1 Introduction

In the 1960s, Mil’man and Myshkis [10] first discussed the stability of motions in the presence of impulses. Since then, the theory of impulsive differential equations has been an attractive topic. The highlight of impulsive differential equations is that it can show us the variety of the system when it takes a sudden change. Hence, this type of equations plays an important part in the studies of Demography, Physics, Biology, Control Theory and so on.

An impulsive differential system can usually be written as

$$\begin{aligned}&x'=f(t,x), \nonumber \\&\varDelta x|_{t={\tau _i}(x)} =I_i (x),~~i\in {\mathbb {Z}}^1, \end{aligned}$$
(1)

where

$$\begin{aligned} \varDelta x|_{t={\tau _i}(x)} = x(\tau _i (x))-x(\tau _i (x)-0), \end{aligned}$$

\(\tau _i: {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{1}\) are the time of impulsive effects and \(I_i: {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{n}\) the impulsive effects. When considering the actual meaning of the systems, we usually require additional conditions about the time of impulsive effects. For important periodicity problems, \(\{\tau _{i}\}\) is assumed to be T-periodic (see (\(H_{2}\)) below).

Generally speaking, impulsive effects include two kinds of forms: the case of fixed time, i.e., all the \(\tau _{i}(x)=\tau _{i}\) independent of space variables x, for some recent work, see [5, 11, 19]; and the case of varying times, i.e., some \(\tau _{i}(x)\) dependent on the space variables. For the former, the operator \((x', \varDelta x|_{t=\tau _{i}(x)})\) is linear and hence it is easy to use the usual nonlinear analysis formulism. However, this becomes very difficult due to the nonlinearity of the operator \((x', \varDelta x|_{t=\tau _{i}(x)})\) in the latter. Actually, impulsive systems with various time take place in many problems from physical sciences. The general theory can be found in the book [3]. For some earlier research on the periodic and general boundary value problems in varying times, see [7], and for the situation of fixed impulsive effects, see [12].

Recently, a new type of periodicity, affine periodicity, is introduced in [20], which describes those natural phenomena that exhibit certain symmetry in space rather than periodicity in time. The existence of affine-periodic solutions for such equations, for instance, dissipative systems and nonlinear systems on time scales etc, was discussed (see [6, 14,15,16]). Further, to know about the recent work, we refer to [8, 9, 13, 17].

The concept of affine-periodic systems is defined as follows.

Definition 1

The system

$$\begin{aligned} x'=f(t,x),~~~~'=\frac{d}{dt} \end{aligned}$$
(2)

is said to be a (Q, T)-affine-periodic system (APS, for short), if there exist \(Q\in GL(n)\) and \(T>0\) such that

$$\begin{aligned} f(t+T,x)=Qf(t,Q^{-1}x) \end{aligned}$$

holds for all \((t,x)\in {\mathbb {R}}^{1}\times {\mathbb {R}}^{n}\).

We make throughout the paper the following assumptions:

  1. (H1):

    \(f: {\mathbb {R}}^{1} \times {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{n}\), \(I_i: {\mathbb {R}}^{n}\rightarrow {\mathbb {R}}^{n}\) are continuous, (Q, T)-affine periodic and satisfy the local Lipschitz conditions on x.

  2. (H2):

    \(\tau _{i}(x)\) is periodic, that is, there exists a positive integer \(k_{0}\) and \(0 \le \tau _{0}(x)< \cdot \cdot \cdot < \tau _{k_{0}-1}(x) \le T\), such that

    $$\begin{aligned}&\tau _{k_0} (x)=\tau _0 (x) +T, \\&\tau _{k_0 +1} (x)=\tau _1 (x) +T,\\&\cdots ,\\&\tau _{k_0 + k_0 - 1} (x)=\tau _{k_0 - 1} (x) +T,\\&\tau _{k_0 + k_0} (x)=\tau _0 (x) +2T,\\&\cdots ,\\&\tau _{-1} (x)=\tau _{k_0 -1} (x),\\&\tau _{-2} (x)=\tau _{k_0 -2} (x),\\&\cdots ,\\&\tau _{-k_0} (x)=\tau _0 (x)-T,\\&\tau _{-k_0 -1} (x)=\tau _{-1} (x) -T,\\&\cdots ,\\&\tau _{-k_0 -k_0 -1} (x)=\tau _{-k_0 -1} (x) -T,\\&\tau _{-2k_0} (x)=\tau _0 (x) -2T,\\&\cdots . \end{aligned}$$
  3. (H3):

    \(\tau _{i}: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{1}\) is \(C^{1}\) and

    $$\begin{aligned} {\frac{\partial \tau _i}{\partial x}}^\top f(t,x) \ne 1~~~~\forall ~i, t, x, \end{aligned}$$

    which confirms the existence of solutions of impulsive system (1) with initial values.

  4. (H4):

    \(\left\{ I_{i}(x)\right\} \) is (Q, T)-affine-periodic. It means that

    $$\begin{aligned} I_{i+ lk_0 } (x)=Q^l I_i (Q^{-l} x)~~~~\forall ~i. \end{aligned}$$

Also, for APS (2), we have the definition of affine-periodic solutions.

Definition 2

If x(t) is a solution of APS (2) on \({\mathbb {R}}^{1}\) and

$$\begin{aligned} x(t+T)=Qx(t)~~~~\forall ~t\in {\mathbb {R}}^{1}, \end{aligned}$$
(3)

then x(t) is said to be a (Q, T)-affine-periodic one.

Remark 1

By the definition above, we can easily get some examples of affine-periodic solutions:

  1. (1)

    When \(Q=id\), a (Q, T)-affine-periodic solution is the usual periodic one.

  2. (2)

    When \(Q=-id\), a (Q, T)-affine-periodic solution is an anti-periodic one.

  3. (3)

    If there exists a positive integer \(N>1\) such that

    $$\begin{aligned}&Q^{i} \ne id,~~1 \le i \le N-1,\\&Q^{N} = id, \end{aligned}$$

    then a (Q, T)-affine-periodic solution is a subharmonic one.

  4. (4)

    If for any \(i \in {\mathbb {Z}}^{1}{\backslash }\left\{ 0\right\} \), we always have

    $$\begin{aligned} Q^{i} \ne id ~~~~\hbox {and}~~~~Q \in O(n), \end{aligned}$$

    then a (Q, T)-affine-periodic solution is a quasi-periodic one.

Many dynamical systems are dissipative with respect to the energy, that is, their solutions, wherever they start, enter into bounded ranges ultimately in future. Started by Levinson [4], through a series of efforts, the following fundamental result is obtained: A dissipative system admits a periodic solution. See, for instance, literature [1, 18]. A natural question is:

Whether does an impulsive system with varying times admit a periodic or an affine-periodic solution if it is dissipative?

Generally speaking, any periodicity result in the usual differential equations is not always true for impulsive differential equations, especially in the systems with varying times for impulsive effects. Also, since affine-periodicity is a generalization of pure periodicity, a natural idea is how to use this new type of periodicity in the study of impulsive equations.

In this paper, we will discuss the existence of affine-periodic solutions for impulsive systems. In Sect. 2, we consider the dissipative impulsive differential equations. We first prove that a dissipative impulsive differential equation admits periodic solutions, then, we extend it to the existence of affine-periodic solutions. In Sect. 3, we consider the dissipative impulsive functional differential equations, and prove that this type of systems also admits affine-periodic solutions. Finally, in Sect. 4, we discuss the existence of affine-periodic solutions for dissipative impulsive differential systems by applying Lyapunov’s methods.

2 Impulsive Differential Equations

In this section, we first discuss the standard periodic solution situation, which has, as previously mentioned, itself right in the qualitative theory.

For system (1), we first give some basic definitions.

Definition 3

For \(I = (a, b)\subset {\mathbb {R}}^{1}\), and \(t_{0} \in (a, b)\), we call \(x: I \rightarrow {\mathbb {R}}^{n}\) a solution of system (1) with the initial value condition

$$\begin{aligned} x(t_{0})=x_{0}, \end{aligned}$$
(4)

if the following hold:

  1. (I):

    x satisfies Eq. (1) on \(I~\backslash \mathop {\bigcup }\nolimits _{k\in {\mathbb {Z}}^{1}} \{\tau _k (x)\}\);

  2. (II):

    x is continuous on the left, has limits right on \(\mathop {\bigcup }\nolimits _{k\in {\mathbb {Z}}^{1}} \{\tau _k (x)\}\), and satisfies that

    $$\begin{aligned} \varDelta x|_{t=\tau _i (x)} = I_i (x) \end{aligned}$$

    for \(\tau _i (x) \in I\);

  3. (III):

    \(x(t_{0}) = x_{0}\).

Definition 4

We call function \(x: {\mathbb {R}}^{1} \rightarrow {\mathbb {R}}^{n}\) a T-periodic solution of system (1), if x is a solution of system (1) and satisfies

$$\begin{aligned} x(t+T) = x(t)~~~~\forall t \in {\mathbb {R}}^{1}. \end{aligned}$$

Definition 5

For system (1), if:

  1. (I):

    for any \(x_{0} \in {\mathbb {R}}^{n}\), a solution x(t) of system (1) with the initial value condition \(x(0) = x_{0}\) is well defined on \({\mathbb {R}}^{1}\); and

  2. (II):

    there exists a positive number \(B>0\), such that for any \(R>0\), there is \(l(R)>0\), such that

    $$\begin{aligned} |x(t)|\le B~~~~\forall t \ge l(R) \end{aligned}$$

    whenever \(|x(0)| \le R\), then system (1) is said to be dissipative,

For periodic system (1) with \(Q=\mathrm{id}\), we define a Poincar\(\acute{\mathrm{e}}\) map \(P: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n}\) by

$$\begin{aligned} P(x_{0}) = x(T, x_{0}), \end{aligned}$$

where \(x(t, x_{0})\) is a solution of system (1) with the initial condition \(x(0) = x_{0}\). The following lemma shows the relationship between the fixed points of this Poincar\(\acute{\mathrm{e}}\) map and the T-periodic solutions of system (1).

Lemma 1

System (1) admits a T-periodic solution x(t) if and only if \(x(0) \in {\mathbb {R}}^{n}\) is a fixed point of Poincar\(\acute{\mathrm{e}}\) map P. In other words,

$$\begin{aligned} P(x(0)) = x(0). \end{aligned}$$

Proof

The necessity is obvious. It suffices to prove the sufficiency.

For a solution x(t) of system (1), let \(x_0\) be a fixed point of Poincar\(\acute{\mathrm{e}}\) map P. Hence, we have

$$\begin{aligned} x(T, x_0)=P(x_0)=x_0. \end{aligned}$$

Let \(u(t) = x(t+T, x_0)\). It is easy to prove that u(t) is also a solution of system (1).

Notice that

$$\begin{aligned} u(0)=x(T, x_0)=x_0. \end{aligned}$$

By the uniqueness of solutions with the initial value condition, we have

$$\begin{aligned} x(t+T, x_0)=u(t) \equiv x(t, x_0)~~~~\forall t \in {\mathbb {R}}^{1}. \end{aligned}$$

Hence x(t) is a T-periodic solution of system (1). \(\square \)

By using Lemma 1 and Horn’s fixed point theorem [2], we can prove the following Yoshizawa type theorem for dissipative periodic impulsive systems.

Theorem 1

Let (1) be a periodic impulsive system, and assume f and \(\left\{ I_{i}\right\} _{i \in {\mathbb {Z}}^{1}}\) satisfy (H1)-(H3). Then, if system (1) is dissipative, it admits a T-periodic solution.

Proof

Define a Poincar\(\acute{\mathrm{e}}\) map P by

$$\begin{aligned} P(x_0)=x(T, x_0), \end{aligned}$$

where \(x(t, x_{0})\) is a solution of (1) with the initial value condition \(x(0) = x_{0}\).

It is easy to see that \(x(t, x_{0})\) exists locally. Then, by condition (I) in Definition 5 and T-periodicity of system (1), we know that

$$\begin{aligned} P^i (x_0)=x(iT, x_0)~~~~\forall i\in {\mathbb {N}}. \end{aligned}$$
(5)

Also, by condition (II) in Definition 5, if \(|x_{0}| \le B+1\), there exists a positive number \(l(B+1)\) such that

$$\begin{aligned} |x(t, x_0)|\le B~~~~\forall t \ge l(B+1). \end{aligned}$$
(6)

Let N be a positive integer satisfying \(NT \ge l(B+1)\). Define three convex sets:

$$\begin{aligned}&S_0=\{x_0 \in {\mathbb {R}}^n: |x_0|\le B\},\\&S_1=\{x_0 \in {\mathbb {R}}^n: |x_0|< B+1\},\\&S_2=\{x_0 \in {\mathbb {R}}^n: |x_0|\le M\}, \end{aligned}$$

where M satisfies

$$\begin{aligned} |x(t, x_0)|\le M~~~~\forall t \in [0, NT] \end{aligned}$$

when \(|x_{0}| \le B+1\).

By (5), (6) and the choice of N, we know that

$$\begin{aligned}&P^i (S_1) \subset S_2,~~0\le i \le N-1,\\&P^i (S_1) \subset S_0,~~N\le i \le 2N-1. \end{aligned}$$

Hence, by Horn’s fixed point theorem, the map P admits a fixed point \(x_{*}\) in \(S_{0}\). In other words,

$$\begin{aligned} P(x_{*})=x(T, x_{*})=x_{*}. \end{aligned}$$

By the uniqueness of solutions with the initial value condition, we have

$$\begin{aligned} x(t+T, x_{*})\equiv x(t, x_{*})~~~~\forall t. \end{aligned}$$

It means that \(x(t, x_{*})\) is a T-periodic solution of system (1). \(\square \)

Now we consider an affine-periodic impulsive system.

Definition 6

Affine-periodic system (1) is called to be Q-dissipative, if:

  1. (I):

    Any solution x(t) of system (1) is well defined on \({\mathbb {R}}^{1}_{+}\);

  2. (II):

    There exists a positive number \(B>0\), such that for any \(R>0\), we have \(l(R)>0\), such that

    $$\begin{aligned} |Q^{-k} x(t+kT, x_0)|\le B~~~~\forall ~t+kT \ge l(R),~~k\in {\mathbb {Z}}^{1}_{+} \end{aligned}$$

    whenever \(|x(0)| \le R\).

For an affine-periodic impulsive system (1), we can easily prove the following lemma.

Lemma 2

Affine-periodic system (1) admits a (Q, T)-affine-periodic solution x(t), if and only if its initial value x(0) satisfies

$$\begin{aligned} P(x(0))=Qx(0), \end{aligned}$$

where

$$\begin{aligned} P(x(0))=x(T, x(0)). \end{aligned}$$

Proof

The necessity is obvious, and we only prove the sufficiency.

Let \(u(t) = Q^{-1}x(t+T)\). Then

$$\begin{aligned} \frac{\mathrm{d} u(t)}{\mathrm{d} t}&=\frac{\mathrm{d} Q^{-1} x(t+T)}{\mathrm{d} (t+T)}\\&=\frac{Q^{-1}\mathrm{d}x(t+T)}{\mathrm{d}(t+T)}\\&=Q^{-1} f(t+T, x(t+T))\\&=Q^{-1} Qf(t, Q^{-1}x(t+T))\\&=f(t, u(t)), \end{aligned}$$

which together with (H4) implies that u(t) is also a solution of (1). Obviously,

$$\begin{aligned} u(0)&= Q^{-1} x(T)\\&= Q^{-1} x(T, x(0))\\&= x(0). \end{aligned}$$

By the uniqueness of solutions with the initial value condition, we know that

$$\begin{aligned} Q^{-1} x(t+T)&\equiv u(t)\\&= x(t)~~~~\forall t. \end{aligned}$$

In other words,

$$\begin{aligned} x(t+T)= Q x(t)~~~~\forall t. \end{aligned}$$

Hence, x(t) is a (Q, T)-affine-periodic solution of system (1). \(\square \)

Now we give a Yoshizawa type theorem for (Q, T)-affine-periodic impulsive systems.

Theorem 2

Let (1) be a (Q, T)-affine-periodic impulsive system, and assume (H1)–(H4) hold. Then, if system (1) is Q-dissipative, it admits a (Q, T)-affine-periodic solution.

Proof

Define a map \({\widetilde{P}}: {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n}\) by

$$\begin{aligned} {\widetilde{P}} (x_0)= Q^{-1} P(x_0) = Q^{-1} x(T, x_0)~~~~\forall x_0 \in {\mathbb {R}}^n. \end{aligned}$$

Notice that \(x(t, Q^{-1} x(T, x_0))\) is a solution of (1) with the initial value condition \(x(0)=Q^{-1} x(T, x_0)\). Moreover, \(Q^{-1} x(t+T, x_0)\) is also a solution of (1) with the same initial value condition. Hence, by the uniqueness of solution with initial value condition, we have

$$\begin{aligned} Q^{-1} x(t+T, x_0)\equiv x(t, Q^{-1} x(T, x_0))~~~~\forall t. \end{aligned}$$

Hence

$$\begin{aligned} Q^{-2} x(t+T, x_0)=Q^{-1} x(t, Q^{-1} x(T, x_0))~~~~\forall t. \end{aligned}$$

Let \(t = T\), and we have

$$\begin{aligned} {\widetilde{P}}^2 (x_0)&={\widetilde{P}} ({\widetilde{P}} (x_0))\\&={\widetilde{P}}(Q^{-1} x(T, x_0))\\&=Q^{-1} x(T, Q^{-1} x(T, x_0))\\&=Q^{-2} x(2T, x_0). \end{aligned}$$

Similarly, we can prove that

$$\begin{aligned} {\widetilde{P}}^i (x_0)=Q^{-i} x(iT, x_0)~~~~\forall i=0, 1, \ldots . \end{aligned}$$
(7)

Since system (1) is Q-dissipative, there exists a positive integer N such that \(NT \ge l(B+1)\), such that

$$\begin{aligned} |Q^{-k} x(kT, x_0)| \le B \end{aligned}$$
(8)

for \(|x_0|\le B+1\), \(k\ge N\).

By (7) and (8) we have

$$\begin{aligned} |{\widetilde{P}}^k (x_0)| \le B,~~~~~~\forall ~k\ge N,~|x_0|\le B+1. \end{aligned}$$
(9)

And it is easy to prove that there exists a positive number \(M > B+1\) such that

$$\begin{aligned} |{\widetilde{P}}^k (x_0)| \le M,~~~~~~\forall ~0\le k < N,~~|x_0|\le B+1. \end{aligned}$$
(10)

Define three convex sets:

$$\begin{aligned}&S_0 =\{x_0 \in {\mathbb {R}}^n: |x_0|\le B\},\\&S_1 =\{x_0 \in {\mathbb {R}}^n: |x_0|< B+1\},\\&S_2 =\{x_0 \in {\mathbb {R}}^n: |x_0|\le M\}. \end{aligned}$$

By (9) and (10) we have

$$\begin{aligned}&{\widetilde{P}}^i (S_1)\subset S_2~~~~\forall ~0\le i \le N-1,\\&{\widetilde{P}}^i (S_1)\subset S_0~~~~\forall ~N\le i \le 2N-1. \end{aligned}$$

Hence, by Horn’s fixed point theorem, we know that map \({\widetilde{P}}\) admits a fixed point \(x_{*}\) in \(S_{0}\). In other words,

$$\begin{aligned} {\widetilde{P}}(x_{*}) = Q^{-1} x(T, x_*) = x_{*}. \end{aligned}$$

By the uniqueness of solution with initial value condition, we have

$$\begin{aligned} Q^{-1} x(t+T, x_*) \equiv x(t, x_*), \end{aligned}$$

which means that

$$\begin{aligned} x(t+T, x_*) \equiv Q x(t, x_*)~~~~\forall ~t. \end{aligned}$$

Hence \(x(t, x_{*})\) is a (Q, T)-affine-periodic solution of system (1). \(\square \)

3 Functional Differential Equations

Let \(\left\{ \tau _{i}\right\} , i\in {\mathbb {Z}}^{1}\) be a partition of \({\mathbb {R}}^{1}\) such that

$$\begin{aligned}&\tau _{0}=0,~~~~\tau _{k_{0}}=T,\\&\tau _{i+k_{0}}=\tau _{i}+T. \end{aligned}$$

Obviously, for a constant \(-r\), there exists \(\tau _{i_{r}}\) such that \(-r\in (\tau _{i_{r}}, \tau _{i_{r}+1}]\). Hence

$$\begin{aligned} {[}-r, 0] = [-r, \tau _{i_{r}+1}]\cup (\tau _{i_{r}+1}, \tau _{i_{r}+2}]\cup \cdot \cdot \cdot \cup (\tau _{-1}, \tau _{0}]. \end{aligned}$$

Let

$$\begin{aligned} \varUpsilon _{k} = \left\{ \begin{array}{ll} {[}-r, \tau _{i_{r}+1}], &{} ~~~k=i_{r}+1,\\ \\ (\tau _{k-1}, \tau _{k}], &{}~~~i_{r}+1<k\le 0. \end{array} \right. \end{aligned}$$

Denote by \(C_r=C([-r, 0], {\mathbb {R}}^{n})\) the set of all the functions from \([-r, 0]\) to \({\mathbb {R}}^{n}\) which is continuous on each interval \({\varUpsilon }_{k}\), \(i_{r}+1 \le k \le 0\) .

Consider an impulsive functional differential system

$$\begin{aligned} \begin{array}{l} x'=F(t,x_{t}), \\ \varDelta x\vert _{t = \tau _{i}} = I_{i}(x),~~i\in {\mathbb {Z}}^{1}, \end{array} \end{aligned}$$
(11)

where \(x_{t} = x(s + t)\), \(s\in [-r, 0]\). We make the following hypothesis.

(H5):\(F:{\mathbb {R}}^{1}\times C_r \rightarrow {\mathbb {R}}^{n}\)is continuous and satisfies local Lipschitz condition on the second variable. Hence the solution of (11) is unique with initial value condition.

Similar to Sect. 2, we have the following definitions.

Definition 7

If

$$\begin{aligned} F(t + T, \varphi ) = QF(t, Q^{-1}\varphi ), \end{aligned}$$

for any \((t, \varphi )\in {\mathbb {R}}^{1}\times C_r\) and (H5) holds, then we call \(F(t, \varphi )\) is a (Q, T)-affine-periodic functional differential equation.

Definition 8

Affine-periodic system (11) is said to be (Q, T)-dissipative, if for some positive number \(B_{0}>0\), and for any \(B>0\), there are \(M = M(B) > 0\), and \(L = L(B) > 0\) such that

$$\begin{aligned}&\vert x(t, \varphi )\vert \le M~~~~\forall t\in [0, L],\\&\vert Q^{-m}x(t + mT, \varphi )\vert \le B_{0}~~~~\forall t+mT\in [L, \infty ) \end{aligned}$$

for all \(\vert \vert \varphi \vert \vert = \mathop {\max }\nolimits _{s\in [-r, 0]}\vert \varphi (s)\vert \le B\), where \(x(t, \varphi )\) is a solution of (11) with initial value condition \(x_{0}=\varphi \).

We have the following theorem.

Theorem 3

If system (11) is (Q, T)-dissipative, then it admits a (Q, T)-affine-periodic solution.

Proof

Let \(x(t, t_{0}, \varphi )\) be the solution of (11) with the initial value condition \(x_{t_{0}} = \varphi \). Denote \(x(t, \varphi )=x(t, 0 ,\varphi )\).

Define a Poincar\(\acute{\mathrm{e}}\) map \(P: C_r\rightarrow C_r\) by

$$\begin{aligned} P(\varphi ) = Q^{-1}x_{T}(\cdot , \varphi ). \end{aligned}$$

We have

$$\begin{aligned} P^{2}(\varphi )&= P(P(\varphi ))\\&= P(Q^{-1}x_{T}(\cdot , \varphi ))\\&= Q^{-1}x_{T}(\cdot , Q^{-1}x_{T}(\cdot , \varphi )). \end{aligned}$$

Since \(x(t, Q^{-1}x_{T}(\cdot , \varphi ))\) is a solution of (11) with the initial value condition \(x_{0}=Q^{-1}x_{T}(\cdot , \varphi )\) and \(Q^{-1}x(t+T, \varphi )\) is also a solution with the same initial value condition, we know that

$$\begin{aligned} x(t, Q^{-1}x_{T}(\cdot , \varphi )) = Q^{-1}x(t+T, \varphi ). \end{aligned}$$

Hence

$$\begin{aligned} x_{T}(t, Q^{-1}x_{T}(\cdot , \varphi )) = Q^{-1}x_{T}(t+T, \varphi ). \end{aligned}$$

We have

$$\begin{aligned} P^{2}(\varphi )&= Q^{-1}x_{T}(\cdot , Q^{-1}x_{T}(\cdot , \varphi ))\\&= Q^{-1}Q^{-1}x_{T}(\cdot +T, \varphi )\\&= Q^{-2}x_{2T}(\cdot , \varphi ). \end{aligned}$$

Similarly, it is easy to prove that

$$\begin{aligned} P^{k}(\varphi ) = Q^{-k}x_{kT}(\cdot , \varphi ). \end{aligned}$$

Since system (11) is (Q, T)-dissipative, there exist \(B_{0} > 0\), \(B_{1}=B_{0}+1\) and \(L=L(B_{1})>0\) such that

$$\begin{aligned} \vert Q^{-k}x(t+kT, \varphi )\vert \le B_{0}~~~~\forall ~t+kT\in [L, \infty ) \end{aligned}$$

whenever \(\vert \vert \varphi \vert \vert \le B_{1}\).

Let K be the smallest positive integer satisfying \(KT-r > L+1\), and set

$$\begin{aligned} B_{2}&= \sup \left\{ \vert \vert Q^{-i}x_{iT}(\cdot , \varphi )\vert \vert :~~i = 0, 1, \ldots , K-1, \vert \vert \varphi \vert \vert \le B_{1}\right\} +B_{1}+1,\\ h_{0}&= \sup \left\{ \vert F(t, \varphi )\vert :~~t\in {\mathbb {R}}^{1}, \vert \vert \varphi \vert \vert \le B_{0}\right\} ,\\ h_{1}&= \sup \left\{ \vert F(t, \varphi )\vert :~~t\in {\mathbb {R}}^{1}, \vert \vert \varphi \vert \vert \le B_{1}\right\} ,\\ h_{2}&= \sup \left\{ \vert F(t, \varphi )\vert :~~t\in {\mathbb {R}}^{1}, \vert \vert \varphi \vert \vert \le B_{2}\right\} . \end{aligned}$$

Define

$$\begin{aligned} S_{0}&= \left\{ \varphi \in C_r:\begin{array}{c} \vert \vert \varphi \vert \vert \le B_{0}, \vert \varphi (s_{1})-\varphi (s_{2})\vert \le h_{0}\vert s_{1}-s_{2}\vert ,\\ \forall s_{1}, s_{2}\in \varUpsilon _{j}, j= i_{r}+1, i_{r}+2,\ldots ,0 \end{array} \right\} ,\\ S_{1}&= \left\{ \varphi \in C_r:\begin{array}{c} \vert \vert \varphi \vert \vert< B_{1}, \vert \varphi (s_{1})-\varphi (s_{2})\vert < h_{1}\vert s_{1}-s_{2}\vert ,\\ \forall s_{1}, s_{2}\in \varUpsilon _{j}, j= i_{r}+1, i_{r}+2,\ldots ,0 \end{array} \right\} ,\\ S_{2}&= \left\{ \varphi \in C_r:\begin{array}{c} \vert \vert \varphi \vert \vert \le B_{2}, \vert \varphi (s_{1})-\varphi (s_{2})\vert \le h_{2}\vert s_{1}-s_{2}\vert ,\\ \forall s_{1}, s_{2}\in \varUpsilon _{j}, j= i_{r}+1, i_{r}+2,\ldots ,0 \end{array} \right\} . \end{aligned}$$

It is easy to prove that

$$\begin{aligned} P^{i}(S_{1})&\subset S_{2},\,\,\,\,\,\,\,\,i=1, 2, \ldots , K-1,\\ P^{i}(S_{1})&\subset S_{0},\,\,\,\,\,\,\,\,i=K, K+1, \cdot \cdot \cdot , 2K-1. \end{aligned}$$

By Horn’s fixed point theorem, P admits a fixed point \(\varphi _{*}\) on \(S_{0}\). It means that

$$\begin{aligned} P(\varphi _{*})&= Q^{-1}x_{T}(\cdot , \varphi _{*})\\&= \varphi _{*}. \end{aligned}$$

In other words, both \(x(t, \varphi _{*})\) and \(Q^{-1}x(t+T, \varphi _{*})\) are solutions of (11) with initial value condition \(x_{0} = \varphi _{*}\).

Hence

$$\begin{aligned} x(t+T, \varphi _{*}) = Qx(t, \varphi _{*}). \end{aligned}$$

Namely, system (11) admits a (Q, T)-affine-periodic solution \(x(t, \varphi _{*})\). \(\square \)

4 Applications

Lyapunov’s method is a fundamental tool to study dissipative and stable properties of solutions for differential equations. Hence in this section, we give some sufficiency conditions of the results in previous sections by applying Lyapunov’s second method.

First let us consider system (1). We recall some definitions and preliminaries.

Definition 9

A function \(V: {\mathbb {R}}^{1}_{+} \times {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}_{+}\) is said to be Lyapunov type if it is \(C^1\) and satisfies that

$$\begin{aligned} a(|x|) \le V(t, x) \le b(|x|), \end{aligned}$$
(12)

\(a(\cdot )\) and \(b(\cdot )\) are wedges, which means that they are continuous, strictly increasing to \(\infty \) and \(a(0)=b(0)=0\).

Definition 10

The following is the directional derivative of Lyapunov type function V(t, x):

$$\begin{aligned} V'_{(1)}(t, x)=\frac{\partial V}{\partial t}+\frac{\partial V^\top }{\partial x}f(t, x)~~~~\forall (t, x). \end{aligned}$$

Definition 11

A Lyapunov type function V(t, x) is said to be nonincreasing in large to impulsive effects \(\{I_i (x)\}\), if for some \(K \ge 0\), one has

$$\begin{aligned} V(t, x+I_i (x))\le V(t, x) \end{aligned}$$

for all \(t \in {\mathbb {R}}^{1}_{+}\), \(i=0, 1, \ldots \), and \(x \in {\mathbb {R}}^n\) with \(|x|\ge K\).

Now we are in a position to state the main result of this section.

Theorem 4

Let \(Q\in O(n)\), and let \(V: {\mathbb {R}}^{1} \times {\mathbb {R}}^{n} \rightarrow {\mathbb {R}}_{+}\) be a nonincreasing Lyapunov type function. Assume besides (H1)-(H4) the directional derivative of V(t, x) along (1) satisfies

$$\begin{aligned} V'_{(1)}(t, x)\le \alpha (t)V(t, x)~~~~\forall |x|\ge K,~t\ge 0, \end{aligned}$$

where \(\alpha : {\mathbb {R}}^{1}_{+} \rightarrow {\mathbb {R}}^{1}\) is locally \(L^1\)-integrable and satisfies that

$$\begin{aligned} \int ^{t_0+kT}_{t_0} \alpha (s) \mathrm{d}s \rightarrow -\infty ~~~~\hbox {as}~~~~k \rightarrow \infty ~~~\hbox {uniformly in } t_0\ge 0. \end{aligned}$$

Then (1) is Q-dissipative and hence it admits a (Q, T)-affine-periodic solution.

Proof

Fix any \(R>0\). Let \(x(t, x_{0})\) be a possible solution of system (1) with initial value condition \(x(0)=x_{0}\) with \(|x_0|\le R\).

  1. I.

    There exists a positive constant \(T_{0}\) such that

    $$\begin{aligned} |x(t, x_{0})| < K~~~~~~~~~\forall ~t \ge T_{0}, \end{aligned}$$

    as desired.

  2. II.

    There exists a positive constant \(T_{1}\) such that

    $$\begin{aligned} |x(t, x_{0})| \ge K~~~~~~~~~\forall ~t \in [0, T_{1}]. \end{aligned}$$

We have

$$\begin{aligned} V'_{(1)}(t, x)\le \alpha (t)V(t, x)~~~~~~~~~\forall ~t \in [0, T_{1}]. \end{aligned}$$

Thus

figure a

Set \((\tau _{m-1}, \tau _{m}]\subset [0,T_1]\). Integrating \((*)\) from \(\tau _{m-1}\) to \(\tau _{m}\) yields

$$\begin{aligned} \int ^{\tau _{m}}_{\tau _{m-1}}\frac{V'_{(1)}(t, x)}{V(t, x)}\mathrm{d}t \le \int ^{\tau _{m}}_{\tau _{m-1}}\alpha (t)\mathrm{d}t. \end{aligned}$$

Then

$$\begin{aligned} \ln V(\tau _{m}, x(\tau _{m}))-\ln V(\tau _{m-1}, x(\tau _{m-1})+I_{m-1}(x(\tau _{m-1}-0))) \le \int ^{\tau _{m}}_{\tau _{m-1}}\alpha (t)\mathrm {d}t.\nonumber \\ \end{aligned}$$
(13)

By Definition 11,

$$\begin{aligned} V(t, x+I_i (x))\le V(t, x) \end{aligned}$$

whenever \(|x| \ge K\). Thus

$$\begin{aligned}&\ln V(\tau _{m}, x(\tau _{m}))-\ln V(\tau _{m-1}, x(\tau _{m-1})) \\&\quad \le \ln V(\tau _{m}, x(\tau _{m}))-\ln V(\tau _{m-1}, x(\tau _{m-1})+I_{m-1}(x(\tau _{m-1}-0)))\\&\quad \le \int ^{\tau _{m+1}}_{\tau _{m}}\alpha (t)\mathrm {d}t. \end{aligned}$$

It follows from (12) that

$$\begin{aligned} a(|x(t)|) \le V(t, x(t)) \le V(0, x(0)) \cdot e^{\int ^{t}_{0}\alpha (s)\mathrm {d}s}\le b(R)\cdot e^{\int ^{t}_{0}\alpha (s)\mathrm {d}s}. \end{aligned}$$

Since \(\int ^{kT}_0 \alpha (s) \mathrm{d}s \rightarrow -\infty \) as \(k \rightarrow \infty \), we have

$$\begin{aligned} a^{-1}(b(R)e^{\int ^{kT}_{0}\alpha (s)\mathrm {d}s})\rightarrow 0~~~(k\rightarrow \infty ). \end{aligned}$$

Hence there exists \(T_1\) such that

$$\begin{aligned} |x(T_1)|<K. \end{aligned}$$

Put

$$\begin{aligned} B=\sup _{{\mathbb {R}}_+}\left\{ K,\max _{[0,NT]} a^{-1}(b(K+1)e^{\int ^{t_0+t}_{t_0}\alpha (s)\mathrm {d}s})\right\} , \end{aligned}$$

where the integer \(N>0\) is chosen so that \(a^{-1}(b(K+1)e^{\int ^{t_0+NT}_{t_0}\alpha (s)\mathrm {d}s})<K\). We obtain that

$$\begin{aligned} |x(t)| \le B~~~~~~~~~\forall ~t\ge T_1. \end{aligned}$$

To sum up, it means that system (1) is Q-dissipative. By Theorem 2, it admits a (Q, T)-affine-periodic solution. \(\square \)

As applications, we have the following:

Corollary 1

Let (1) be a (Q, T)-affine-periodic impulsive system, where \(Q\in O(n)\). Assume (H1)-(H4) hold, and for some \(K\ge 0\), we have

$$\begin{aligned} x^\top f(t, x) \le \alpha (t)|x|^2~~~~\forall |x|\ge K,~~t\ge 0, \end{aligned}$$
(14)

where \(\alpha : {\mathbb {R}}^{1}_{+} \rightarrow {\mathbb {R}}^{1}\) is locally \(L^1\)-integrable and satisfies that

$$\begin{aligned} \int ^{t_0+kT}_{t_0} \alpha (s) \mathrm{d}s \rightarrow -\infty ~~~~\hbox {as}~~~~k \rightarrow \infty ~~~\hbox {uniformly in } t_0\ge 0. \end{aligned}$$

Moreover, assume

$$\begin{aligned} |x+I_{i}(x)|\le |x| \end{aligned}$$

for all \(i=0, 1, \ldots ,\) and \(x\in {\mathbb {R}}^{n}\) with \(|x|\ge K\).

Then (1) is Q-dissipative and hence it admits a (Q, T)-affine-periodic solution.

Proof

Let \(V(t, x) = \frac{1}{2}|x|^{2}\). Obviously, V(t, x) is a Lyapunov type function by Definition 9. Since

$$\begin{aligned} |x+I_{i}(x)|\le |x| \end{aligned}$$

for all \(t\in {\mathbb {R}}^{1}_{+}\), \(i=0, 1, \ldots ,\) and \(x\in {\mathbb {R}}^{n}\) with \(|x|\ge K\). It is easy to see that

$$\begin{aligned} V(t, x+I_i(x))\le V(t,x) \end{aligned}$$

whenever \(|x|\ge K\). Also, we have

$$\begin{aligned} V'_{(1)}(t, x) = x^{\top } x' = x^{\top } f(t,x). \end{aligned}$$

By (14), it is easy to prove that

$$\begin{aligned} V'_{(1)}(t, x)&= x^{\top } f(t, x)\\&\le \alpha (t)|x|^2\\&= 2\alpha (t)V(t, x) \end{aligned}$$

whenever \(|x|\ge K\).

By Theorem 4, system (1) is Q-dissipative and hence it admits a (Q, T)-affine-periodic solution. \(\square \)

As a simple use of Corollary 1, consider a linear system

$$\begin{aligned} \begin{array}{l} x'=A(t)x\equiv f(t, x), \\ \varDelta x\vert _{t = \tau _{i}} = I_{i}(x),~~i\in {\mathbb {Z}}^{1}, \end{array} \end{aligned}$$
(15)

where \(A(t)=(A_{ij}(t))_{n\times n}\) is an \(n\times n\) matrix with respect to t. The function \(A_{ij}(t)\) is continuous for any index i and j. Assume that \((H1)-(H4)\) hold and there exists \(Q\in O(n)\) and \(T>0\) which make \(A(t+T)=QA(t)Q^{-1}\) for any \(t\in {\mathbb {R}}\).

Corollary 2

For system (15), assume the following hold:

  1. (I)

    \(\mathop {\inf }\nolimits _{\vert x\vert =1, t \in {\mathbb {R}}}\vert x^{\top }A(t)x\vert > 0\).

  2. (II)

    There exsits \(k\in \left\{ 1,2,......,n\right\} \) which makes \(A_{kk}(t)<0\) for some \(t\in {\mathbb {R}}\).

  3. (III)

    For some \(K \ge 0\), \(\vert x+I_{i}(x)\vert \le \vert x\vert \) for all i and \(x\in {\mathbb {R}}^{n}\) with \(\vert x\vert \ge K\).

Then, system (15) admits a (Q, T)-affine-periodic solution.

Proof

Let \(V(t, x)=\frac{1}{2}\vert x\vert ^{2}\). Obviously, V(t, x) is a Lyapunov type function by Definition 9. Since

$$\begin{aligned} |x+I_{i}(x)|\le |x| \end{aligned}$$

for all \(t\in {\mathbb {R}}^{1}_{+}\), \(i=0, 1, \ldots ,\) and \(x\in {\mathbb {R}}^{n}\) with \(|x|\ge K\), it is easy to see that

$$\begin{aligned} V(t, x+I_i(x))\le V(t,x) \end{aligned}$$

whenever \(|x|\ge K\). Also, we have

$$\begin{aligned} V'_{(1)}(t, x) = x^{\top } x' = x^{\top }A(t)x. \end{aligned}$$

Notice that by (II), there exsit \(k\in \left\{ 1,2,......,n\right\} \) and \(t_{0}\in {\mathbb {R}}\) which make \(A_{kk}(t_{0})<0\). Denote \(e_{k}=(0,0,...,1,...,0)\) an n-dimensional unit vector with the k-th element equal to 1.

Obviously, we have \(e_{k}^{\top }A(t_{0})e_{k}<0\). Since \(A_{ij}(t)\) is continuous and

$$\begin{aligned} \underset{\vert x\vert =1, t \ge 0}{\inf }\vert x^{\top }A(t)x\vert > 0, \end{aligned}$$

there exsits \(\alpha < 0\) which makes \(x^{\top }A(t)x<\alpha <0\) holds for any \(t\in {\mathbb {R}}^{1}_{+}\) and \(\vert x\vert =1\).

Hence,

$$\begin{aligned} V'_{(1)}(t, x)=x^{\top }A(t)x=\left[ \frac{x^{\top }}{\vert x\vert }A(t)\frac{x}{\vert x\vert }\right] \cdot \vert x\vert ^{2}<\alpha \vert x\vert ^{2}<2\alpha V(t, x). \end{aligned}$$

By Corollary 1, system (15) admits a (Q, T)-affine-periodic solution. \(\square \)

Also, consider Newtonian systems with friction of the form

$$\begin{aligned}&x''+C(t)x'+\nabla U(x)=e(t),\nonumber \\&x^{(j)}(\tau _i+0)=cx^{(j)}(\tau _i),~~j=0,1, \end{aligned}$$
(16)

where \(c\in (0,1];C: {\mathbb {R}}^1\rightarrow {\mathbb {R}}^{1}\), \(U: {\mathbb {R}}^m \rightarrow {\mathbb {R}}^1\), \(e: {\mathbb {R}}^1 \rightarrow {\mathbb {R}}^m\) are continuous, \(C^1\) with respect to x, and satisfy the following (Q, T)-affine-periodicity:

$$\begin{aligned} C(t+T)&=C(t),\\ \nabla U(x)&=Q\nabla U(Q^{-1}x),\\ e(t+T)&=Qe(t). \end{aligned}$$

Here and in what follows \(Q\in O(m)\) is given.

We have the following.

Theorem 5

Assume besides continuity, smoothness, and (Q, T)-affine-periodicity that there exist \(\alpha , \eta ,l>0\) such that

$$\begin{aligned} C(t)&\ge \alpha ;\\ x^\top \nabla U(x)&\ge \eta |x|^2~~~~~\hbox {for all}~(t,x)~\hbox {with}~|x|\ge l. \end{aligned}$$

Then system (16) admits a (Q, T)-affine-periodic solution.

We make some comments to condition in Theorem 5.

Remark 2

When \(m=1\) and \(Q=\pm id\), (Q, T)-affine-periodic solutions correspond periodic or anti-periodic solutions. In this case, U and C may have the following forms:

$$\begin{aligned} U(x)&=a_0 +a_1 x+a_2 x^2 + \cdots + a_{2p} x^{2p},\\ C(t+T)&=C(t), \end{aligned}$$

where \(p>0\) is a positive integers, \(a_{2p}>0\), and \(C(t)\ge \alpha >0\). Then corresponding system (16) becomes Lienard and Duffing equations.

Thus, we have the following corollary:

Corollary 3

Assume U(x) has the form in Remark 2. Then the system

$$\begin{aligned}&x''+C(t)x'+b_0 +b_1 x+\cdots +b_{2p-1}x^{2p-1}=e(t)=\pm e(t+T),\\&x^{(j)}(\tau _i+0)=cx^{(j)}(\tau _i),~~j=0,1, \end{aligned}$$

where \(C: {\mathbb {R}}^1 \rightarrow {\mathbb {R}}^1\) is continuous and T-periodic, \(C(t)\ge \alpha >0\), \(b_{2p-1}>0\), and \(c\in (0,1]\), admits a T-periodic or anti-periodic solution.

Proof of Theorem 5

Let

$$\begin{aligned} V(x, y)=\lambda |y|^2 + |x+y|^2 + (2\lambda +2)U(x), \end{aligned}$$

where \(\lambda \gg 1\). Then along (16) for \((x, y)=(x, x')\),

$$\begin{aligned} V'_{(16)} =\,&2\lambda x'^\top (e(t)-\nabla U(x)-C(t)x')+2(x'+x)^\top (e(t)-\nabla U(x)-C(t)x'+x')\\&+(2\lambda +2)x'^\top \nabla U(x)\\ =&\,2\lambda x'^\top e(t)+2x^\top e(t)+2x^\top e(t)-2\lambda C(t)|x'|^{2}-2C(t)|x'|^{2}+2|x'|^{2}\\&-2C(t)x^\top x'+2x^\top x'-2x^\top \nabla U(x)\\ =&\,-(\lambda +1)C(t)|x'|^{2}-2x^\top \nabla U(x)\\&-(\lambda +1)\left[ C(t)|x'|^{2}-2x'^\top e(t)+\frac{1}{C(t)}|e(t)|^{2}\right] \\&+\left[ \frac{\lambda +1}{C(t)}|e(t)|^{2}+2x^\top e(t)+\frac{C(t)}{\lambda +1}|x^\top |^{2}\right] -\frac{C(t)}{\lambda +1}|x^\top |^{2}\\&+2\left[ |x'|^{2}-[C(t)-1]x^\top x'+\frac{[C(t)-1]^{2}}{4}|x|^{2}\right] -\frac{[C(t)-1]^{2}}{2}|x|^{2}\\ \le&-(\lambda +1)C(t)|x'|^{2}-2x^\top \nabla U(x)-\frac{C(t)}{\lambda +1}|x^\top |^{2}-\frac{[C(t)-1]^{2}}{2}|x|^{2}\\&+\left[ \sqrt{\frac{\lambda +1}{C(t)}}|e(t)|+\sqrt{\frac{C(t)}{\lambda +1}}|x^\top |\right] ^{2}+2\left[ |x'|-\frac{[C(t)-1]}{2}|x|\right] ^{2}\\ \le&-(\lambda +1)C(t)|x'|^{2}-\left( 2\eta +\frac{C(t)}{\lambda +1}+\frac{[C(t)-1]^{2}}{2}\right) |x|^{2}\\&+\left[ \sqrt{\frac{\lambda +1}{C(t)}}|e(t)|+\sqrt{\frac{C(t)}{\lambda +1}}|x^\top |\right] ^{2}+2\left[ |x'|-\frac{[C(t)-1]}{2}|x|\right] ^{2}. \end{aligned}$$

Hence, for \(\lambda \) large and \(\mu (>0)\) small, and some \(L > 0\),

$$\begin{aligned} V'\le -\mu (|x|^2+|x'|^2)+L . \end{aligned}$$
(17)

Clearly, there exists \(\sigma >0\) such that

$$\begin{aligned} V(x,y)\ge \sigma (|x|^2 +|y|^2)~~~\hbox {for all}~x~~\hbox {with}~|x|\ge l. \end{aligned}$$
(18)

From (17), (18), we obtain that along any solution \(z(t, z_0)=(x(t, x_0, y_0), y(t, x_0, y_0))\) of (16) with the initial value \(z(0)=(x(0), y(0))=(x_0, y_0)\) for \(y(t, x_0, y_0)= x'(t, x_0, y_0)\),

$$\begin{aligned} \sigma |z(t, z_0)|^2 \le V(z(t, z_0)) \le V(z_0) e^{\int ^t_0 (-\mu |z(s, z_0)|^2 +L)\mathrm{d}s}, \end{aligned}$$

which implies that for any \(z_0 =(x_0, y_0)\), the solution \(z(t, z_0)\) exists on \({\mathbb {R}}^1_+\). Let

$$\begin{aligned}&M=\max \left\{ V(x,y):|x|^2+|y|^2 \le \frac{L+1}{\mu }\right\} ,\\&N_1 \in {\mathbb {N}}~~\hbox {with}~~Me^{-N_1 T}< \frac{\sigma L}{\mu },\\&B=\left( \frac{M}{\sigma } e^{LN_1 T}\right) ^{\frac{1}{2}}. \end{aligned}$$

We claim that for any \(R>0\) with \(R^2 \ge \frac{L+1}{\mu }\), there is a positive integer N depending on R such that

$$\begin{aligned} |z(t, z_0)|\le B~~~~\forall t\ge NT, \end{aligned}$$
(19)

whenever \(|z_0|\le R\). In fact, for

$$\begin{aligned} M_R=\max \{V(x,y): |x|^2+|y|^2 \le R\}. \end{aligned}$$

We set a positive integer N such that

$$\begin{aligned} M_R e^{-NT}<\frac{\sigma L}{\mu }. \end{aligned}$$
(20)

Obviously,

$$\begin{aligned} V(x(\tau _i+0),y(\tau _i+0))\le V(x(\tau _i),y(\tau _i)). \end{aligned}$$

Then it follows from (17), (18), (20) that (19) holds, and hence system (16) is dissipative. Now the conclusion follows from Theorem 2. \(\square \)

Finally, we give an example.

Example 1

Consider the impulsive differential system

$$\begin{aligned}&x'=-|x|^{2m}x+ \left( \begin{array}{c} e(t) \\ \cos \omega _1t\\ \sin \omega _1t\\ \vdots \\ \cos \omega _lt\\ \sin \omega _lt \end{array} \right) \equiv f(t,x),\nonumber \\&x(\tau _{i}+0)=cx(\tau _{i}), \end{aligned}$$
(21)

where \(m\ge 1\), \(0<c<1\); \(e:{\mathbb {R}}^1\rightarrow {\mathbb {R}}^{n-2l}\) is continuous and \(e(t+T)=e(t)\); \(\omega _i>0,i=1,\dots ,l\).

Set

$$\begin{aligned} Q= diag (I_{n-2l}, diag\left( \begin{array}{ccc} \left( \begin{array}{cc} cos \omega _1 T &{}\quad -sin \omega _1 T \\ sin \omega _1 T &{}\quad cos \omega _1 T \\ \end{array} \right) &{} \cdots &{} \left( \begin{array}{cc} cos \omega _l T &{}\quad -sin \omega _l T \\ sin \omega _l T &{}\quad cos \omega _l T \\ \end{array} \right) \\ \end{array} \right) ). \end{aligned}$$

It is easy to verify that

$$\begin{aligned} f(t+T, x)=Qf(t,Q^{-1}x). \end{aligned}$$

Set

$$\begin{aligned} V(x)=\frac{1}{2}|x|^2. \end{aligned}$$

Then

$$\begin{aligned} V'(x)=x^\top x'=-|x|^{2m+2}+x^\top \left( \begin{array}{c} e(t) \\ \cos \omega _1t\\ \sin \omega _1t\\ \vdots \\ \cos \omega _lt\\ \sin \omega _lt \end{array}\right) \le -V(x) \end{aligned}$$

whenever \(|x|\gg 1\). Note \(0<c<1\). Obviously, for any i, we have

$$\begin{aligned} V(x(\tau _i+0))=V(x(\tau _i)+I_i(x(\tau _i)))=\frac{1}{2}|c||x(\tau _{i})|^2\le V(x(\tau _{i})). \end{aligned}$$

Let \(\alpha (t)=-\frac{1}{2}\). Then system (21) satisfies all the conditions of Corollary 1. Thus system (21) has a (Q, T)-affine periodic solution.

5 Conclusion

In this paper, the existence of affine-periodic solutions for impulsive differential system has been investigated. We first prove Theorem 1, which is a Yoshizawa type theorem for dissipative periodic impulsive systems. Then, Theorem 2 concerning the existence of affine-periodic solutions for dissipative impulsive differential systems is derived similarly. Also, for functional differential equations, Theorem 3 shows that the existence of (Q, T)-affine-periodic solution is confirmed as long as the system is (Q, T)-dissipative. Theorem 4 as well as Corollaries 1, 2 provide us a simple way to study the dissipativity of the system by using Lyapunov’s second method. Finally, Newtonian systems with friction are discussed and a sufficiency condition is given in Theorem 5. We give an example at last, which shows that the proposed methods are effective.