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


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.


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:

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

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.


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

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

is given by

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

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

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


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

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

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

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

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


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:

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

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

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


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


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

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

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

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


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

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

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


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

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

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


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

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


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 \(f_{0}=\max_{t\in [0,\omega ]}|f(t,0)|\).


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

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

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 \(x\in \varPsi _{\omega ,c}\).


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 \(x\in \varPsi _{\omega ,c}\). □


Example 4.1

We consider the following semilinear impulsive equation:

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

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:

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

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 \(|\rho |<\frac{1}{6e}\).


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


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

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Wang, J.R., Ren, L. & Zhou, Y. \((\omega ,c)\)-Periodic solutions for time varying impulsive differential equations. Adv Differ Equ 2019, 259 (2019).

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