Periodic solution for neutral-type differential equation with piecewise impulses on time scales

Abstract

In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.

1 Introduction

In 1988, Hilger [1] first proposed time scale theory and calculation. By using time scale theory discrete and continuous analysis can be unified. Thus it provides us with new methods for studying discrete and continuous systems. For more details of time scale theory, we refer to books [2, 3], where the authors give important overviews of the basic theory of time scales and their applications to differential equations. For recent research on differential equations on time scales, see [4–8] and the references therein.

In this paper, we focus on the problem of periodic solutions to differential equations on time scales. Let us briefly review the research progress on the above issues. In 2007, Kaufmann and Raffoul [9] introduced a new definition of periodic time scale and studied nonlinear neutral dynamic equations with delay of the form

$$ x^{\Delta}(t)=-a(t)x(\sigma (t))+\big(F(t,x(t),x(t-g(t)))\big)^{ \Delta }+G(t,x(t),x(t-g(t))),~t\in \mathbb{T}, $$

(1.1)

where $\mathbb{T}$ is a periodic time scale, $a\in \mathcal{R}^{+}$ with $a>0$, and the functions $F(t,x,y)$ and $G(t,x,y)$ are continuous and periodic in t and Lipschitz continuous in x and y. Based on Krasnoselskii’s fixed point theorem and contraction mapping principle, some sufficient conditions were obtained guaranteeing the existence and stability of periodic solutions to equation (1.1). Ardjouni and Djoudi [10] considered a nonlinear dynamic equation with variable delay of the form

$$ x^{\Delta}(t)=-a(t)x^{3}(\sigma (t))+c(x)x^{\tilde{\Delta}}(t-\tau (t)) +G(t,x^{3}(t),x^{3}(t-\tau (t)),~t\in \mathbb{T}, $$

(1.2)

where $\mathbb{T}$ is a periodic time scale, $x^{\Delta}$ is the Δ-derivative on $\mathbb{T}$, and $x^{\tilde{\Delta}}$ is the Δ-derivative on $(id-\tau )(\mathbb{T})$. Using the Burton–Krasnoselskii theorem, they obtained the existence of periodic solutions to equation (1.2). Alam, Abbas, and Nieto [11] investigated the existence of periodic solutions of modified version of the Leslie–Gower predator–prey model on time scales by using the continuation theorem of coincidence degree theory. Bohner et al. [12] studied the existence of periodic solutions for predator–prey dynamic systems with Beddington–DeAngelis functional response on time scales. Li and Li [13] studied almost periodic time scales and almost periodic functions on time scales. Most of results about periodic solutions on time scales can be found in [14–18].

On the other hand, the impulsive dynamic systems have been extensively studied. Kumar and Djemai [19] established the existence of solutions, stability, and controllability results for piecewise nonlinear impulsive dynamic systems on time scales by using the Banach fixed point theorem, Gramian-type matrices, and the theory of time scales. For the controllability, stability, and observability problems for linear impulsive dynamic systems on time scales, see [20, 21]; for the approximate controllability of semilinear dynamic equations, see [22]; for the controllability of neutral differential equations with impulses on time scales, see [23]; for the stability and controllability results of evolution impulsive dynamic systems, see [24]; for the total controllability results for a class of time-varying switched dynamical systems with impulses, see [25]; and for the abstract integro-hybrid evolution systems with impulses on time scales, see [26].

However, a few authors studied periodic solution problems for neutral-type differential equations with piecewise impulses on time scales. Motivated by the above discussion, we study the existence and stability of solutions of the following piecewise impulsive neutral-type equation on time scales:

$$ \left \{ \textstyle\begin{array}{l} (x(t)+c(t)x(t-\gamma ))^{\Delta}=-a(t)x(t) +f(t,x(t-\tau (t))) \\ \hphantom{(x(t)+c(t)x(t-\gamma ))^{\Delta}=}{}+ \int _{0}^{t} K(s)g(t,x(s))\Delta s,~t\geq 0,~t\neq t_{j}, \\ x(t_{j}^{-})-x(t_{j}^{+})=I_{j}(x(t_{j})),~~t= t_{j},~j=1,2,\ldots ,q, \end{array}\displaystyle \right . $$

(1.3)

where $\mathbb{T}$ is a periodic time scale, $t\in \mathbb{T}^{+}=[0,\infty )_{\mathbb{T}}$, $\gamma >0$ is a constant, $~a\in C_{rd}(\mathbb{T}^{+},\mathbb{R})$ with $a(t)<0$, $~c,\tau \in C_{rd}(\mathbb{T}^{+},\mathbb{R})$ with $|c(t)|\neq 1 \text{ and } \tau (t)>0$, $K\in C_{rd}(\mathbb{T}^{+}, \mathbb{R})$, $f, g\in C_{rd}(\mathbb{T}^{+}\times \mathbb{R}, \mathbb{R})$ with $f(t+\omega ,\cdot )=f(t,\cdot )$ and $g(t+\omega ,\cdot )=g(t,\cdot )$, the impulse points $t_{j}$ satisfy the relation $0< t_{1}<\cdots <t_{q}$, $x(t_{j}^{-})$ and $x(t_{j}^{+})$ represent the right and the left limits of $x(t_{j})$, respectively, $I_{j}\in C(\mathbb{R},\mathbb{R})$, and there exists a positive constant q such that $t_{j+q}=t_{j}+\omega$ and $I_{j}(x(t_{j}))=I_{j+q}(x(t_{j+q}))$. Throughout this paper, we assume that a, c, and τ are all ω-periodic functions.

We list the following main contributions of this paper:

(1) We first study a class of neutral-type differential equations with piecewise impulses on time scales. The model in the present paper is different from that in [27]. Since (1.3) is a time scale system unifying discrete and continuous systems, our results have wider applicability.

(2) We develop a fixed point theorem for differential systems on time scales.

(3) We develop two different sufficient conditions for the existence of periodic solutions to the considered system by using the Banach and Schauder fixed point theorems.

The rest of this paper is organized as follows. Section 2 gives preliminaries including some definitions and lemmas. In Sect. 3, some sufficient conditions for existence of periodic solution of system (1.3) are obtained. Section 4 gives some stability results for system (1.3). In Sect. 5, an example is given to show the effectiveness of the main results. Finally, we give some conclusions and discussions.

2 Preliminaries

A time scale $\mathbb{T}$ is a closed subset of $\mathbb{R}$ with the topology and ordering inherited from $\mathbb{R}$. For the definitions and means for the forward jump operator σ, backward jump operator ρ, graininess function μ, regressive function set $\mathcal {R}$, and positive regressive set $\mathcal {R}^{+}$, see [2]. The interval $[s,t]_{\mathbb{T}}$ means $[s,t]\cap{\mathbb{T}}$. The intervals $[s,t)_{\mathbb{T}}$, $(s,t)_{\mathbb{T}}$, and $(s,t]_{\mathbb{T}}$ are defined similarly. A function $f:\mathbb{T}\rightarrow \mathbb{R}$ is said to be rd-continuous if it is continuous at all right-dense points of $\mathbb{T}$ and the finite left-hand limit exists at left-dense points of $\mathbb{T}$. The set of all rd-continuous functions is denoted by $C_{rd}(\mathbb{T},\mathbb{R})$.

Definition 2.1

[28] A time scale $\mathbb{T}$ is periodic if for each $t\in \mathbb{T}$, there is $l>0$ such that $t\pm l\in \mathbb{T}$. For $\mathbb{T}\neq \mathbb{R}$, the period of the time scale is the smallest positive l.

Definition 2.2

[28] Let $\mathbb{T}\neq \mathbb{R}$ be a periodic time scale with period l. The function $\psi :\mathbb{T}\rightarrow \mathbb{R}$ is periodic with period τ if there exists a natural number n such that $\tau =nl$ and $\psi (t\pm \tau )=\psi (t)$ for all $t\in \mathbb{T}$. When $\mathbb{T}=\mathbb{R}$, ψ is a periodic function if τ is the smallest positive number such that $\psi (t\pm \tau )=\psi (t)$.

The exponential function on $\mathbb{T}$ is defined by $e_{h}(t,s)=\exp \bigg(\int _{s}^{t}\xi _{\mu (r)}(h(r))\Delta r \bigg)$, where

$$ \xi _{\mu (r)}(h(r))=\left \{ \textstyle\begin{array}{l} \frac{1}{\mu (r)}Log(1+\mu (r)\alpha (r)),~\mu (r)>0, \\ h(r),~~\mu (r)=0. \end{array}\displaystyle \right . $$

Lemma 2.1

[2] Let $u,v\in \mathcal {R}$. Then

• [1] ${e}_{0}(t,s)\equiv 1$ and ${e}_{u}(t,t)\equiv 1$;

• [2] ${e}_{u}(\rho (t),s)=(1-\mu (t)u(t)){e}_{u}(t,s)$;

• [3] ${e}_{u}(t,s)=\frac{1}{{e}_{u}(s,t)}=e_{\ominus u}(s,t)$, where $\ominus u(t)=-\frac{u(t)}{1+\mu (t)u(t)}$;

• [4] ${e}_{u}(t,s){e}_{u}(s,r)={e}_{u}(t,r)$.

Lemma 2.2

[29] For a nonnegative function ρ with $-\rho \in \mathcal{R}^{+}$, we have

$$ 1-\int _{s}^{t}\rho (u)\Delta u\leq e_{-\rho}(t,s)\leq \exp \bigg\{ - \int _{s}^{t}\rho (u)\Delta u\bigg\} ~\mathrm{for~all}~t\geq s. $$

For a nonnegative function ρ with $\rho \in \mathcal{R}^{+}$, we have

$$ 1+\int _{s}^{t}\rho (u)\Delta u\leq e_{\rho}(t,s)\leq \exp \bigg\{ \int _{s}^{t}\rho (u)\Delta u\bigg\} ~\mathrm{for~all}~t\geq s. $$

Lemma 2.3

(Banach contraction mapping principle) [30] Let X be a Banach space with norm $||\cdot ||$. Let $T: \Gamma \rightarrow \Gamma$, where $\Gamma \subseteq X$ is a nonempty closed set. If there exists $k\in [0,1)$ such that $||Tx-Ty||< k||x-y||$ for all $x,y\in \Gamma$, then T has unique fixed point $x^{*}\in \Gamma$ satisfying $Tx^{*}=x^{*}$.

Lemma 2.4

(Schauder’s fixed theorem) [31] Let X be a nonempty convex compact subset of a Banach space Y, and let $T: X\rightarrow X$ be a continuous map. Then T has a fixed point.

Let

$$ PC(\mathbb{T})=\{x(t):x(t)\in C_{rd}((t_{j},t_{j+1})_{\mathbb{T}}, \mathbb{R}),~j=1,2,\ldots ,q\} $$

and

$$ \Xi =\{x(t):x\in PC(\mathbb{T}),~x(t+\omega )=x(t)\} $$

with the norm $||x||=\sup _{t\in [0,\omega ]_{\mathbb{T}}}|x(t)|$. Obviously, Ξ is a Banach space. Let

$$ \mathcal{A}:\Xi \rightarrow \Xi ,~(\mathcal{A}x)(t)=x(t)+c(t)x(t- \gamma ). $$

Lemma 2.5

[32] If $|c(t)|\neq 1$, then $\mathcal{A}$ has a bounded inverse $\mathcal{A}^{-1}$ on Ξ, and for all $x\in \Xi$,

$$ ||\mathcal{A}^{-1}x||\leq \frac{1}{1-c_{0}}~for~c_{0}< 1 $$

or

$$ ||\mathcal{A}^{-1}x||\leq \frac{1}{c_{1}-1}~for~c_{1}>1, $$

where $c_{0}=\sup _{t\in [0,\omega ]_{\mathbb{T}}}|c(t)|$ and $c_{1}=\inf _{t\in [0,\omega ]_{\mathbb{T}}}|c(t)|$.

Obviously, $\mathcal{A}^{-1}$ and $\mathcal{A}$ are both linear operators. Letting $(\mathcal{A}x)(t)=u(t)$, we can use $\mathcal{A}^{-1}$ to change (1.3) into the form

$$ \left \{ \textstyle\begin{array}{l} u^{\Delta}(t)=-a(t)(\mathcal{A}^{-1}u)(t) +f(t,\mathcal{A}^{-1}u(t- \tau (t))) \\ \hphantom{u^{\Delta}(t)=}{}+\int _{0}^{t} K(s)g(t,\mathcal{A}^{-1}u(s))\Delta s,~t\in \mathbb{T},~t\neq t_{j}, \\ \mathcal{A}^{-1}u(t_{j}^{-})-\mathcal{A}^{-1}u(t_{j}^{+})=I_{j}( \mathcal{A}^{-1}u(t_{j})),~~t= t_{j},~j=1,2,\ldots ,q. \end{array}\displaystyle \right . $$

(2.1)

Using $(\mathcal{A}^{-1}u(t)=u(t)-c(t)(\mathcal{A}^{-1}u)(t-\gamma )$, (2.1) can be changed into

$$ \left \{ \textstyle\begin{array}{l} u^{\Delta}(t)=-a(t)u(t)+a(t)c(t)(\mathcal{A}^{-1}u)(t) +f(t, \mathcal{A}^{-1}u(t-\tau (t))) \\ \hphantom{u^{\Delta}(t)=}{}+\int _{0}^{t} K(s)g(t,\mathcal{A}^{-1}u(s))\Delta s,~t\in \mathbb{T},~t\neq t_{j}, \\ u(t_{j}^{-})-u(t_{j}^{+})=(\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j}),~~t= t_{j},~j=1,2,\ldots ,q. \end{array}\displaystyle \right . $$

(2.2)

Since system (1.3) is equivalent to system (2.2), we only study the existence of periodic solutions for system (2.2).

Lemma 2.6

A function $u\in \Xi$ is called a solution of system (2.2) if it satisfies the equation

$$ \begin{aligned} u(t)&=\int _{t}^{t+\omega}\Theta (t,s)\bigg(-a(s)c(s)(\mathcal{A}^{-1}u)(s) +f(s,\mathcal{A}^{-1}u(s-\tau (s))) \\ &\quad {}+\int _{0}^{s} K(\xi )g(s, \mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\Delta s \\ &\quad {}+\sum _{t_{j}\in [t,t+\omega ]_{\mathbb{T}}}\Theta (t,t_{j})e_{-a}( \sigma (t_{j}),t_{j}) (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j}), \end{aligned} $$

where $\Theta (t,s)=\frac{e_{-a}(t,\sigma (s))}{1-e_{-a}(0,\omega )}$, $t \in \mathbb{T}$, $s\in [t,t+\omega ]_{\mathbb{T}}$.

Proof

The proof of Lemma 2.6 is similar to that of Lemma 2.8 in [27], so we omit it. □

In view of Lemma 2.2 and $a(t)<0$, it is easy to see that

$$ \Theta (t,s)\leq \frac{1}{1-\alpha}, $$

(2.3)

where $\alpha =\inf \{ e_{-a(t)}(0,\omega ),~t\in [0,\omega ]_{\mathbb{T}} \}<1$.

In this paper, we need the following assumptions:

(H₁) For all $t\in \mathbb{T}^{+}$ and $x,y\in \mathbb{R}$, the nonlinear functions f and g satisfy the following inequalities:

$$\begin{aligned}& ||f(t,x)-f(t,y)||\leq l_{f}||x-y||~~||g(t,x)-g(t,y)||\leq l_{g}||x-y||,\\& ||f(t,0)||\leq m_{f},~~||g(t,0)||\leq m_{g}, \end{aligned}$$

where $l_{f}$, $l_{g}$, $m_{f}$, and $m_{g}$ are positive constants.

(H₂) For all $j=1,2,\ldots ,q$ and $x,y\in \mathbb{R}$, the impulsive term $I_{j}$ satisfies the inequality

$$ ||I_{j}(t_{j},x)-I_{j}(t_{j},y)||\leq l_{j}||x-y||~~||I_{j}(t_{j},0)|| \leq m_{j}, $$

where $l_{j}$ and $m_{j}$ are positive constants.

(H₃) For all $s,t\in \mathbb{T}^{+}$, there exists a positive constant M such that

$$ \int _{0}^{t} |K(s)|\Delta s\leq M. $$

(H₄) For all $t\in \mathbb{T}^{+}$ and $x\in \mathbb{R}$, the nonlinear functions f and g satisfy the inequalities

$$ ||f(t,x)||\leq \tilde{l}_{f} \quad \text{and}\quad ||g(t,x)||\leq \tilde{l}_{g}, $$

where $\tilde{l}_{f}$ and $\tilde{l}_{g}$ are positive constants.

(H₅) For all $j=1,2,\ldots ,q$ and $x\in \mathbb{R}$, the impulsive term $I_{j}$ satisfies the inequality

$$ ||I_{j}(t_{j},x)||\leq \tilde{l}_{j}, $$

where $\tilde{l}_{j}$ is a positive constant.

3 Existence of periodic solutions

In this section, we prove the existence of periodic solutions of system (1.3). First, we prove the existence of a unique periodic solution by using the Banach contraction mapping principle, and after that, we prove the existence of at least one periodic solution by using the Schauder fixed point theorem.

Theorem 3.1

If assumptions (H₁)–(H₃) hold, then system (1.3) has a unique periodic solution, provided that

$$ \frac{\omega (\alpha c_{0}+l_{f}+Ml_{g})}{(1-\alpha )(1-c_{0})} + \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}< 1~~\mathrm{for}~c_{0}< 1 $$

(3.1)

or

$$ \frac{\omega (\alpha c_{0}+l_{f}+Ml_{g})}{(1-\alpha )(c_{1}-1)} + \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}< 1~~\mathrm{for}~c_{1}>1, $$

(3.2)

where $\beta =\exp \bigg\{-\int _{0}^{\omega }a(u)\Delta u\bigg\}$.

Proof

For a positive constant r, let $B_{r}=\{x:x\in \Xi ,~||x||\leq r\}$, where r satisfies

$$ \begin{aligned} r&\geq \bigg[\frac{\omega}{1-\alpha}\big(m_{f}+Mm_{g}\big) + \frac{\beta}{1-\alpha}\sum _{j=1}^{q}m_{j}\bigg] \bigg[1- \frac{\omega (\alpha c_{0}+l_{f}+Ml_{g})}{(1-\alpha )(1-c_{0})} - \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\bigg]^{-1} \\ &>0~~~~~\mathrm{for}~c_{0}< 1, \end{aligned} $$

(3.3)

or

$$ \begin{aligned} r&\geq \bigg[\frac{\omega}{1-\alpha}\big(m_{f}+Mm_{g}\big) + \frac{\beta}{1-\alpha}\sum _{j=1}^{q}m_{j}\bigg] \bigg[1- \frac{\omega (\alpha c_{0}+l_{f}+Ml_{g})}{(1-\alpha )(c_{1}-1)} - \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\bigg]^{-1} \\ &>0~~~~~\mathrm{for}~c_{1}>1. \end{aligned} $$

(3.4)

Here we use (3.1) and (3.2) tor guarantee $r>0$. Define the operator $\Gamma : B_{r}\rightarrow B_{r}$ by

$$ \begin{aligned} (\Gamma u)(t)&=\int _{t}^{t+\omega}\Theta (t,s)\bigg(-a(s)c(s)( \mathcal{A}^{-1}u)(s) +f(s,\mathcal{A}^{-1}u(s-\tau (s))) \\ &\quad {}+\int _{0}^{s} K(\xi )g(s,\mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\Delta s \\ &\quad {}+\sum _{t_{j}\in [t,t+\omega ]_{\mathbb{T}}}\Theta (t,t_{j})e_{-a}( \sigma (t_{j}),t_{j}) (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j}). \end{aligned} $$

From Lemma 2.6, if there exists a fixed point of Γ, i.e., there exists $u\in \Xi$ such that $\Gamma u=u$, then the fixed point u is a periodic solution of system (2.2). Hence we prove the existence of a fixed point of Γ by using the Banach contraction mapping principle. For better readability, we divide the proof into two steps.

Step 1: We show that $\Gamma (B_{r})\subset B_{r}$. If $c_{0}<1$, then for any $u\in B_{r}$, using assumptions (H₁)–(H₃), (3.3), and Lemma 2.5, we have

$$ \begin{aligned} ||(\Gamma u)(t)||&\leq \frac{1}{1-\alpha}\int _{t}^{t+\omega}\bigg(a_{0}c_{0}||( \mathcal{A}^{-1}u)(s) || +||f(s,\mathcal{A}^{-1}u(s-\tau (s)))|| \\ &+\int _{0}^{s} ||K(\xi )g(s,\mathcal{A}^{-1}u(\xi ))||\Delta \xi \bigg)\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{t_{j} \in [t,t+\omega ]_{\mathbb{T}}}|| (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j})|| \\ &\leq \frac{\omega }{1-\alpha}\bigg(\frac{a_{0}c_{0}}{1-c_{0}}r + \frac{l_{f}r}{1-c_{0}}+m_{f}+\frac{Ml_{g}r}{1-c_{0}}+Mm_{g}\bigg) \\ &+\frac{\beta}{1-\alpha}\sum _{j=1}^{q}(l_{j}r+m_{j}) \\ &\leq r. \end{aligned} $$

(3.5)

If $c_{1}>1$, then for any $u\in B_{r}$, using assumptions (H₁)–(H₃), (3.4), and Lemma 2.5, we have

$$ \begin{aligned} ||(\Gamma u)(t)||&\leq \frac{1}{1-\alpha}\int _{t}^{t+\omega}\bigg(a_{0}c_{0}||( \mathcal{A}^{-1}u)(s)|| +||f(s,\mathcal{A}^{-1}u(s-\tau (s)))|| \\ &+\int _{0}^{s} ||K(\xi )g(s,\mathcal{A}^{-1}u(\xi ))||\Delta \xi \bigg)\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{t_{j} \in [t,t+\omega ]_{\mathbb{T}}}|| (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j})|| \\ &\leq \frac{\omega }{1-\alpha}\bigg(\frac{a_{0}c_{0}}{c_{1}-1}r + \frac{l_{f}r}{c_{1}-1}+m_{f}+\frac{Ml_{g}r}{c_{1}-1}+Mm_{g}\bigg) \\ &+\frac{\beta}{1-\alpha}\sum _{j=1}^{q}(l_{j}r+m_{j}) \\ &\leq r. \end{aligned} $$

(3.6)

From (3.5) and (3.6) it follows that the operator Γ maps $B_{r}$ into $B_{r}$.

Step 2: We show that the operator Γ is a contraction. If $c_{0}<1$, then for any $u,v\in B_{r}$, using assumptions (H₁)–(H₃), we have

$$\begin{aligned} ||(\Gamma u)(t)-(\Gamma v)(t)|| &\leq \frac{1}{1-\alpha}\int _{t}^{t+ \omega}\bigg(a_{0}c_{0}||(\mathcal{A}^{-1}u)(s) -(\mathcal{A}^{-1}v)(s)|| \\ &+||f(s,\mathcal{A}^{-1}u(s-\tau (s)))-f(s,\mathcal{A}^{-1}v(s-\tau (s)))|| \\ &+\int _{0}^{s} ||K(\xi )|||| g(s,\mathcal{A}^{-1}u(\xi ))-g(s, \mathcal{A}^{-1}v(\xi ))||\Delta \xi \bigg)\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{t_{j} \in [t,t+\omega ]_{\mathbb{T}}}|| (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j})-(\mathcal{A}I_{j} \mathcal{A}^{-1})v(t_{j})|| \\ &\leq \frac{\omega }{1-\alpha}\bigg(\frac{a_{0}c_{0}}{1-c_{0}} + \frac{l_{f}}{1-c_{0}}+\frac{Ml_{g}}{1-c_{0}}\bigg)||u-v|| \\ &+\frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}||u-v|| \\ &=\bigg[ \frac{\omega (\alpha c_{0}+l_{f}+Ml_{g})}{(1-\alpha )(1-c_{0})} + \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\bigg]||u-v||. \end{aligned}$$

(3.7)

If $c_{1}>1$, then for any $u,v\in B_{r}$, using assumptions (H₁)–(H₃), we have

$$ \begin{aligned} ||(\Gamma u)(t)-(\Gamma v)(t)|| &\leq \frac{1}{1-\alpha}\int _{t}^{t+ \omega}\bigg(a_{0}c_{0}||(\mathcal{A}^{-1}u)(s) -(\mathcal{A}^{-1}v)(s)|| \\ &+||f(s,\mathcal{A}^{-1}u(s-\tau (s)))-f(s,\mathcal{A}^{-1}v(s-\tau (s)))|| \\ &+\int _{0}^{s} ||K(\xi )|||| g(s,\mathcal{A}^{-1}u(\xi ))-g(s, \mathcal{A}^{-1}v(\xi ))||\Delta \xi \bigg)\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{t_{j} \in [t,t+\omega ]_{\mathbb{T}}}|| (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j})-(\mathcal{A}I_{j} \mathcal{A}^{-1})v(t_{j})|| \\ &\leq \frac{\omega }{1-\alpha}\bigg(\frac{a_{0}c_{0}}{c_{1}-1} + \frac{l_{f}}{c_{1}-1}+\frac{Ml_{g}}{c_{1}-1}\bigg)||u-v|| \\ &+\frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}||u-v|| \\ &=\bigg[ \frac{\omega (\alpha c_{0}+l_{f}+Ml_{g})}{(1-\alpha )(c_{1}-1)} + \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\bigg]||u-v||. \end{aligned} $$

(3.8)

In view of (3.1), (3.2), (3.7), and (3.8), the operator Γ is a contraction.

From the proof of steps 1 and 2 we can see that the operator Γ fulfills all the requirements of the Banach contraction mapping principle, and hence system (2.2) has a unique periodic solution u, i.e., system (1.3) has a unique periodic solution $\mathcal{A}^{-1}u$. □

Next, we establish the existence of at least one periodic solution of system (1.3) by using the Schauder fixed point theorem.

Theorem 3.2

If assumptions (H₃)–(H₅) hold, then system (1.3) has at least one periodic solution, provided that

$$ \frac{\omega \alpha c_{0}}{(1-\alpha )(1-c_{0})} < 1~~~~~\mathrm{for}~c_{0}< 1 $$

(3.9)

or

$$ \frac{\omega \alpha c_{0}}{(1-\alpha )(c_{1}-1)} < 1~~~~~\mathrm{for}~c_{1}>1. $$

(3.10)

Proof

In view of (3.9) and (3.10), let $B_{\tilde{r}}=\{x:x\in \Xi ,~||x||\leq \tilde{r}\}$, where r̃ satisfies

$$ \begin{aligned} \tilde{r}\geq \bigg[\frac{\omega}{1-\alpha}\big(\tilde{l}_{f}+M \tilde{l}_{g}\big) +\frac{\beta}{1-\alpha}\sum _{j=1}^{q}\tilde{l}_{j} \bigg] \bigg[1-\frac{\omega \alpha c_{0}}{(1-\alpha )(1-c_{0})}\bigg]^{-1} >0~~~~~\mathrm{for}~c_{0}< 1 \end{aligned} $$

(3.11)

or

$$ \begin{aligned} \tilde{r}\geq \bigg[\frac{\omega}{1-\alpha}\big(\tilde{l}_{f}+M \tilde{l}_{g}\big) +\frac{\beta}{1-\alpha}\sum _{j=1}^{q}\tilde{l}_{j} \bigg] \bigg[1-\frac{\omega \alpha c_{0}}{(1-\alpha )(c_{1}-1)}\bigg]^{-1} >0~~~~~\mathrm{for}~c_{1}>1, \end{aligned} $$

(3.12)

where $\beta =\exp \bigg\{-\int _{0}^{\omega }a(u)\Delta u\bigg\}$. Define the operator $\Gamma : B_{\tilde{r}}\rightarrow B_{\tilde{r}}$ by

$$ \begin{aligned} (\Gamma u)(t)&=\int _{t}^{t+\omega}\Theta (t,s)\bigg(-a(s)c(s)( \mathcal{A}^{-1}u)(s) +f(s,\mathcal{A}^{-1}u(s-\tau (s))) \\ &\quad{} +\int _{0}^{s} K(\xi )g(s,\mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\Delta s \\ &\quad {}+\sum _{t_{j}\in [t,t+\omega ]_{\mathbb{T}}}\Theta (t,t_{j})e_{-a}( \sigma (t_{j}),t_{j}) (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j}). \end{aligned} $$

We divide the proof into three steps.

Step 1: We show that $\Gamma (B_{\tilde{r}})\subset B_{\tilde{r}}$. If $c_{0}<1$, then for any $u\in B_{\tilde{r}}$, using assumptions (H₃)–(H₅), (3.11), and Lemma 2.5, we have

$$ \begin{aligned} ||(\Gamma u)(t)||&\leq \frac{1}{1-\alpha}\int _{t}^{t+\omega}\bigg(a_{0}c_{0}||( \mathcal{A}^{-1}u)(s) || +||f(s,\mathcal{A}^{-1}u(s-\tau (s)))|| \\ &+\int _{0}^{s} ||K(\xi )g(s,\mathcal{A}^{-1}u(\xi ))||\Delta \xi \bigg)\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{t_{j} \in [t,t+\omega ]_{\mathbb{T}}}|| (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j})|| \\ &\leq \frac{\omega }{1-\alpha}\bigg(\frac{a_{0}c_{0}}{1-c_{0}} \tilde{r} +\tilde{l}_{f}+M\tilde{l}_{g}\bigg) +\frac{\beta}{1-\alpha} \sum _{j=1}^{q}\tilde{l}_{j} \\ &\leq \tilde{r}. \end{aligned} $$

(3.13)

If $c_{1}>1$, then for any $u\in B_{\tilde{r}}$, using assumptions (H₃)–(H₅), (3.12), and Lemma 2.5, we have

$$ \begin{aligned} ||(\Gamma u)(t)||&\leq \frac{1}{1-\alpha}\int _{t}^{t+\omega}\bigg(a_{0}c_{0}||( \mathcal{A}^{-1}u)(s) || +||f(s,\mathcal{A}^{-1}u(s-\tau (s)))|| \\ &+\int _{0}^{s} ||K(\xi )g(s,\mathcal{A}^{-1}u(\xi ))||\Delta \xi \bigg)\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{t_{j} \in [t,t+\omega ]_{\mathbb{T}}}|| (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j})|| \\ &\leq \frac{\omega }{1-\alpha}\bigg(\frac{a_{0}c_{0}}{c_{1}-1} \tilde{r} +\tilde{l}_{f}+M\tilde{l}_{g}\bigg) +\frac{\beta}{1-\alpha} \sum _{j=1}^{q}\tilde{l}_{j} \\ &\leq \tilde{r}. \end{aligned} $$

(3.14)

From (3.13) and (3.14) it follows that the operator Γ maps $B_{\tilde{r}}$ into $B_{\tilde{r}}$.

Step 2: We prove that the operator Γ is continuous. Let $\{u_{n}\}_{n=1}^{\infty}$ be an arbitrary sequence in $B_{\tilde{r}}$ with $\lim _{n\rightarrow \infty}||u_{n}-u||=0$, where $~u\in B_{\tilde{r}}$. We claim that $\lim _{n\rightarrow \infty}||\Gamma u_{n}-\Gamma u||=0$. Indeed, for $t\in \mathbb{T}^{+}$, we have

$$ \begin{aligned} ||(\Gamma u_{n})(t)-(\Gamma u)(t)|| &\leq \frac{1}{1-\alpha}\int _{t}^{t+ \omega}\bigg(a_{0}c_{0}||(\mathcal{A}^{-1}u_{n})(s) -(\mathcal{A}^{-1}u)(s)|| \\ &+||f(s,\mathcal{A}^{-1}u_{n}(s-\tau (s)))-f(s,\mathcal{A}^{-1}u(s- \tau (s)))|| \\ &+\int _{0}^{s} ||K(\xi )|||| g(s,\mathcal{A}^{-1}u_{n}(\xi ))-g(s, \mathcal{A}^{-1}u(\xi ))||\Delta \xi \bigg)\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{t_{j} \in [t,t+\omega ]_{\mathbb{T}}}|| (\mathcal{A}I_{j}\mathcal{A}^{-1})u_{n}(t_{j})-(\mathcal{A}I_{j} \mathcal{A}^{-1})u(t_{j})||. \end{aligned} $$

(3.15)

From the continuity of f, g, K, $I_{j}$, and $~\mathcal{A}$ in their definition domains it follows by (3.15) that

$$ \lim _{n\rightarrow \infty}||\Gamma u_{n}-\Gamma u||=0 ~\mathrm{as}~ \lim _{n\rightarrow \infty}|| u_{n}- u||=0. $$

Hence the operator Γ is continuous.

Step 3: We prove that $\Gamma (B_{\tilde{r}})$ is relatively compact. We first show that Γ is an equicontinuous operator. For $t_{1},t_{2}\in [0,\omega ]_{\mathbb{T}}$ with $t_{1}< t_{2}$ and $u\in B_{\tilde{r}}$, if $c_{0}<1$, then we have

$$ \begin{aligned} &||(\Gamma u)(t_{2})-(\Gamma u)(t_{1})|| \\ &\leq \int _{t_{1}}^{t_{2}}||\Theta (t_{1},s)||\bigg|\bigg(-a(s)c(s)( \mathcal{A}^{-1}u)(s) \\ &+f(s,\mathcal{A}^{-1}u(s-\tau (s)))+\int _{0}^{s} K(\xi )g(s, \mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\bigg|\Delta s \\ &+\int _{t_{2}}^{t_{1}+\omega}||\Theta (t_{1},s)-\Theta (t_{2},s)|| \bigg|\bigg(-a(s)c(s)(\mathcal{A}^{-1}u)(s) \\ &+f(s,\mathcal{A}^{-1}u(s-\tau (s)))+\int _{0}^{s} K(\xi )g(s, \mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\bigg|\Delta s \\ &+\int _{t_{1}+\omega}^{t_{2}+\omega}||\Theta (t_{2},s)||\bigg|\bigg(-a(s)c(s)( \mathcal{A}^{-1}u)(s) \\ &+f(s,\mathcal{A}^{-1}u(s-\tau (s)))+\int _{0}^{s} K(\xi )g(s, \mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\bigg|\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{j=1}^{q}\tilde{l}_{j}||\Theta (t_{2},t_{j})- \Theta (t_{1},t_{j})|| \\ &\leq \frac{2}{1-\alpha}\bigg(\frac{a_{0}c_{0}\tilde{r}}{1-c_{0}} + \tilde{l}_{f}+\tilde{l}_{g}M\bigg)(t_{2}-t_{1}) \\ &+\bigg(\frac{a_{0}c_{0}\tilde{r}}{1-c_{0}} +\tilde{l}_{f}+\tilde{l}_{g}M \bigg)(t_{1}+\omega -t_{2})||\Theta (t_{1},s)-\Theta (t_{2},s)||. \end{aligned} $$

(3.16)

Also, for $t_{1},t_{2}\in [0,\omega ]_{\mathbb{T}}$ with $t_{1}< t_{2}$ and $u\in B_{\tilde{r}}$, if $c_{1}>1$, then we have

$$ \begin{aligned} &||(\Gamma u)(t_{2})-(\Gamma u)(t_{1})|| \\ &\leq \int _{t_{1}}^{t_{2}}||\Theta (t_{1},s)||\bigg|\bigg(-a(s)c(s)( \mathcal{A}^{-1}u)(s) \\ &+f(s,\mathcal{A}^{-1}u(s-\tau (s)))+\int _{0}^{s} K(\xi )g(s, \mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\bigg|\Delta s \\ &+\int _{t_{2}}^{t_{1}+\omega}||\Theta (t_{1},s)-\Theta (t_{2},s)|| \bigg|\bigg(-a(s)c(s)(\mathcal{A}^{-1}u)(s) \\ &+f(s,\mathcal{A}^{-1}u(s-\tau (s)))+\int _{0}^{s} K(\xi )g(s, \mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\bigg|\Delta s \\ &+\int _{t_{1}+\omega}^{t_{2}+\omega}||\Theta (t_{2},s)||\bigg|\bigg(-a(s)c(s)( \mathcal{A}^{-1}u)(s) \\ &+f(s,\mathcal{A}^{-1}u(s-\tau (s)))+\int _{0}^{s} K(\xi )g(s, \mathcal{A}^{-1}u(\xi ))\Delta \xi \bigg)\bigg|\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{j=1}^{q}\tilde{l}_{j}||\Theta (t_{2},t_{j})- \Theta (t_{1},t_{j})|| \\ &\leq \frac{2}{1-\alpha}\bigg(\frac{a_{0}c_{0}\tilde{r}}{c_{1}-1} + \tilde{l}_{f}+\tilde{l}_{g}M\bigg)(t_{2}-t_{1}) \\ &+\bigg(\frac{a_{0}c_{0}\tilde{r}}{c_{1}-1} +\tilde{l}_{f}+\tilde{l}_{g}M \bigg)(t_{1}+\omega -t_{2})||\Theta (t_{1},s)-\Theta (t_{2},s)||. \end{aligned} $$

(3.17)

Since $\Theta (t,s)$ is continuous, $||\Theta (t_{1},s)-\Theta (t_{2},s)||\rightarrow 0$ as $t_{2}\rightarrow t_{1}$. Hence, in view of (3.16) and (3.17), we obtain that

$$ ||(\Gamma u)(t_{2})-(\Gamma u)(t_{1})||\rightarrow 0~\mathrm{as}~t_{2} \rightarrow t_{1} $$

and thus $\Gamma (B_{\tilde{r}})$ is equicontinuous. From step 1, $\Gamma (B_{\tilde{r}})$ is uniformly bounded. Thus $\Gamma (B_{\tilde{r}})$ is relatively compact.

From the above proof, by the Arzelà–Ascoli theorem we can conclude that $\Gamma : B_{\tilde{r}}\rightarrow B_{\tilde{r}}$ is completely continuous. Hence all the conditions of Lemma 2.4 hold, and system (2.2) has at least one periodic solution u, i.e., system (1.3) has at least one periodic solution $\mathcal{A}^{-1}u$. □

Remark 3.1

In [27], when $|c|<1$, the authors obtained the existence of three positive periodic solutions to the nonlinear neutral-type equations with impulses and parameters on time scales by using the Leggett–Williams fixed point theorem. In this paper, we have expanded the scope of c for ensuring the existence of periodic solutions.

4 Stability of periodic solutions

Definition 4.1

A periodic solution u of system (2.2) is called stable if for any $\delta >0$, there exists $\varepsilon =\varepsilon (\delta )>0$ such that

$$ ||u(t)-\tilde{u}(t)||\leq \delta ~\mathrm{for~all}~t\in \mathbb{T}^{+} $$

whenever $||u(0)-\tilde{u}(0)||\leq \varepsilon$, where ũ is the solution of system (2.2) with $\tilde{u}(t_{j}^{-})-\tilde{u}(t_{j}^{+})=(\mathcal{A}I_{j} \mathcal{A}^{-1})\tilde{u}(t_{j})$, $j=1,2,\ldots ,q$.

Theorem 4.1

If all the conditions of Theorem 3.1are satisfied, then system (1.3) is stable.

Proof

Since all the conditions of Theorem 3.1 are satisfied, system (2.2) has a unique periodic solution u. Let ũ be another solution of system (2.2). If $c_{0}<1$, then for $u,\tilde{u}\in B_{r}$, using assumptions (H₁)–(H₃) and (3.1), we have

$$ \begin{aligned} || u(t)-\tilde{u}(t)|| &\leq \frac{1}{1-\alpha}\int _{t}^{t+\omega} \bigg(a_{0}c_{0}||(\mathcal{A}^{-1}u)(s) -(\mathcal{A}^{-1}\tilde{u})(s)|| \\ &+||f(s,\mathcal{A}^{-1}u(s-\tau (s)))-f(s,\mathcal{A}^{-1}\tilde{u}(s- \tau (s)))|| \\ &+\int _{0}^{s} ||K(\xi )|||| g(s,\mathcal{A}^{-1}u(\xi ))-g(s, \mathcal{A}^{-1}\tilde{u}(\xi ))||\Delta \xi \bigg)\Delta s \\ &+\frac{\beta}{1-\alpha}\sum _{t_{j} \in [t,t+\omega ]_{\mathbb{T}}}|| (\mathcal{A}I_{j}\mathcal{A}^{-1})u(t_{j})-(\mathcal{A}I_{j} \mathcal{A}^{-1})\tilde{u}(t_{j})|| \\ &\leq \frac{\omega }{1-\alpha}\bigg(\frac{a_{0}c_{0}}{1-c_{0}}||u- \tilde{u}|| +\frac{4l_{f}r}{1-c_{0}}+\frac{4Ml_{g}r}{1-c_{0}}+ \frac{l_{f}+Ml_{g}}{1-c_{0}}||u(0)-\tilde{u}(0)||\bigg) \\ &+\frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}||u-\tilde{u}|| \end{aligned} $$

and

$$ \begin{aligned} || u(t)-\tilde{u}(t)||&\leq \frac{(4l_{f}r+4Ml_{g}r)\omega}{(1-c_{0})(1-\alpha )} \bigg[1- \frac{\omega a_{0}c_{0}}{(1-c_{0})(1-\alpha )}-\frac{\beta}{1-\alpha} \sum _{j=1}^{q}l_{j}\bigg]^{-1} \\ &+\frac{(l_{f}+Ml_{g})\omega}{(1-c_{0})(1-\alpha )}\bigg[1- \frac{\omega a_{0}c_{0}}{(1-c_{0})(1-\alpha )}- \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\bigg]^{-1}||u(0)-\tilde{u}(0)||. \end{aligned} $$

(4.1)

Also, similarly to the proof of (4.1), if $c_{1}>1$, then we have

$$ \begin{aligned} || u(t)-\tilde{u}(t)||&\leq \frac{(4l_{f}r+4Ml_{g}r)\omega}{(c_{1}-1)(1-\alpha )} \bigg[1- \frac{\omega a_{0}c_{0}}{(c_{1}-1)(1-\alpha )}-\frac{\beta}{1-\alpha} \sum _{j=1}^{q}l_{j}\bigg]^{-1} \\ &+\frac{(l_{f}+Ml_{g})\omega}{(c_{1}-1)(1-\alpha )}\bigg[1- \frac{\omega a_{0}c_{0}}{(c_{1}-1)(1-\alpha )}- \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\bigg]^{-1}||u(0)-\tilde{u}(0)||. \end{aligned} $$

(4.2)

From (4.1) and (4.2) we have

$$ || u(t)-\tilde{u}(t)||\leq \Delta _{1}+\Delta _{2}\varepsilon , $$

(4.3)

where

$$\begin{aligned}& \begin{aligned} \Delta _{1}=\max &\bigg\{ \frac{(4l_{f}r+4Ml_{g}r)\omega}{(1-c_{0})(1-\alpha )} \bigg[1- \frac{\omega a_{0}c_{0}}{(1-c_{0})(1-\alpha )}-\frac{\beta}{1-\alpha} \sum _{j=1}^{q}l_{j}\bigg]^{-1}, \\ &\frac{(4l_{f}r+4Ml_{g}r)\omega}{(c_{1}-1)(1-\alpha )} \bigg[1- \frac{\omega a_{0}c_{0}}{(c_{1}-1)(1-\alpha )}-\frac{\beta}{1-\alpha} \sum _{j=1}^{q}l_{j}\bigg]^{-1}\bigg\} , \end{aligned} \\& \begin{aligned} \Delta _{2}=\max &\bigg\{ \frac{(l_{f}+Ml_{g})\omega}{(1-c_{0})(1-\alpha )}\bigg[1- \frac{\omega a_{0}c_{0}}{(1-c_{0})(1-\alpha )}- \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\bigg]^{-1}, \\ &\frac{(l_{f}+Ml_{g})\omega}{(c_{1}-1)(1-\alpha )}\bigg[1- \frac{\omega a_{0}c_{0}}{(c_{1}-1)(1-\alpha )}- \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\bigg]^{-1}\bigg\} . \end{aligned} \end{aligned}$$

Now by choosing $\varepsilon \leq \frac{\delta -\Delta _{1}}{\Delta _{2}}$, in view of (4.3), we get

$$ ||u(t)-\tilde{u}(t)||\leq \delta ~\mathrm{for~all}~t\in \mathbb{T}^{+}, $$

and thus system (2.2) is stable. Furthermore, by Lemma 2.5 we get

$$ ||x(t)-\tilde{x}(t)||=||(\mathcal{A}^{-1}u)(t) -(\mathcal{A}^{-1} \tilde{u})(t)|| \leq \frac{\delta}{1-c_{0}}~\mathrm{for~all}~t\in \mathbb{T}^{+},~c_{0}< 1, $$

or

$$ ||x(t)-\tilde{x}(t)||=||(\mathcal{A}^{-1}u)(t) -(\mathcal{A}^{-1} \tilde{u})(t)|| \leq \frac{\delta}{c_{1}-1}~\mathrm{for~all}~t\in \mathbb{T}^{+},~c_{1}>1. $$

Hence system (1.3) is stable. □

5 An example

In this section, we give an example for different time scales to verify our theoretical outcomes. Consider the following neutral-type differential equation with piecewise impulses on time scales:

$$ \left \{ \textstyle\begin{array}{l} (x(t)+c(t)x(t-\gamma ))^{\Delta}=-a(t)x(t) +f(t,x(t-\tau (t))) \\ \hphantom{(x(t)+c(t)x(t-\gamma ))^{\Delta}=}{} +\int _{0}^{t} K(s)g(t,x(s))\Delta s,~t\geq 0,~t\neq t_{j}, \\ x(t_{j}^{-})-x(t_{j}^{+})=I_{j}(x(t_{j})),~~t= t_{j},~j=1,2,\ldots ,q, \end{array}\displaystyle \right . $$

(5.1)

where

$$\begin{aligned}& c(t)=0.1\sin t,~\gamma =\pi ,~a(t)=\cos t-1.5,~\tau (t)=e_{0.2}( \sigma (t),t),\\& f(t,x)=\frac{1}{30}+\frac{1}{30}\cos x,~g(t,x)=\frac{1}{50}+ \frac{1}{50}\sin x,~K(t)=e^{-t}, \\& I_{j}(x(t_{j}))=8\times 10^{-4} \bigg(\frac{2}{3}\bigg)^{j}x(t_{j}). \end{aligned}$$

When $\mathbb{T}=\mathbb{R}$, it is easy to see that assumptions (H₁)–(H₃) are satisfied. After simple calculations, we can conclude that

$$ \omega =2\pi ,~ c_{0}=0.1,~l_{f}=\frac{1}{30},~l_{g}=\frac{1}{50},~M=1,~ \alpha =e^{-5\pi}, ~\beta =e^{3\pi}, $$

and

$$ l_{j}=0.08\bigg(\frac{2}{3}\bigg)^{j}, ~\sum _{j=1}^{q}l_{j}\leq \sum _{j=1}^{\infty }l_{j}=1.6\times 10^{-3}. $$

Thus

$$ \frac{\omega (\alpha c_{0}+l_{f}+Ml_{g})}{(1-\alpha )(1-c_{0})} + \frac{\beta}{1-\alpha}\sum _{j=1}^{q}l_{j}\approx 0.94< 1, $$

and all conditions of Theorems 3.1 and 4.1 are satisfied. Hence system (5.1) has a unique stable periodic solution. When $\mathbb{T}=\mathbb{Z}$, we have $\alpha =\frac{1}{3.5^{2\pi}}$, and other parameters remain unchanged. Similarly to the above proof, we also obtain that system (5.1) has a unique stable periodic solution.

6 Conclusions

In this paper, we considered the existence and stability of a periodic solution for a neutral-type differential equation with piecewise impulses on time scales. By using time scale theory, the properties of neutral-type operator, and the Banach contraction mapping principle, we prove the existence of a unique periodic solution. We also prove the existence of at least one periodic solution by using the Schauder fixed point theorem. Further, we establish the stability results.

Another topic for a future work related to this paper would be the controllability of piecewise impulsive dynamic systems on time scales. This would require extending the class of systems to systems involving control.