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\((\omega ,c)\)-Periodic solutions for time varying impulsive differential equations

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Abstract

In this paper, we study a class of \((\omega ,c)\)-periodic time varying impulsive differential equations and establish the existence and uniqueness results for \((\omega ,c)\)-periodic solutions of homogeneous problem as well as nonhomogeneous problem.

Introduction

It is well known that the concept of \((\omega ,c)\)-periodic functions is the same of “affine-periodic functions” or “periodic of second kind”, which were introduced by Floquet [1] and have been studied in the past decades. Recently, Alvarez et al. [2] introduced a new concept of \((\omega ,c)\)-periodic function by considering Mathieu’s equation \(z''+[\alpha -2\beta \cos (2t)]z=0\), and its solution satisfies \(z(t+\omega )=cz(t)\), \(c\in \mathbb {C}\). Clearly, \((\omega ,c)\)-periodic functions become the standard ω-periodic functions when \(c=1\) and ω-antiperiodic functions when \(c=-1\). For these particular cases, we refer readers to [3,4,5,6].

Meanwhile, Alvarez et al. [7] transferred the same idea to study \((N,\lambda )\)-periodic discrete functions and established the existence and uniqueness of \((N,\lambda )\)-periodic solutions to a class of Volterra difference equations with infinite delay. Next, Agaoglou et al. [8] applied the concept of \((\omega ,c)\)-periodic to semilinear evolution equations in complex Banach spaces and studied its existence and uniqueness of \((\omega ,c)\)-periodic solutions. Li et al. [9] transferred the similar idea to consider \((\omega ,c)\)-periodic solutions impulsive differential systems.

Although, Floquet [1] studied a homogenous linear periodic system \(x'(t)=A(t)x(t)\) with \(A(t+\omega )=A(t)\), \(t\in \mathbb {R}\), there are quite few analogous results to Floquet’s theory for \((\omega ,c)\)-periodic systems with impulse. Motivated by [1, 2, 8, 9], we consider the following time varying impulsive differential equation:

$$ \textstyle\begin{cases} x'(t)=a(t)x(t)+f (t,x(t) ), \quad t\neq t_{i}, i\in \mathbb {N}=\{1,2,\ldots \}, \\ \Delta x|_{t=t_{i}}=x (t_{i}^{+} )-x (t_{i}^{-} )=b_{i}x (t_{i}^{-} )+c _{i}, \end{cases} $$
(1)

where \(a\in C(\mathbb {R},\mathbb {R})\), \(f\in C(\mathbb {R}\times \mathbb {R},\mathbb {R})\), \(b_{i}, c_{i} \in \mathbb {R}\), and \(t_{i}< t_{i+1}\), \(i\in \mathbb {N}\). The symbols \(x(t_{i}^{+})\) and \(x(t_{i}^{-})\) represent the right and left limits of \(x(t)\) at \(t=t_{i}\).

The main purpose of this paper is to derive existence and uniqueness results for \((\omega ,c)\)-periodic solutions of nonhomogeneous linear problem as well as homogeneous linear problem.

Preliminaries

We introduce a Banach space \(\operatorname{PC}(\mathbb {R},\mathbb {R})=\{x: \mathbb {R}\to \mathbb {R}:x\in C((t _{i},t_{i+1}],\mathbb {R}), \text{and } x(t_{i}^{-})=x(t_{i}), x(t_{i} ^{+}) \text{ exists } \forall i\in \mathbb {N}\}\) endowed with the norm \(\|x\|=\sup_{t\in \mathbb {R}}|x(t)|\).

Lemma 2.1

(See [10, p.9])

Suppose that \(f\in C(\mathbb {R},\mathbb {R})\). A solution \(x\in \operatorname{PC}(\mathbb {R},\mathbb {R})\) of the following nonhomogeneous linear impulsive equation

$$ \textstyle\begin{cases} x'(t)=a(t)x(t)+f(t),\quad t \neq t_{i}, i\in \mathbb {N}, \\ \Delta x|_{t=t_{i}}=b _{i}x (t_{i}^{-} )+c_{i}, \\ x(t_{0})=x_{t_{0}}, \end{cases} $$
(2)

is given by

$$ x(t)=W(t,t_{0})x(t_{0})+ \int _{t_{0}}^{t}W(t,s)f(s)\,ds+\sum _{t_{0}< t _{i}< t}W(t,t_{i})c_{i},\quad {t\geq t_{0}}, $$
(3)

where (see [10, p.8])

$$ W(t,t_{0})=e^{\int _{t_{0}}^{t}a(s)\,ds}\prod_{t_{0} < t_{i}< t}(1+b_{i}), \quad {t\geq t_{0}}. $$

Lemma 2.2

For any \(t, t_{0}\in \mathbb {R}\), \(\tau \in \mathbb {R}\setminus \{t_{i}\}_{i\in \mathbb {N}}\), and \({t\geq \tau \geq t_{0}}\), we have

$$ W(t,t_{0})=W(t,\tau )W(\tau ,t_{0}). $$
(4)

Proof

Since \(\tau \notin \{t_{i}\}_{i\in \mathbb {N}}\), we derive

$$\begin{aligned} W(t,t_{0}) =& e^{\int _{t_{0}}^{t}a(s)\,ds}\prod_{t_{0}< t_{i}< t}(1+b _{i}) \\ =& \biggl(e^{\int _{t_{0}}^{\tau }a(s)\,ds}\prod_{t_{0}< t_{i}< \tau }(1+b _{i}) \biggr)e^{\int _{\tau }^{t}a(s)\,ds}\prod_{\tau \leq t_{i}< t}(1+b_{i}) \\ =& \biggl(e^{\int _{t_{0}}^{\tau }a(s)\,ds}\prod_{t_{0}< t_{i}< \tau }(1+b _{i}) \biggr)e^{\int _{\tau }^{t}a(s)\,ds}\prod_{\tau < t_{i}< t}(1+b_{i})= W(t, \tau )W(\tau ,t_{0}). \end{aligned}$$

 □

Definition 2.3

(See [2])

Let \(c\in \mathbb {R}\setminus \{0\}\) and \(\omega >0\). A function \(f:\mathbb {R}\to \mathbb {R}\) is said to be \((\omega ,c)\)-periodic if \(f(t+\omega )=cf(t)\) for all \(t\in \mathbb {R}\).

Lemma 2.4

(See [8, Lemma 2.2])

Set \(\varPsi _{\omega ,c}:=\{x:x\in \operatorname{PC}(\mathbb {R},\mathbb {R}) \text{ and } cx(\cdot )=x(\cdot +\omega )\}\). Let \(x\in \varPsi _{\omega ,c}\), that is, x is a piecewise continuous and \((\omega ,c)\)-periodic function. Then \(x\in \varPsi _{\omega ,c}\) is equivalent to

$$\begin{aligned} x(\omega )=cx(0). \end{aligned}$$
(5)

Lemma 2.5

Assume that the following conditions hold:

\((A_{1})\) :

\(a(\cdot )\) is ω-periodic, i.e., \(a(t+\omega )=a(t)\), \(\forall t\in \mathbb {R}\).

\((A_{2})\) :

Set \(t_{0}=0\) and \(t_{i}< t_{i+1}\), \(i\in \mathbb {N}\). There exists \(N\in \mathbb {N}\) such that \(t_{i+N}=t_{i}+\omega \), \(b_{i+N}=b _{i}\), and \(c_{i+N}=c_{i}\), \(\forall i\in \mathbb {N}\).

Then the following homogeneous linear impulsive equation

$$ \textstyle\begin{cases} x'(t)=a(t)x(t),\quad t\neq t_{i}, i\in \mathbb {N}, \\ \Delta x|_{t=t_{i}}=b _{i}x (t_{i}^{-} ), \\ x(0)=x_{0}, \end{cases} $$
(6)

has a solution \(x\in \varPsi _{\omega ,c}\) if and only if \(x_{0}(c-W( \omega ,0))=0 \).

Proof

The solution \(x\in PC(\mathbb{R},\mathbb{R})\) of (6) is given by

$$ x(t)=x_{0}W(t,0)= x_{0}e^{\int _{t_{0}}^{t}a(s)\,ds}\prod _{0< t_{i}< t}(1+b _{i}), \quad t\geq 0. $$

If there exists \(t_{i}\in (0,t)\) such that \(1+b_{i}=0\), obviously, \(x(t+\omega )=cx(t)=0\), and the result holds.

If \(1+b_{i}\neq 0\), \(\forall t_{i}\in (0,t)\) and \(t\in [0,\infty ) \setminus \{t_{i}\}_{i\in \mathbb {N}}\), we derive

$$\begin{aligned} x(t+\omega )=cx(t)\quad \Longleftrightarrow &\quad x_{0}e^{\int _{0}^{t+\omega }a(s)\,ds} \prod_{0< t_{i}< t+\omega }(1+b_{i})=c x_{0}e^{\int _{0}^{t}a(s)\,ds} \prod_{0< t_{i}< t}(1+b_{i}) \\ \Longleftrightarrow &\quad x_{0}e^{\int _{t}^{t+\omega }a(s)\,ds} \prod _{t< t_{i}< t+\omega }(1+b_{i})=cx_{0} \\ \Longleftrightarrow &\quad x_{0} \biggl(c-e^{\int _{t}^{t+\omega }a(s)\,ds} \prod _{t< t_{i}< t+\omega }(1+b_{i}) \biggr)=0 \\ \Longleftrightarrow &\quad x_{0} \biggl(c-e^{\int _{0}^{\omega }a(s)\,ds} \prod _{0< t_{i}< \omega }(1+b_{i}) \biggr)=0 \\ \Longleftrightarrow &\quad x_{0} \bigl(c-W(\omega ,0) \bigr)=0. \end{aligned}$$

In addition, since \(x(t_{i})=x(t_{i}^{-})\), we obtain \(x(t_{i}+\omega )=cx(t_{i})\). □

Main results

We consider the \((\omega ,c)\)-periodic solutions of the following nonhomogeneous linear problem:

$$ \textstyle\begin{cases} x'(t)=a(t)x(t)+f(t),\quad t\neq t_{i}, i\in \mathbb {N}, \\ \Delta x|_{t=t_{i}}=b _{i}x (t_{i}^{-} )+c_{i}, \\ x(0)=x_{0}, \end{cases} $$
(7)

where \(f\in C(\mathbb {R},\mathbb {R})\) and f is \((\omega ,c)\)-periodic. We give the following assumption:

\((A_{3})\) :

\(c\neq W(\omega ,0)\).

Lemma 3.1

Assume that \((A_{1})\), \((A_{2})\), and \((A_{3})\) hold. Then the solution \(x\in \varUpsilon :=\operatorname{PC}([0,\omega ],\mathbb {R})\) of (7) satisfying (5) is given by

$$ x(t)= \int _{0}^{\omega }F(t,s)f(s)\,ds+\sum _{i=1}^{N} F(t,t_{i})c_{i}, $$
(8)

where

$$ F(t,s)= \textstyle\begin{cases} c (c-W(\omega ,0) )^{-1}W(t,s),\quad 0\leq s< t, \\ W(t,0) (c-W(\omega ,0) )^{-1}W( \omega ,s),\quad t\leq s< \omega . \end{cases} $$
(9)

Proof

The solution \(x\in \varUpsilon \) of (7) is given by

$$ x(t)=W(t,0)x_{0}+ \int _{0}^{t}W(t,s)f(s)\,ds+\sum _{0< t_{i}< t}W(t,t_{i})c _{i}. $$
(10)

Thus \(x(\omega )=W(\omega ,0)x_{0}+\int _{0}^{\omega }W(\omega ,s)f(s)\,ds+ \sum_{0< t_{i}<\omega }W(\omega ,t_{i})c_{i}=cx_{0} \), which is equivalent to \(x_{0}=(c-W(\omega ,0))^{-1} (\int _{0}^{\omega }W( \omega ,s)f(s)\,ds+\sum_{0< t_{i}<\omega }W(\omega ,t_{i})c_{i} ) \) due to \(c\neq W(\omega ,0)\).

Then we have

$$\begin{aligned} x(t) =& W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} \biggl( \int _{0}^{\omega }W(\omega ,s)f(s)\,ds+\sum _{0< t_{i}< \omega }W(\omega ,t_{i})c_{i} \biggr) \\ &{} + \int _{0}^{t}W(t,s)f(s)\,ds+\sum _{0< t_{i}< t}W(t,t_{i})c_{i} :=I _{1}+I_{2}, \end{aligned}$$

where

$$\begin{aligned}& I_{1}:= W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} \int _{0}^{\omega }W(\omega ,s)f(s)\,ds+ \int _{0} ^{t}W(t,s)f(s)\,ds, \\& I_{2} := W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} \sum _{0< t_{i}< \omega }W( \omega ,t_{i})c_{i}+ \sum_{0< t_{i}< t}W(t,t_{i})c_{i}. \end{aligned}$$

If \(t\in [0,\omega ]\setminus \{t_{1},\ldots ,t_{N}\}\), by (4) and condition \((A_{3})\), we derive

$$\begin{aligned} I_{1} =& W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} \int _{0}^{t} W(\omega ,t)W(t,s)f(s)\,ds+ \int _{0}^{t}W(t,s)f(s)\,ds \\ & {}+W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} \int _{t}^{\omega }W(\omega ,s)f(s)\,ds \\ =& \bigl(W(\omega ,0) \bigl(c-W(\omega ,0) \bigr)^{-1}+1 \bigr) \int _{0}^{t}W(t,s)f(s)\,ds \\ &{}+ \int _{t}^{\omega }W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} W(\omega ,s) f(s)\,ds \\ =&c \int _{0}^{t} \bigl(c-W(\omega ,0) \bigr)^{-1}W(t,s)f(s)\,ds+ \int _{t}^{\omega }W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} W(\omega ,s) f(s)\,ds \\ =& \int _{0}^{\omega }F(t,s)f(s)\,ds, \end{aligned}$$

and

$$\begin{aligned} I_{2} =& W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} \sum _{0< t_{i}< t}W(\omega ,t)W(t,t _{i})c_{i}+ \sum_{0< t_{i}< t}W(t,t_{i})c_{i} \\ &{} +W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} \sum _{t< t_{i}< \omega }W(\omega ,t _{i})c_{i} \\ =& \bigl(W(\omega ,0) \bigl(c-W(\omega ,0) \bigr)^{-1}+1 \bigr) )\sum _{0< t_{i}< t}W(t,t _{i})c_{i} \\ &{} +W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1} \sum _{t< t_{i}< \omega }W(\omega ,t _{i})c_{i} \\ =&c\sum_{0< t_{i}< t} \bigl(c-W(\omega ,0) \bigr)^{-1}W(t,t_{i})c_{i}+ \sum _{t< t_{i}< \omega }W(t,0) \bigl(c-W(\omega ,0) \bigr)^{-1}W(\omega ,t_{i})c _{i} \\ =&\sum_{0< t_{i}< \omega }F(t,t_{i})c_{i} \\ =& \sum_{i=1}^{N}F(t,t_{i})c _{i}. \end{aligned}$$

Thus we get (8). Since \(x(t_{i})=x(t_{i}^{-})\), we can also get the same result for \(t\in \{t_{1},\ldots ,t_{N}\}\). □

Lemma 3.2

Let \(\tilde{a}:=\max_{t\in [0,\omega ]}\{a(t)\}\) and \(\tilde{b}:=\max_{1\leq i\leq N}\{|1+b_{i}|\}\). Then, for any \(t\in [0,\omega ]\), we have

$$ \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \,ds \leq P_{\tilde{a}}:= \textstyle\begin{cases} \vert (c-W( \omega ,0) )^{-1} \vert e^{\tilde{a}\omega }\omega \tilde{b}^{N} ( \vert c \vert +1 ), &\tilde{a}>0, \\ \vert (c-W(\omega ,0) )^{-1} \vert \omega \tilde{b}^{N} ( \vert c \vert +1 ) , &\tilde{a}\leq 0. \end{cases} $$

Proof

Using (9), we derive

$$\begin{aligned} \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \,ds \leq & \bigl\vert \bigl(c-W(\omega ,0) \bigr)^{-1} \bigr\vert \biggl( \int _{0}^{t} \bigl\vert cW(t,s) \bigr\vert \,ds+ \int _{t}^{\omega } \bigl\vert W(t,0)W(\omega ,s) \bigr\vert \,ds \biggr) \\ \leq & \bigl\vert \bigl(c-W(\omega ,0) \bigr)^{-1} \bigr\vert \biggl( \vert c \vert \int _{0}^{t}e^{\int _{s}^{t}a( \tau )\,d\tau }\prod _{s< t_{i}< t} \vert 1+b_{i} \vert \,ds \\ &{}+ \int _{t}^{\omega }e^{(\int _{0}^{t}+\int _{s}^{\omega })a(\tau )\,d\tau }\prod _{0< t_{i}< t\cup s< t_{i}< \omega } \vert 1+b_{i} \vert \,ds \biggr). \end{aligned}$$

If \(\tilde{a}>0\), we get

$$ \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \,ds \leq \bigl\vert \bigl(c-W(\omega ,0) \bigr)^{-1} \bigr\vert e^{\tilde{a} \omega }\omega \tilde{b}^{N} \bigl( \vert c \vert +1 \bigr). $$

If \(\tilde{a}\leq 0\), we get

$$ \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \,ds \leq \bigl\vert \bigl(c-W(\omega ,0) \bigr)^{-1} \bigr\vert \omega \tilde{b}^{N} \bigl( \vert c \vert +1 \bigr). $$

The proof is finished. □

Lemma 3.3

For any \(t\in [0,\omega ]\), we have

$$ \sum_{i=1}^{N} \bigl\vert F(t,t_{i})c_{i} \bigr\vert \leq Q_{\tilde{a}}:= \textstyle\begin{cases} \vert (c-W(\omega ,0) )^{-1} \vert ( \vert c \vert +1 )e^{\tilde{a}\omega }\tilde{b}^{N} \sum_{i=1}^{N} \vert c_{i} \vert &\tilde{a}>0, \\ \vert (c-W(\omega ,0) )^{-1} \vert ( \vert c \vert +1 ) \tilde{b}^{N}\sum_{i=1}^{N} \vert c_{i} \vert & \tilde{a}\leq 0. \end{cases} $$

Proof

By (9), we have

$$\begin{aligned} \sum_{i=1}^{N} \bigl\vert F(t,t_{i})c_{i} \bigr\vert \leq & \bigl\vert \bigl(c-W(\omega ,0) \bigr)^{-1} \bigr\vert \biggl( \sum _{0< t_{i}< t} \bigl\vert cW(t,t_{i})c_{i} \bigr\vert + \sum_{t\leq t_{i}< \omega } \bigl\vert W(t,0)W( \omega ,t_{i})c_{i} \bigr\vert \biggr) \\ \leq & \bigl\vert \bigl(c-W(\omega ,0) \bigr)^{-1} \bigr\vert \biggl(\sum_{0< t_{i}< t} \vert c_{i} \vert \vert c \vert e^{ \int _{t_{i}}^{t}a(\tau )\,d\tau } \prod_{t_{i}< t_{k}< t} \vert 1+b_{k} \vert \\ &{}+\sum_{t\leq t_{i}< \omega } \vert c_{i} \vert e^{(\int _{0}^{t}+\int _{t_{i}}^{ \omega })a(\tau )\,d\tau }\prod_{0< t_{k}< t \cup t_{i}< t_{k}< \omega } \vert 1+b _{k} \vert \biggr). \end{aligned}$$

If \(\tilde{a}> 0\), we obtain

$$ \sum_{i=1}^{N} \bigl\vert F(t,t_{i})c_{i} \bigr\vert \leq \bigl\vert \bigl(c-W( \omega ,0) \bigr)^{-1} \bigr\vert \bigl( \vert c \vert +1 \bigr)e ^{\tilde{a}\omega }\tilde{b}^{N}\sum_{i=1}^{N} \vert c_{i} \vert . $$

If \(\tilde{a}\leq 0\), we obtain

$$ \sum_{i=1}^{N} \bigl\vert F(t,t_{i})c_{i} \bigr\vert \leq \bigl\vert \bigl(c-W( \omega ,0) \bigr)^{-1} \bigr\vert \bigl( \vert c \vert +1 \bigr) \tilde{b}^{N}\sum_{i=1}^{N} \vert c_{i} \vert . $$

The proof is complete. □

Now we are ready to study the existence of semilinear impulsive problems. We make the following hypotheses:

\((A_{4})\) :

For any \(t\in \mathbb {R}\) and \(x\in \mathbb {R}\), it holds \(f(t+\omega ,cx)=cf(t,x)\).

\((A_{5})\) :

There exists \(L>0\) such that \(|f(t,x)-f(t,y)|\leq L|x-y|\) for any \(t\in \mathbb {R}\) and \(x,y\in \mathbb {R}\).

\((A_{6})\) :

There exist constants \(K,J>0\) such that \(|f(t,x)|\leq K |x|+J\) for any \(t\in \mathbb {R}\) and \(x\in \mathbb {R}\).

Theorem 3.4

Suppose that \((A_{1})\), \((A_{2})\), \((A_{3})\), \((A_{4})\), and \((A_{5})\) hold. If \(0< LP_{\tilde{a}}<1\), then (1) has a unique \((\omega ,c)\)-periodic solution \(x\in \varPsi _{\omega ,c}\). Moreover, it holds \(\|x\|\leq \frac{f_{0}P_{\tilde{a}}+Q_{\tilde{a}}}{1-LP_{ \tilde{a}}} \), where \(f_{0}=\max_{t\in [0,\omega ]}|f(t,0)|\).

Proof

For any \(x\in \varPsi _{\omega ,c}\), i.e., \(x(\cdot +\omega )=cx)\), we have \(f(t+\omega ,x(t+\omega ))=f(t,cx(t))\), \(t\in \mathbb {R}\). Further, by assumption \((A_{4})\), \(f(t+\omega ,x(t+\omega ))=f(t,cx(t))=cf(t,x)\), \(t\in \mathbb {R}\). Thus, \(f(\cdot ,x(\cdot ))\in \varPsi _{\omega ,c}\). For more characterization of the \((\omega ,c)\)-periodic functions, see [2, Sect. 2].

Let \(\mathbb {G}:\varUpsilon \to \varUpsilon \) be the operator given by

$$ (\mathbb {G}x) (t)= \int _{0}^{\omega }F(t,s)f \bigl(s,x(s) \bigr)\,ds+\sum _{i=1}^{N}F(t,t_{i})c _{i}. $$
(11)

By Lemma 2.4 and Lemma 3.1, the existence of \((\omega ,c)\)-periodic solutions of (1) is equivalent to the existence of the fixed point of (11).

It is easy to show that \(\mathbb {G}(\varUpsilon )\subseteq \varUpsilon \). For any \(x,y\in \varUpsilon \), we derive

$$\begin{aligned} \bigl\vert (\mathbb {G}x) (t)-(\mathbb {G}y) (t) \bigr\vert \leq& L \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \bigl\vert x(s)-y(s) \bigr\vert \,ds \\ \leq& L \Vert x-y \Vert \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \,ds \leq LP_{\tilde{a}} \Vert x-y \Vert , \end{aligned}$$

which implies \(\|\mathbb {G}x-\mathbb {G}y\|\leq LP_{\tilde{a}}\|x-y\| \). Noticing \(0< LP_{\tilde{a}}<1\), \(\mathbb {G}\) is a contraction mapping. Thus, \(\mathbb {G}\) defined in (11) has a unique fixed point satisfying \(x(\omega )=cx(0)\) due to Lemma 3.1. Further, by Lemma 2.4, one has \(x\in \varPsi _{\omega ,c}\). From the above, there exists a unique \((\omega ,c)\)-periodic solution \(x\in \varPsi _{\omega ,c}\) of (1).

Moreover, we have

$$\begin{aligned} \bigl\vert x(t) \bigr\vert \leq& L \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \bigl\vert x(s) \bigr\vert \,ds+ \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \bigl\vert f(s,0) \bigr\vert \,ds+ \sum_{i=1}^{N} \bigl\vert F(t,t_{i})c_{i} \bigr\vert \\ \leq& LP_{\tilde{a}} \Vert x \Vert +f_{0}P_{ \tilde{a}}+Q_{\tilde{a}}, \end{aligned}$$

which implies

$$ \Vert x \Vert \leq \frac{f_{0}P_{\tilde{a}}+Q_{\tilde{a}}}{1-LP_{\tilde{a}}}. $$

The proof is finished. □

Theorem 3.5

Suppose that \((A_{1})\), \((A_{2})\), \((A_{3})\), \((A_{4})\), and \((A_{6})\) hold. If \(KP_{\tilde{a}}<1\), then (1) has at least one \((\omega ,c)\)-periodic solution \(x\in \varPsi _{\omega ,c}\).

Proof

Let \(\mathbb {B}_{r}=\{x\in \varUpsilon :\|x\|\leq r\}\), where \(r\geq \frac{J P _{\tilde{a}}+Q_{\tilde{a}}}{1-KP_{\tilde{a}}} \). We consider \(\mathbb {G}\) defined in (11) on \(\mathbb {B}_{r}\). For all \(x\in \mathbb {B}_{r}\) and \(t\in [0,\omega ]\), using Lemmas 3.2 and 3.3, we derive

$$\begin{aligned} \bigl\vert (\mathbb {G}x) (t) \bigr\vert \leq K \Vert x \Vert \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \,ds+J \int _{0}^{\omega } \bigl\vert F(t,s) \bigr\vert \,ds+Q_{\tilde{a}}\leq KP_{\tilde{a}} \Vert x \Vert +JP_{\tilde{a}}+Q _{\tilde{a}}\leq r, \end{aligned}$$

which implies \(\|\mathbb {G}x\|\leq r\). Thus \(\mathbb {G}(B_{r})\subset B_{r}\). In addition, it is easy to see that \(\mathbb {G}\) is continuous and \(\mathbb {G}(\mathbb {B}_{r})\) is pre-compact. By Schauder’s fixed point theorem, we obtain that (1) has at least one \((\omega ,c)\)-periodic solution \(x\in \varPsi _{\omega ,c}\). □

Examples

Example 4.1

We consider the following semilinear impulsive equation:

$$ \textstyle\begin{cases}x'(t)=(\cos 2t) x(t)+ \rho \sin t \cos x(t),\quad t\neq t_{i}, i=1,2, \ldots , \\ \Delta x|_{t=t_{i}}=\frac{1}{2}\sin {\frac{(2i-1)\pi }{2}}x (t _{i}^{-} )+\cos i\pi , \end{cases} $$
(12)

where \(\rho \in \mathbb {R}\), \(t_{i}=\frac{(3i-1)\pi }{6}\), \(\omega =\pi \), \(c=-1\), \(a(t)=\cos 2t\), \(f(t,x)=\rho \sin t\cos x\), \(b_{i}= \frac{1}{2}\sin {\frac{(2i-1)\pi }{2}}\), and \(c_{i}=\cos i\pi \). Clearly, \(t_{i+2}=t_{i}+\pi \), \(b_{i+2}=b_{i}\), \(c_{i+2}=c_{i}\) for all \(i\in \mathbb {N}\), then we obtain \(N=2\), \((A_{1})\) and \((A_{2})\) hold. Since \(W(\omega ,0)=\frac{3}{4}\neq -1=c\), we get \((A_{3})\) holds. Note that \(f(\cdot +\omega ,cx)=f(\cdot +\pi ,-x)=-\rho \sin \cdot \cos x=-f( \cdot ,x)=cf(\cdot ,x)\), we get \((A_{4})\) holds. \(|f(t,x)-f(t,y)| \leq |\rho ||x-y|\), then we get \(L=|\rho |\) and \((A_{5})\) holds. In addition, \(\tilde{a}=1\), \(\tilde{b}=\frac{3}{2}\), \(P_{\tilde{a}}=\frac{18 \pi e^{\pi }}{7}\doteq 186.939334\), and \(Q_{\tilde{a}}= \frac{36e^{ \pi }}{7}\doteq 119.009276\).

Letting \(0<|\rho |<\frac{7}{18\pi e^{\pi }}\doteq 0.005349\), we get \(0< LP_{\tilde{a}}<1\), then all the assumptions of Theorem 3.4 hold. So if \(0<|\rho |<\frac{7}{18\pi e^{\pi }}\), problem (12) has a unique π-antiperiodic solution \(x\in \operatorname{PC}([0,\infty )),\mathbb {R})\).

Since \(|f(t,x)|\leq |\rho |\), we get \(K=0\), \(J=|\rho |\), \((A_{6})\) holds, and \(KP_{\tilde{a}}=0<1\). Then all the assumptions of Theorem 3.5 hold for any \(\rho \in \mathbb {R}\). So (12) has at least one π-antiperiodic solution for any \(\rho \in \mathbb {R}\).

Example 4.2

We consider the following semilinear impulsive equation:

$$ \textstyle\begin{cases} x'(t)=(\sin 2\pi t) x(t)+\rho x(t) \cos (2^{-t}x(t) ),\quad t\neq t_{i}, i=1,2,\ldots , \\ \Delta x|_{t=t_{i}}=x (t_{i}^{-} )+1, \end{cases} $$
(13)

where \(\rho \in \mathbb {R}\), \(t_{i}=\frac{3i-1}{6}\), \(\omega =1\), \(c=2\), \(a(t)=\sin 2\pi t\), \(f(t,x)=\rho x \cos (2^{-t}x)\), \(b_{i}=1\) and \(c_{i}=1\). Clearly, \(t_{i+2}=t_{i}+1\), \(b_{i+2}=b_{i}\), \(c_{i+2}=c_{i}\) for all \(i\in \mathbb {N}\), then we obtain \(N=2\), \((A_{1})\) and \((A_{2})\) hold. Since \(W(\omega ,0)=4\neq 2=c\), we get \((A_{3})\) holds. Note that \(f(\cdot +\omega ,cx)=f(\cdot +1,2x)=2\rho x \cdot \cos (2^{-t}x)=2f(\cdot ,x)=cf(\cdot ,x)\), we get \((A_{4})\) holds. Now \(f(\cdot ,x)\) does not satisfy the Lipschitz condition. Since \(|f(t,x)|\leq |\rho ||x|\), we get \(K=|\rho |\), \(J=0\), and \((A_{6})\) holds. Moreover, \(\tilde{a}=1\), \(\tilde{b}=2\), and \(P_{\tilde{a}}=6e\).

Set \(|\rho |<\frac{1}{6e}\doteq 0.061313\). Then \(KP_{\tilde{a}}<1\). Now all the assumptions of Theorem 3.5 hold. Thus,(13) has at least one \((1,2)\)-periodic solution \(x\in PC([0,\infty )),\mathbb {R})\) if \(|\rho |<\frac{1}{6e}\).

Conclusion

Existence and uniqueness of \((\omega ,c)\)-periodic solutions for homogeneous problem and nonhomogeneous as well as semilinear time varying impulsive differential equations are established. In a forthcoming work, we shall extend the study to \((\omega ,c)\)-periodic solutions for nonlinear impulsive evolution systems in infinite dimensional spaces as follows:

$$ \textstyle\begin{cases} \dot{y}=C(t)y+h(t,y),\quad t\neq \tau _{i}, i\in \mathbb{N},\\ \triangle y\mid _{t=\tau _{i}}=y(\tau _{i}^{+})-y(\tau _{i}^{-})=Dy(\tau _{i}^{-})+d_{i}, \end{cases} $$

where the linear operator \(\{C(t):t\geq 0\}\) generates a strongly continuous evolutionary process \(\{U(t,s),t\geq s\geq 0\}\) on a Banach space X. D is a bounded linear operator and \(d_{i}\in X\). Motivated by [11,12,13,14,15], we shall also consider \((\omega ,c)\)-periodic delay differential equations with non-instantaneous impulses.

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Acknowledgements

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

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This work is partially supported by the National Natural Science Foundation of China (11671339).

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Correspondence to Jin Rong Wang.

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Keywords

  • \((\omega ,c)\)-periodic solutions
  • Impulsive differential equation
  • Existence and uniqueness