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

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.

1 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.

2 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 $x(t_{i}+\omega )=cx(t_{i})$. □

3 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 $x\in \varPsi _{\omega ,c}$. □

4 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 $|\rho |<\frac{1}{6e}$.

5 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.