{i}} B_{i} ∥ \leq K (ρ (B_{i}) + ε) e^{(- Re p_{1}) ϱ_{i}}$ . Choosing $α_{i} \geq max {K (ρ (B_{i}) + ε), - Re p_{1}}, ε > 0$ , then $∥ {\tilde{B}}_{i} ∥ \leq α_{i} e^{α_{i} ϱ_{i}}$ .

The following property is very useful for our next stability results.

Lemma 3. Assume $(H_{0})$ is satisfied. For any $ε > 0$ , we have

\begin{matrix} ∥ X (t, s) ∥ \leq (\underset{s - ϱ < τ_{j} \leq t}{Π} (ρ (C_{j}) + 1 + ε)) e^{α (t + ϱ - s)}, \\ α = α_{1} + \dots + α_{n} \end{matrix}

(8)

and

∥ Y (t, s) ∥ \leq K (\underset{s < τ_{j} \leq t}{Π} (ρ (C_{j}) + 1 + ε)) e^{(α - k) (t - s)}

(9)

Proof. Without loss of generality, we suppose that $τ_{m} \leq s - ϱ < τ_{m + 1}$ and $τ_{m + r} \leq t < τ_{m + r + 1}, m, r = 0, 1, 2, \dots$ We will use mathematical induction.

For $r = 0$ , by Medveď and Pospíšil (2012: Lemma 13)

\begin{array}{l} ‖ X (t, s) ‖ \leq ‖ ℰ_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - s)} ‖ \\ \leq e^{(α_{1} + \dots + α_{n}) (t + ϱ - s)} = e^{α (t + ϱ - s)} \end{array}

For $r = 1$ , by Medveď and Pospíšil (2012: Lemma 13) and Ortega and Rheinboldt (1970: Chapter 2), we have

\begin{matrix} ∥ X (t, s) ∥ \leq ∥ E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - s)} ∥ \\ + ∥ C_{m + 1} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ - τ_{m + 1})} X (τ_{m + 1}, s) ∥ \\ \leq e^{α (t + ϱ - s)} \\ + (ρ (C_{m + 1}) + ε) e^{(α_{1} + \dots + α_{n}) (t - ϱ + ϱ_{n} - τ_{m + 1})} e^{(α_{1} + \dots + α_{n}) (τ_{m + 1} + ϱ - s)} \\ = e^{α (t + ϱ - s)} + (ρ (C_{m + 1}) + ε) e^{α (t + ϱ_{n} - s)} \\ \leq (ρ (C_{m + 1}) + 1 + ε) e^{α (t + ϱ - s)} \end{matrix}

For $r = k$ , we suppose that

∥ X (t, s) ∥ \leq (\underset{s - ϱ < τ_{j} \leq t}{Π} (ρ (C_{j}) + 1 + ε)) e^{α (t + ϱ - s)}

For $r = k + 1$ , by Medveď and Pospíšil (2012: Lemma 13) and Ortega and Rheinboldt (1970: Chapter 2), again

\begin{matrix} ∥ X (t, s) ∥ \leq ∥ E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - s)} ∥ \\ + \sum_{s - ϱ < τ_{j} \leq t} ∥ C_{j} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ - τ_{j})} X (τ_{j}, s) ∥ \\ \leq e^{α (t + ϱ - s)} + \sum_{j = m + 1}^{m + k + 1} (ρ (C_{j}) + ε) e^{α (t - ϱ + ϱ_{n} - τ_{j})} \\ \times (Π_{z = m + 1}^{j - 1} (ρ (C_{z}) + 1 + ε)) e^{α (τ_{j} + ϱ - s)} \\ \leq (1 + \sum_{j = m + 1}^{m + k + 1} (ρ (C_{j}) + ε) \times Π_{z = m + 1}^{j - 1} (ρ (C_{z}) + 1 + ε)) e^{α (t + ϱ - s)} \\ = (\underset{s - ϱ < τ_{j} \leq t}{Π} (ρ (C_{j}) + 1 + ε)) e^{α (t + ϱ - s)} \end{matrix}

From mathematical induction, we obtain equation (8).

Finally, using equations (6) and (8) via $(H_{0})$ , one derives equation (9) immediately. The proof is finished.

Representation of solutions

In this section, we seek an explicit formula of solutions to linear impulsive nonhomogeneous multi-delay differential equations by adopting similar ideas to those presented by You and Wang (2016).

Firstly, we drive an explicit formula of solutions to linear impulsive homogeneous multi-delay differential equations.

Theorem 1. The solution of equation (3) has the form

\begin{matrix} u (t) = Y (t, - ϱ) ψ (- ϱ) + \int_{- ϱ}^{0} Y (t, s) [ψ' (s) - A_{1} ψ (s)] d s \\ - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} Y (t, s + ϱ_{m}) ψ (s) d s, t \geq - ϱ \end{matrix}

(10)

Proof. We will divide the proof into the following three steps:

Step 1. For $t \in (τ_{i}, τ_{i + 1}]$ , note

\begin{array}{l} u^{'} (t) = A_{1} {Y (t, - ϱ) ψ (- ϱ) + \int_{- ϱ}^{0} Y (t, s) [ψ^{'} (s) - A_{1} ψ (s)] d s - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} Y (t, s + ϱ_{m}) ψ (s) d s} \\ + B_{1} {Y (t - ϱ_{1}, - ϱ) ψ (- ϱ) + \int_{- ϱ}^{0} Y (t - ϱ_{1}, s) [ψ^{'} (s) - A_{1} ψ (s)] d s \\ - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} Y (t - ϱ_{1}, s + ϱ_{m}) ψ (s) d s} + \dots + B_{n} {Y (t - ϱ_{n}, - ϱ) ψ (- ϱ) \\ + \int_{- ϱ}^{0} Y (t - ϱ_{n}, s) [ψ^{'} (s) - A_{1} ψ (s)] d s - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} Y (t - ϱ_{n}, s + ϱ_{m}) ψ (s) d s} \\ = A_{1} u (t) + B_{1} u (t - ϱ_{1}) + \dots + B_{n} u (t - ϱ_{n}) \end{array}

Step 2. Without loss of generality, we suppose that $0 < ϱ_{1} < \dots < ϱ_{n} = ϱ$ . For $- ϱ_{n} \leq t \leq 0$ , we have

Y (t, - ϱ_{n}) = e^{A_{1} (t + ϱ_{n})} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} t} = e^{A_{1} (t + ϱ_{n})} X_{n - 1} (t + ϱ_{n})

For $- ϱ_{n} \leq s \leq 0$ , i.e. $- 2 ϱ_{n} \leq t - ϱ_{n} - s \leq 0$

\begin{matrix} Y (t, s) = e^{A_{1} (t - s)} E_{ϱ_{1}, \dots, ϱ_{n}}^{{\tilde{B}}_{1}, \dots, {\tilde{B}}_{n} (t - ϱ_{n} - s)} \\ = \begin{matrix} {\begin{matrix} Θ, & s \in (t, 0] \\ e^{A_{1} (t - s)} X_{n - 1} (t - s), & s \in [- ϱ_{n}, t] \end{matrix} \end{matrix} \end{matrix}

For $- ϱ_{n} \leq s \leq - ϱ_{m}$ , i.e. $- 2 ϱ_{n} \leq t - ϱ_{n} - ϱ_{m} - s \leq - ϱ_{m}$

Y (t, s + ϱ_{m}) = \begin{matrix} {\begin{matrix} Θ, & s \in (t - ϱ_{m}, - ϱ_{m}] \\ e^{A_{1} (t - ϱ_{m} - s)} X_{n - 1} (t - ϱ_{m} - s), & s \in [- ϱ_{n}, t - ϱ_{m}] \end{matrix} \end{matrix}

Next, from equation (10) and these facts, we have

\begin{matrix} u (t) = e^{A_{1} (t + ϱ_{n})} X_{n - 1} (t + ϱ_{n}) ψ (- ϱ_{n}) \\ + \int_{- ϱ_{n}}^{t} [e^{A_{1} (t - s)} X_{n - 1} (t - s) ψ (s)]' d s \\ + \sum_{m = 1}^{n - 1} B_{m} \int_{- ϱ_{n}}^{t} e^{A_{1} (t - ϱ_{m} - s)} X_{n - 1} (t - ϱ_{m} - s) ψ (s) d s \\ - \sum_{m = 1}^{n - 1} B_{m} \int_{- ϱ_{n}}^{t - ϱ_{m}} e^{A_{1} (t - ϱ_{m} - s)} X_{n - 1} (t - ϱ_{m} - s) ψ (s) d s \\ = e^{A_{1} (t + ϱ_{n})} X_{n - 1} (t + ϱ_{n}) ψ (- ϱ_{n}) \\ + \int_{- ϱ_{n}}^{t} e^{A_{1} (t - s)} X_{n - 1} (t - s) [ψ' (s) - A ψ (s)] d s \\ - \sum_{m = 1}^{n - 1} B_{m} \int_{- ϱ_{n}}^{t - ϱ_{m}} e^{A_{1} (t - ϱ_{m} - s)} X_{n - 1} (t - ϱ_{m} - s) ψ (s) d s \end{matrix}

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Note that

\begin{matrix} \frac{d}{d s} [e^{A_{1} (t - s)} X_{n - 1} (t - s) ψ (s)] = - A_{1} e^{A_{1} (t - s)} X_{n - 1} (t - s) ψ (s) \\ - \sum_{m = 1}^{n - 1} {\tilde{B}}_{m} e^{A_{1} (t - s)} X_{n - 1} (t - ϱ_{m} - s) ψ (s) \\ + e^{A_{1} (t - s)} X_{n - 1} (t - s) ψ' (s) \\ = e^{A_{1} (t - s)} X_{n - 1} (t - s) [ψ' (s) - A_{1} ψ (s)] \\ - \sum_{m = 1}^{n - 1} B_{m} e^{A_{1} (t - ϱ_{m} - s)} X_{n - 1} (t - ϱ_{m} - s) ψ (s) \end{matrix}

and for $s \in (t - ϱ_{m}, t]$ , i.e. $- ϱ_{m} \leq t - ϱ_{m} - s < 0$ , we have $X_{n - 1} (t - ϱ_{m} - s) = Θ .$

Then, equation (11) becomes

\begin{matrix} u (t) = e^{A_{1} (t + ϱ_{n})} X_{n - 1} (t + ϱ_{n}) ψ (- ϱ_{n}) \\ + \int_{- ϱ_{n}}^{t} [e^{A_{1} (t - s)} X_{n - 1} (t - s) ψ (s)]' d s \\ + \sum_{m = 1}^{n - 1} B_{m} \int_{- ϱ_{n}}^{t} e^{A_{1} (t - ϱ_{m} - s)} X_{n - 1} (t - ϱ_{m} - s) ψ (s) d s \\ - \sum_{m = 1}^{n - 1} B_{m} \int_{- ϱ_{n}}^{t - ϱ_{m}} e^{A_{1} (t - ϱ_{m} - s)} X_{n - 1} (t - ϱ_{m} - s) ψ (s) d s \\ = e^{A_{1} (t + ϱ_{n})} X_{n - 1} (t + ϱ_{n}) ψ (- ϱ_{n}) \\ + \int_{- ϱ_{n}}^{t} [e^{A_{1} (t - s)} X_{n - 1} (t - s) ψ (s)]' d s \\ + \sum_{m = 1}^{n - 1} B_{m} \int_{t - ϱ_{m}}^{t} e^{A_{1} (t - ϱ_{m} - s)} X_{n - 1} (t - ϱ_{m} - s) ψ (s) d s \\ = e^{A_{1} (t + ϱ_{n})} X_{n - 1} (t + ϱ_{n}) ψ (- ϱ_{n}) \\ + e^{A_{1} (t - s)} X_{n - 1} (t - s) ψ (s) |_{- ϱ_{n}}^{t} \\ = ψ (t) \end{matrix}

Step 3. Note that $Y (τ_{i}^{+}, s) = Y (τ_{i}^{-}, s) + C_{i} Y (τ_{i}, s)$ , and so

\begin{matrix} u (τ_{i}^{+}) = Y (τ_{i}^{+}, - ϱ) ψ (- ϱ) + \int_{- ϱ}^{0} Y (τ_{i}^{+}, s) [ψ' (s) - A_{1} ψ (s)] d s \\ - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} Y (τ_{i}^{+}, s + ϱ_{m}) ψ (s) d s \\ = (Y (τ_{i}^{-}, - ϱ) + C_{i} Y (τ_{i}, - ϱ)) ψ (- ϱ) \\ + \int_{- ϱ}^{0} (Y (τ_{i}^{-}, s) + C_{i} Y (τ_{i}, s)) [ψ' (s) - A_{1} ψ (s)] d s \\ - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} (Y (τ_{i}^{-}, s + ϱ_{m}) + C_{i} Y (τ_{i}, s + ϱ_{m})) ψ (s) d s \\ = u (τ_{i}^{-}) + C_{i} u (τ_{i}) \end{matrix}

From this, equation (10) is the solution of equation (3).

Next, one can repeat the proof in Lemma 1 to drive an explicit formula of solutions to linear impulsive nonhomogeneous multi-delay differential equations (we omit the details).

Lemma 4.The solution of equation (4) has the form

\begin{matrix} u (t) = Y (t, - ϱ) ψ (- ϱ) + \int_{- ϱ}^{0} Y (t, s) [ψ' (s) - A_{1} ψ (s)] d s \\ - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} Y (t, s + ϱ_{m}) ψ (s) d s \\ + \sum_{j = 0}^{i - 1} \int_{τ_{j}}^{τ_{j + 1}} Y (t, s) f (s) d s + \int_{τ_{i}}^{t} Y (t, s) f (s) d s \end{matrix}

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Asymptotic stability

Local asymptotic stability of the trivial solution to equation (3)

To prove our main results on the stability of the trivial solution, we introduce the following conditions:

(H₁) Suppose that $0 < θ_{1} \leq τ_{i + 1} - τ_{i} \leq θ_{2}, i = 0, 1, 2 \dots,$ and set

θ = {\begin{matrix} θ_{1}, & α - k < 0 \\ θ_{2}, & α - k \geq 0 \end{matrix}

(H₂) Suppose that

lim_{t \to \infty} \frac{i (0, t)}{t} = p

where $p > 0$ is a finite number.

(H₃) Assume

η_{1} : = α - k + \frac{1}{θ} \ln (ρ (C) + 1) < 0

where $ρ (C) : = max {ρ (C_{j}) : j = 1, \dots, i (0, t)}$

(H₄) Suppose that $η_{2} < 0$ , where

η_{2} : = α - k + p \ln (ρ (C) + 1)

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Now we are ready to state our stability result for the trivial solution of equation (3).

Theorem 2. If $(H_{0})$ , $(H_{1})$ and $(H_{3})$ are satisfied, then the trivial solution of equation (3) is locally asymptotically stable.

Proof. Using Lemma 3 and Theorem 1, we have

\begin{matrix} ∥ u (t) ∥ \leq ∥ Y (t, - ϱ) ∥ ∥ ψ (- ϱ) ∥ \\ + \int_{- ϱ}^{0} ∥ Y (t, s) ∥ ∥ ψ' (s) - A_{1} ψ (s) ∥ d s \\ - \sum_{m = 1}^{n} ∥ B_{m} ∥ \int_{- ϱ}^{- ϱ_{m}} ∥ Y (t, s + ϱ_{m}) ∥ ∥ ψ (s) ∥ d s \\ \leq K (\underset{- ϱ < τ_{j} < t}{Π} (ρ (C_{j}) + 1 + ε)) e^{(α - k) (t + ϱ)} ∥ ψ (- ϱ) ∥ \\ + K \int_{- ϱ}^{0} (\underset{s < τ_{j} < t}{Π} (ρ (C_{j}) + 1 + ε)) e^{(α - k) (t - s)} \\ ∥ ψ' (s) - A_{1} ψ (s) ∥ d s \\ + \sum_{m = 1}^{n} K ∥ B_{m} ∥ \int_{- ϱ}^{- ϱ_{m}} (\underset{s + ϱ_{m} < τ_{j} < t}{Π} (ρ (C_{j}) + 1 + ε)) e^{(α - k) (t - ϱ_{m} - s)} \\ ∥ ψ (s) ∥ d s \\ \begin{matrix} \leq M e^{(α - k) t} (ρ (C) + 1 + ε)^{i (0, t)} \\ \leq M e^{(α - k) θ} {[e^{(α - k) θ} (ρ (C) + 1 + ε)]}^{i (0, t)} \end{matrix} \end{matrix}

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where

\begin{matrix} M : = K (e^{(α - k) ϱ} ∥ ψ (- ϱ) ∥ + \int_{- ϱ}^{0} e^{(k - α) s} ∥ ψ' (s) - A_{1} ψ (s) ∥ d s \\ + \sum_{m = 1}^{n} ∥ B_{m} ∥ \int_{- ϱ}^{- ϱ_{m}} e^{(k - α) (ϱ_{m} + s)} ∥ ψ (s) ∥ d s) > 0 \end{matrix}

It follows from condition $(H_{3})$ that one can choose $ε > 0$ such that

e^{(α - k) θ} (ρ (C) + 1 + ε) < e^{\frac{θ η_{1}}{2}} < 1

Then equation (14) gives

∥ u (t) ∥ \leq M e^{(α - k) θ} e^{\frac{θ η_{1}}{2} i (0, t)} \to 0 for t \to \infty

The proof is complete.

Next, we give a second stability result for the trivial solution of equation (3).

Theorem 3. If $(H_{0})$ , $(H_{2})$ and $(H_{4})$ are satisfied, then the trivial solution of equation (3) is locally asymptotically stable.

Proof. The proof is similar to Theorem 2, so we only give details of the main differences. Note, from $(H_{2})$ , we have $i (0, t) < t (p + ε)$ , where $ε > 0$ is arbitrarily small. From equation (14) one has

∥ u (t) ∥ \leq M e^{[α - k + (p + ε) \ln (ρ (C) + 1 + ε)] t}

By virtue of $(H_{4})$ , we can find $ε$ such that

∥ u (t) ∥ \leq M e^{\frac{η_{2}}{2} t} \to 0 as t \to \infty

and thus the proof is finished.

Local asymptotic stability of trivial solution to equation (5)

We need the following additional assumptions:

(H₅) There exists $L_{1} > 0$ . such that $∥ F (u, y_{1}, \dots, y_{n}) ∥ \leq L_{1} ∥ u ∥ .$

(H₆) Suppose $η_{3} : = K L_{1} + η_{2} < 0$ , where $η_{2}$ is defined in equation (13).

(H₇) There exists $L_{2} > 0$ such that $∥ F (u, y_{1}, \dots, y_{n}) ∥ \leq L_{2} (∥ u ∥ + ∥ y_{1} ∥ + \dots + ∥ y_{n} ∥) .$

(H₈) Suppose

η_{4} : = K L_{2} (1 + \sum_{i = 1}^{n} e^{- η_{2} τ_{i}}) + η_{2} < 0

where $η_{2}$ is defined in equation (13).

Case 1

$F (u, u (\cdot - ϱ_{1}), \dots, u (\cdot - ϱ_{n})) = F (u)$ .

Theorem 4. Assume that $(H_{0})$ , $(H_{2})$ , $(H_{5})$ and $(H_{6})$ are satisfied. Then the trivial solution of equation (5) is locally asymptotically stable.

Proof. From $(H_{2})$ , equation (9) reduces to

\begin{matrix} ∥ Y (t, s) ∥ \leq K (ρ (C) + 1 + ε)^{i (s, t)} e^{(α - k) (t - s)} \\ \leq K e^{[α - k + (p + ε) \ln (ρ (C) + 1 + ε)] (t - s)} \\ = K e^{η_{ε} (t - s)} \end{matrix}

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where $η_{ε} : = α - k + (p + ε) \ln (ρ (C) + 1 + ε)$ .

From the variation of constants formula (see Lemma 4), the solution of equation (5) has the form

\begin{matrix} u (t) = Y (t, - ϱ) ψ (- ϱ) + \int_{- ϱ}^{0} Y (t, s) [ψ' (s) - A_{1} ψ (s)] d s \\ - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} Y (t, s + ϱ_{m}) ψ (s) d s \\ + \sum_{j = 0}^{i - 1} \int_{τ_{j}}^{τ_{j + 1}} Y (t, s) F (u (s)) d s \\ + \int_{τ_{i}}^{t} Y (t, s) F (u (s)) d s \end{matrix}

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Taking the norm on two sides of equation (16) via equation (15) and $(H_{5})$ , one has

\begin{matrix} ∥ u (t) ∥ \leq K e^{η_{ε} (t + ϱ)} ∥ ψ (- ϱ) ∥ \\ + \int_{- ϱ}^{0} K e^{η_{ε} (t - s)} ∥ ψ' (s) - A_{1} ψ (s) ∥ d s \\ + \sum_{m = 1}^{n} ∥ B_{m} ∥ \int_{- ϱ}^{- ϱ_{m}} K e^{η_{ε} (t - ϱ_{m} - s)} ∥ ψ (s) ∥ d s \\ + \sum_{j = 0}^{i - 1} \int_{τ_{j}}^{τ_{j + 1}} K e^{η_{ε} (t - s)} L_{1} ∥ u (s) ∥ d s \\ + \int_{τ_{i}}^{t} K e^{η_{ε} (t - s)} L_{1} ∥ u (s) ∥ d s \end{matrix}

This yields

e^{- η_{ε} t} ∥ u (t) ∥ \leq T_{1} + K L_{1} \int_{0}^{t} e^{- η_{ε} s} ∥ u (s) ∥ d s

where

\begin{matrix} T_{1} : = K e^{η_{ε} ϱ} ∥ ψ (- ϱ) ∥ + \int_{- ϱ}^{0} K e^{- η_{ε} s} ∥ ψ' (s) - A_{1} ψ (s) ∥ d s \\ + \sum_{m = 1}^{n} ∥ B_{m} ∥ \int_{- ϱ}^{- ϱ_{m}} K e^{- η_{ε} (ϱ_{m} + s)} ∥ ψ (s) ∥ d s \\ > 0 \end{matrix}

By using Bainov and Simeonov (1992: Theorem 1.1), we have

e^{- η_{ε} t} ∥ u (t) ∥ \leq T_{1} e^{K L_{1} t}

From $(H_{6})$ we can find a $ε > 0$ sufficiently small such that $K L_{1} + η_{ε} \leq η_{3} / 2 < 0,$ and then we have

∥ u (t) ∥ \leq T_{1} e^{(K L_{1} + η_{ε}) t} \leq T_{1} e^{\frac{η_{3}}{2} t} \to 0 as t \to \infty

The proof is finished.

Case 2

$F (u, u (\cdot - ϱ_{1}), \dots, u (\cdot - ϱ_{n}))$ depends on $u (\cdot - ϱ_{i}), i = 1, \dots, n$ .

Theorem 5. Assume that $(H_{0})$ , $(H_{2})$ , $(H_{7})$ and $(H_{8})$ are satisfied. Then the trivial solution of equation (5) is locally asymptotically stable.

Proof. From the variation of constants formula (see Lemma 4), the solution of equation (5) has the form

\begin{matrix} u (t) = Y (t, - ϱ) ψ (- ϱ) + \int_{- ϱ}^{0} Y (t, s) [ψ' (s) - A_{1} ψ (s)] d s \\ - \sum_{m = 1}^{n} B_{m} \int_{- ϱ}^{- ϱ_{m}} Y (t, s + ϱ_{m}) ψ (s) d s \\ + \sum_{j = 0}^{i - 1} \int_{τ_{j}}^{τ_{j + 1}} Y (t, s) F (u (s), u (s - ϱ_{1}), \dots, u (s - ϱ_{n})) d s \\ + \int_{τ_{i}}^{t} Y (t, s) F (u (s), u (s - ϱ_{1}), \dots, u (s - ϱ_{n})) d s \end{matrix}

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Taking the norm on two sides of equation (16) via equation (15) and $(H_{7})$ , one has

\begin{matrix} ∥ u (t) ∥ \leq K e^{η_{ε} (t + ϱ)} ∥ ψ (- ϱ) ∥ \\ + \int_{- ϱ}^{0} K e^{η_{ε} (t - s)} ∥ ψ' (s) - A_{1} ψ (s) ∥ d s \\ + \sum_{m = 1}^{n} ∥ B_{m} ∥ \int_{- ϱ}^{- ϱ_{m}} K e^{η_{ε} (t - ϱ_{m} - s)} ∥ ψ (s) ∥ d s \\ + K L_{2} \int_{0}^{t} e^{η_{ε} (t - s)} (∥ u (s) ∥ \\ + ∥ u (s - ϱ_{1}) ∥ + \dots + ∥ u (s - ϱ_{n}) ∥) d s \end{matrix}

Set $v (t) = e^{- η_{ε} t} ∥ u (t) ∥$ . Then

\begin{matrix} v (t) \leq T_{1} + K L_{2} \int_{0}^{t} v (s) d s \\ + \sum_{i = 1}^{n} K L_{2} \int_{0}^{t} e^{- η_{ε} ϱ_{i}} v (s - ϱ_{i}) d s, t \geq 0 \end{matrix}

where $η_{ε}$ and $T_{1}$ are defined in Theorem 4, and $v (t) = e^{- η_{ε} t} ∥ ψ (t) ∥ : = ϕ (t)$ as $t \in [- τ, 0]$ . Using Remark 1, for any $t \geq ϱ$ , one has

\begin{matrix} v (t) \leq [T_{1} + \sum_{i = 1}^{n} \int_{[0, t] ⋂ E_{i}} K L_{2} e^{- η_{ε} ϱ_{i}} ϕ (s - ϱ_{i}) d s] \\ \times e^{(\int_{0}^{t} K L_{2} d s + \sum_{i = 1}^{n} \int_{[0, t] \ E_{i}} K L_{2} e^{- η_{ε} ϱ_{i}} d s)} \\ \leq T_{2} e^{- K L_{2} \sum_{i = 1}^{n} ϱ_{i} e^{- η_{ε} ϱ_{i}}} e^{K L_{2} (1 + \sum_{i = 1}^{n} e^{- η_{ε} ϱ_{i}}) t} \\ \leq T_{2} e^{K L_{2} (1 + \sum_{i = 1}^{n} e^{- η_{ε} ϱ_{i}}) t} \end{matrix}

where $E_{i} = [0, ϱ_{i}]$ , and

T_{2} : = [T_{1} + \sum_{i = 1}^{n} \int_{0}^{ϱ_{i}} K L_{2} e^{- η_{ε} s} ∥ ψ (s - ϱ_{i}) ∥ d s] e^{- K L_{2} \sum_{i = 1}^{n} τ_{i} e^{- η_{ε} ϱ_{i}}} > 0

which yields

∥ u (t) ∥ \leq e^{η_{ε} t} v (t) \leq T_{2} e^{[K L_{2} (1 + \sum_{i = 1}^{n} e^{- η_{ε} ϱ_{i}}) + η_{ε}] t}

For $ε$ sufficiently small

K L_{2} (1 + \sum_{i = 1}^{n} e^{- η_{ε} ϱ_{i}}) + η_{ε} < η_{4} / 2 < 0

so then

∥ u (t) ∥ \leq T_{2} e^{\frac{η_{4}}{2} t} \to 0

as $t \to \infty$ . The proof is complete.

Numerical examples

In this section, we give numerical examples to demonstrate the effectiveness of our method. Here, we use MATLAB software to compute some parameters and draw the figures for the examples.

Let $[z]$ be the largest integer less than $z$ .

Example 1. Consider equation (3) associated with

\begin{matrix} A_{1} = (\begin{matrix} - 6 & 0 \\ 0 & - 5 \end{matrix}), B_{1} = (\begin{matrix} 0.9 & 0 \\ 0 & 0.5 \end{matrix}) \\ B_{2} = (\begin{matrix} 0.6 & 0 \\ 0 & 0.8 \end{matrix}), B_{3} = (\begin{matrix} 0.4 & 0 \\ 0 & 0.5 \end{matrix}) \\ C_{i} = (\begin{matrix} 2 & 0 \\ 0 & 2.5 \end{matrix}), ψ (t) = (\begin{matrix} 2 \\ 3 \end{matrix}) \\ ϱ_{1} = 0.1, ϱ_{2} = 0.2, ϱ_{3} = 0.25 \\ ϱ = max {ϱ_{1}, ϱ_{2}, ϱ_{3}} = 0.25, i (0, t) = [\frac{t}{2}] \end{matrix}

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Clearly, $A_{1} B_{m} = B_{m} A_{1}$ , $B_{j} B_{m} = B_{m} B_{j}$ , $A_{1} C_{i} = C_{i} A_{1}$ and $B_{m} C_{i} = C_{i} B_{m}$ for $m, j = 1, 2, 3$ , $i = 1, 2, \dots$ . Set $- 6 < - 5 \leq - k < 0$ and $∥ e^{A_{1} t} ∥ = e^{- 5 t} \leq K e^{- kt}$ , where choosing $K = 1, k = 5$ , $ϱ (C) = 2.5$ and $∥ ψ ∥_{1} = 3 = ∥ ψ (- ϱ) ∥ < δ : = 3.1$ and setting $τ_{i + 1} - τ_{i} = 2, i = 0, 1, 2 \dots,$ so $θ = 2$ . Next

\begin{matrix} ∥ e^{- A_{1} ϱ_{1}} B_{1} ∥ = ‖ (\begin{matrix} 1.6399 & 0 \\ 0 & 0.8244 \end{matrix}) ‖ \leq α_{1} e^{0.1 α_{1}} \\ choose α_{1} = 1.4225 \\ ∥ e^{- A_{1} ϱ_{2}} B_{2} ∥ = ‖ (\begin{matrix} 1.9921 & 0 \\ 0 & 2.1748 \end{matrix}) ‖ \leq α_{2} e^{0.2 α_{2}} \\ choose α_{2} = 1.5842 \\ ∥ e^{- A_{1} ϱ_{3}} B_{3} ∥ = ‖ (\begin{matrix} 1.7927 & 0 \\ 0 & 1.7452 \end{matrix}) ‖ \leq α_{3} e^{0.25 α_{3}} \\ choose α_{3} = 1.2965 \end{matrix}

and implies that $α = α_{1} + α_{2} + α_{3} = 4.3032$ , $M e^{(α - k) θ} = 1.6044 > 0$ and

η_{1} = α - k + \frac{1}{θ} \ln (ρ (C) + 1) = - 0.2386 < 0

Now all the conditions of Theorem 2 are satisfied. Then

∥ u (t) ∥ \leq M e^{(α - k) θ} e^{\frac{θ η_{1}}{2} i (0, t)} = 1.6044 e^{- 0.2386 i (0, t)} \to 0

for $t \to \infty$ , since $i (0, t) \to \infty$ as $t \to \infty$ , i.e. the trivial solution of equations (3) and (1) is locally asymptotically stable (see Figure 1).

Figure 1.

State response $u (t)$ of equations (3) and (18).

Example 2. Consider the equation originated from the impulsive two-species competition system

\begin{matrix} {\begin{matrix} u' (t) = A_{1} u (t) + B_{1} u (t - 0.2) + B_{2} u (t - 0.3) \\ + F (u (t), u (t - 0.2), u (t - 0.3)), t > 0, t \neq τ_{i} \\ Δ u (τ_{i}) = C_{i} u (τ_{i}), i = 1, 2, \dots \\ u (t) = ψ (t {) = (0.3, 0.4)}^{T}, - 0.3 \leq t \leq 0 \end{matrix} \end{matrix}

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where we set

i (0, t) = [\frac{t + 1}{2}]

$ϱ_{1} = 0.2$ , $ϱ_{2} = 0.3$ ; then $ϱ = 0.3$ and

\begin{matrix} A_{1} = (\begin{matrix} - 5 & 0.25 \\ 0 & - 4.5 \end{matrix}), B_{1} = (\begin{matrix} 0.4 & 0.05 \\ 0 & 0.5 \end{matrix}) \\ B_{2} = (\begin{matrix} 0.3 & 0.05 \\ 0 & 0.4 \end{matrix}), C_{i} = (\begin{matrix} 2 + 1 / i & 0 \\ 0 & 2 + 1 / i \end{matrix}) \end{matrix}

and

\begin{matrix} F (u (t), u (t - 0.2), u (t - 0.3)) \\ = (\begin{matrix} 1 / 4 \sin (u (t)) + 1 / 3 \sin (u (t - 0.2)) \\ + 1 / 3 \sin (u (t - 0.3)) \\ 1 / 3 \sin (u (t)) + 1 / 4 \sin (u (t - 0.2)) \\ + 1 / 4 \sin (u (t - 0.3)) \end{matrix}) \end{matrix}

Now

\begin{matrix} A_{1} B_{1} = (\begin{matrix} - 2 & - 0.125 \\ 0 & - 2.25 \end{matrix}) = B_{1} A_{1} \\ A_{1} B_{2} = (\begin{matrix} - 1.5 & - 0.15 \\ 0 & - 1.8 \end{matrix}) = B_{2} A_{1} \\ B_{1} B_{2} = (\begin{matrix} 0.12 & 0.04 \\ 0 & 0.2 \end{matrix}) = B_{2} B_{1} \\ A_{1} C_{i} = (\begin{matrix} - 10 - 5 / i & - 0.5 - 0.25 / i \\ 0 & - 9 - 4.5 / i \end{matrix}) = C_{i} A_{1} \\ B_{1} C_{i} = (\begin{matrix} 0.8 + 0.4 / i & 0.1 + 0.05 / i \\ 0 & 1 + 0.5 / i \end{matrix}) = C_{i} B_{1} \\ B_{2} C_{i} = (\begin{matrix} 0.6 + 0.3 / i & 0.1 + 0.05 / i \\ 0 & 0.8 + 0.4 / i \end{matrix}) = C_{i} B_{2} \\ i = 1, 2, \dots \end{matrix}

It is clear that $ρ (C_{i}) = 2 + 1 / i, i = 1, 2, \dots$ , then $ρ (C) = 3$ . Set $- 5 < - 4.5 \leq - k < 0$ and $∥ e^{A_{1} t} ∥ = e^{- 4.5 t} \leq K e^{- kt}$ , where choosing $K = 1, k = 4.5$ . By computation, $∥ ψ ∥_{1} = 0.4 = ∥ ψ (- ϱ) ∥ < δ : = 0.41$

p = lim_{t \to \infty} \frac{i (0, t)}{t} = lim_{t \to \infty} \frac{[\frac{t + 1}{2}]}{t} = \frac{1}{2}

and $∥ F (u (t), u (t - 0.2), u (t - 0.3)) ∥ \leq 1 / 3 (∥ u (t) ∥ + ∥ u (t - 0.2) ∥ + ∥ u (t - 0.3) ∥),$ we can choose $L_{2} = 1 / 3$ . Next

\begin{matrix} ∥ e^{- A_{1} τ_{1}} B_{1} ∥ = ‖ (\begin{matrix} 1.0873 & 0.0712 \\ 0 & 1.2298 \end{matrix}) ‖ \leq α_{1} e^{0.2 α_{1}} \\ choose α_{1} = 1.0058 \\ ∥ e^{- A_{1} τ_{2}} B_{2} ∥ = ‖ (\begin{matrix} 1.3445 & 0.0992 \\ 0 & 1.5430 \end{matrix}) ‖ \leq α_{2} e^{0.3 α_{2}} \\ choose α_{2} = 1.1070 \end{matrix}

By calculation, $α = α_{1} + α_{2} = 2.1128$ , $η_{2} = α - k + p \ln (ρ (C) + 1) = - 1.6940$

\begin{matrix} T_{2} \geq (K e^{η_{2} ϱ} ∥ ψ (- ϱ) ∥ + \int_{- ϱ_{2}}^{0} K e^{- \frac{η_{2}}{2} s} ∥ ψ' (s) - A_{1} ψ (s) ∥ d s \\ + ∥ B_{1} ∥ \int_{- ϱ_{2}}^{- ϱ_{1}} K e^{- \frac{η_{2}}{2} (ϱ_{1} + s)} ∥ ψ (s) ∥ d s \\ + \sum_{i = 1}^{2} \int_{0}^{ϱ_{i}} K L_{2} e^{- \frac{η_{2}}{2} s} ∥ ψ (s - ϱ_{i}) ∥ d s) e^{- K L_{2} \sum_{i = 1}^{2} ϱ_{i} e^{- η_{2} ϱ_{i}}} \\ = 0.6257 \\ > 0 \end{matrix}

and

η_{4} = K L_{2} (1 + \sum_{i = 1}^{2} e^{- η_{2} ϱ_{i}}) + η_{2} = - 0.1888 < 0

Now all the conditions of Theorem 5 are satisfied. Then

∥ u (t) ∥ \leq T_{2} e^{\frac{η_{4}}{2} t} = T_{2} e^{- 0.0944 t} \to 0 as t \to \infty

Then the trivial solution of equation (19) is locally asymptotically stable (see Figure 2).

Figure 2.

State response $u (t)$ of equation (19).

Conclusions

This paper introduces a new concept of an impulsive multi-delayed Cauchy matrix, which is then used to obtain the explicit solution to linear impulsive multi-delayed differential equations using the variation of constants principle. With the help of the representation of solutions, we apply the multi-delayed Grönwall inequality to establish the asymptotic stability of trivial solutions.

Footnotes

Acknowledgements

The authors are grateful to the referees for their careful reading of the manuscript and their valuable comments.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (grant number 11661016), the Training Object of High Level and Innovative Talents of Guizhou Province (grant number (2016)4006), the Unite Foundation of Guizhou Province (grant number [2015]7640), and Graduate ZDKC (grant number [2015]003).

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