1 Introduction
We obtain sufficient conditions for the existence of oscillatory solutions of impulsive linear differential equations of the form
The functions $a(t)>0$ and $b(t)$ are assumed to be piecewise left continuous on $[t_{0},\infty )$ for some $t_{0}\geq 0$ ; $\{a_{k}\}$ , $\{b_{k}\}$ and $\{c_{k}\}$ are real sequences; $\tau _{k+1}> \tau _{k}$ for all $k=1,2,\ldots $ ; $\lim _{k\to \infty }\tau _{k}=\infty $ and $\Delta \varphi (\tau _{k})=\varphi (\tau _{k}^{+})-\varphi (\tau _{k}^{-})$ with $ \varphi (\tau _{k}^{\pm })=\lim _{t\to \tau _{k}^{\pm }}\varphi (t) $ . As the impulse effects contain both the solution and its derivative, they are said to be of mixed type. By separated impulse effects, we mean that the impulse effects contain either only the solution or only the derivative of the solution, for example, $\Delta y+a_{k}y=0$ and $ \Delta a(t)y^{\prime }+c_{k} y^{\prime }=0$ , where $t=\tau _{k}$ .
For the sake of brevity, the notation $\underline {n}(t):=\inf \{k: \tau _{k}\geq t\}$ , $\overline {n}(t):={\rm sup} \{k: \tau _{k}< t\}$ and $\displaystyle \omega _{k}:= ({1-c_{k}/a(\tau _{k})})/({1-a_{k}})$ is used. The following hypotheses are assumed throughout the paper:
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(H1) $1-c_{k}/a(\tau _{k})>0,\ 1-a_{k}>0$ and $b_{k}\leq 0$ , $k=1,2,\ldots $ ;
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(H2) there exists a function $f(t): [t_{0},\infty )\to (0,\infty )$ such that $f^{\prime }(t)$ exists on $[t_{0},\infty )$ and
$$ \begin{align*} g(t):=b(t)+\alpha(t)f^{\prime}(t)+\frac{(\alpha(t)f(t))^{2}}{a(t)}\geq 0,\end{align*} $$where$$ \begin{align*}\alpha(t):=\begin{cases} \displaystyle-\sum\limits_{k={\underline{n}(t_{0})}}^{\overline{n}(t)} \dfrac{b_{k}}{f(\tau_{k})(1-c_{k}/a(\tau_{k}))}\prod\limits_{i=k}^{\overline{n}(t)}{\omega_{i}} &\text{if}\ b_{k}\neq 0,\\[22pt] \displaystyle\prod\limits_{k={\underline{n}(t_{0})}}^{\overline{n}(t)}{\omega_{k}} &\text{if}\ b_{k}= 0. \end{cases}\end{align*} $$
Differential equations containing impulse effects are practical tools to represent many evolutionary processes such as biological models, physical phenomena and engineering problems, and the corresponding theory is quite rich. Their qualitative theory has been investigated deeply by many researchers (see the famous books [Reference Bainov and Simeonov4, Reference Lakshmikantham, Bainov and Simeonov6]). It is well known that impulse effects can cause radical changes in the structure of the solution of a differential equation. For example, a nonoscillatory unforced differential equation may turn out to be oscillatory under impulsive conditions [Reference Akgöl and Zafer2, Reference Luo and Shen7, Reference Sugie9–Reference Wen, Zeng, Peng and Huang11]. Since it is not easy to make a prediction, it is crucial to study the long-time behaviour of impulsive differential equations, in particular, their oscillatory properties. We refer to [Reference Agarwal, Karakoc and Zafer1] for an excellent survey on the oscillation of differential equations under impulse effects and the papers [Reference Bainov, Domshlak and Simeonov3, Reference Özbekler and Zafer8, Reference Sugie9] regarding self-adjoint impulsive differential equations with continuous solutions, namely, equations derived by setting $a_{k}=0=c_{k}$ in (1.1). There are only a few studies of their counterparts having discontinuous solutions (see [Reference Guo and Xu5, Reference Luo and Shen7], where differential equations with separated impulse effects were considered). To the best of our knowledge, the only paper dealing with oscillation of equations with mixed impulse effects of the form (1.1) is [Reference Akgöl and Zafer2], in which a Leighton-type oscillation theorem is produced.
2 Main results
We start with some auxiliary lemmas.
Lemma 2.1. Let
where $y(t)$ is a solution of (1.1). Then, $x(t)$ is a solution of the differential equation with separated impulse effects:
Proof. Let $y(t)$ be a solution of (1.1). For $t\neq \tau _{k}$ ,
and
which clearly implies that
For $t=\tau _{k}$ , $k=1,2,\ldots $ , it is easy to see that
Noting that
we see that
Thus, from (2.3)–(2.5), we conclude that $x(t)$ is a solution of (2.2).
Lemma 2.2. Let $x(t)$ be a nonoscillatory solution of (2.2). If
then $x(t)x^{\prime }(t)$ is ultimately negative.
Proof. Suppose that $x(t)$ is ultimately positive. First, we will show that $x^{\prime }(t)$ is nonoscillatory. We assume on the contrary that $x^{\prime }(t)$ is oscillatory. Then, there is some $k\in \mathbb {N}$ and $t_{a}\in (\tau _{k},\tau _{k+1}]$ such that $x^{\prime }(t_{a})=0$ . Thus, in view of (2.2),
which implies that there is some interval $(t_{a}, t_{a}+\delta )$ , $\delta>0$ , in which $x^{\prime }(t)$ is decreasing. Hence,
Now, assume that $t_{a}$ is the first root and $x^{\prime }$ has another root in the same interval, that is, there is some $t_{b}\in (t_{a}, \tau _{k+1})$ such that $x^{\prime }(t_{b})=0$ . From (2.8), this implies that $x^{\prime \prime }(t_{b})\geq 0$ . However, from (2.7), we see that $a(t_{b})x^{\prime \prime }(t_{b})<0$ which leads to a contradiction. Hence, $x^{\prime }(t)$ cannot have a root in $(t_{a}, \tau _{k+1})$ , that is, $x^{\prime }(t)<0$ there. This implies that
If we again suppose that there is some $t_{c}$ such that $x^{\prime }(t_{c})=0$ , from (2.7), we obtain $x^{\prime \prime }(t_{c})<0$ which implies $x^{\prime }(t)<0$ , $t\in (t_{c}-\delta , t_{c}+\delta )$ , contradicting $x^{\prime }(t_{c})=0$ . Hence, $x^{\prime }(t)<0$ on $(\tau _{k+1}, \tau _{k+2}]$ . By similar arguments, it can be seen that
Fix some $T\geq t_{0}$ and let $\tau _{k}\geq T$ . If $x^{\prime }(t)<0$ on $(\tau _{k}, \tau _{k+1}]$ , from (2.9), $x^{\prime }(\tau _{k+1}^{+})<0$ and, by the above discussion, $x^{\prime }(t)<0$ on $(\tau _{k+i}, \tau _{k+i+1}]$ , for all $i\in \mathbb {N}$ . Thus, $x^{\prime }(t)<0$ on $[T,\infty )$ .
Conversely, if $x^{\prime }(t)>0$ on $(\tau _{k}, \tau _{k+1}]$ , then $x^{\prime }(\tau _{k+1}^{+})(1-{c_{k+1}}/{a(\tau _{k+1})})x^{\prime }(\tau _{k+1})>0$ . However, we know that $x^{\prime }(t)$ has no root in $(\tau _{k+i}, \tau _{k+i+1}]$ , for all $n\in \mathbb {N}$ . Thus, $x^{\prime }(t)>0$ on $[T,\infty )$ . Hence, $x^{\prime }(t)$ is nonoscillatory.
Our next aim is to show that $x^{\prime }(t)$ is ultimately negative, that is, there is some $T_{*}\geq t_{0}$ such that $x^{\prime }(t)<0$ for $t\geq T_{*}$ . Suppose on the contrary, there exists $k\in \mathbb {N}$ such that $x^{\prime }(\tau _{k})>0$ for $\tau _{k}\geq T_{*}$ . Then,
and so $x^{\prime }(t)>0$ for $t\geq \tau _{k}$ . From (H1), $\alpha (t)> 0$ on $[t_{0},\infty )$ . Thus, we can write
which shows that $a(t)x^{\prime }(t)$ is decreasing on each interval $[\tau _{k+i-1},\tau _{k+i}), \ i\in \mathbb {N}$ . Now, we need to prove that, for $n\geq 1$ ,
Integrating (2.10) on $(\tau _{i}, \tau _{i+1}]$ ,
Since $x^{\prime }(t)>0$ for $t\geq \tau _{k}$ , it follows that x is increasing on $(\tau _{k}, \tau _{k+1}]$ . Hence,
Integrating (2.10) on $(\tau _{k+1}, \tau _{k+2}]$ and using (2.12),
Now, suppose that (2.11) holds for $n=N$ . Then, for $t\in (\tau _{k+N}, \tau _{k+N+1}]$ ,
where, in the last line, the estimate $x(\tau _{k+N})\geq x(\tau _{k+N-1}^{+})=(1-a_{k+N-1})x(\tau _{k+N-1})$ is used N times. Thus, by induction on n, we see that (2.11) holds for any $n\geq 1$ .
If we take the limit of both sides of (2.11) as $n\to \infty $ , from (2.6), we see that $x^{\prime }(\tau _{k+n})<0$ for sufficiently large values of n. However, this contradicts the assumption that $x^{\prime }(t)>0$ for $t\geq \tau _{k}$ . Hence, $x^{\prime }(\tau _{k})<0$ for $\tau _{k}\geq T_{*}$ . Since $x^{\prime }(t)$ has a constant sign for $t\geq \tau _{k}$ , it follows that $x^{\prime }(t)<0$ for all $t\neq \tau _{k+n}$ , $t\geq T_{*}$ .
If $x(t)$ is ultimately negative, by repeating all the steps of the proof, it can be shown that there is some $T^{*}\geq t_{0}$ such that $x^{\prime }(t)>0$ for $t\geq T^{*}$ . Thus, the proof is complete.
Theorem 2.3. Suppose that (2.6) holds, and
where
Then, (2.2) is oscillatory.
Proof. Suppose on the contrary that $x(t)$ is a nonoscillatory solution of (2.2). If we assume $x(t)$ is ultimately positive, namely, there exists some $T\geq t_{0}$ such that $x(t)>0$ for $t\geq T$ , in view of Lemma 2.2, $x^{\prime }(t)<0$ for $t\geq T$ and $t\neq \tau _{k}$ . Since $g(t)>0$ , from (2.2),
that is,
Define ${\tau _{k}}:=\min \{\tau _{j}: \tau _{j}\geq T\}$ . For $t\in (\tau _{k}, \tau _{k+1}]$ , integration of (2.14) yields
Since $x^{\prime }(t)<0$ for $t\geq T$ , this implies that
Setting $t=\tau _{k+1}$ ,
Integrating (2.15) on $(\tau _{k}, \tau _{k+1}]$ ,
Now, we apply the same procedure on $(\tau _{k+1}, \tau _{k+2}]$ and similarly obtain
Observe that $\mu (\tau _{k},\tau _{k+1})\mu (\tau _{k+1},s)=\mu (\tau _ k,s) $ . Thus, using (2.16) and (2.17) in (2.18),
Suppose that
for $n=N$ . Then, in a similar way to the proof of (2.18), it can be shown that (2.19) holds for $n=N+1$ . Thus, by induction, the inequality (2.19) is true for any $n\geq 1$ .
Applying (2.13) in (2.19) leads to the contradiction that $x(\tau _{n})<0$ for sufficiently large values of n. Hence, $x(t)$ is oscillatory.
Theorem 2.4. Suppose that (2.6) holds and that there exists a continuous function $h(t): [t_{0},\infty )\to (0,\infty )$ such that $h^{\prime }(t)$ exists on $[t_{0},\infty )$ and $a(t)h^{\prime }(t)\geq 2\alpha (t)f(t)h(t)$ . If
and
then (2.2) is oscillatory.
Proof. Suppose on the contrary that $x(t)$ is a nonoscillatory solution of (2.2). We may assume $x(t)>0$ for $t\geq T$ for some $T\geq t_{0}$ . Then, by Lemma 2.2, $x^{\prime }(t)<0$ for $t\geq T$ . Define ${\tau _{k}}:=\min \{\tau _{j}: \tau _{j}\geq T\}$ . Multiplying the first line of (2.2) by $h(t)/x(t)$ , and then integrating it on $(\tau _{k},t]$ , for $t\in (\tau _{k}, \tau _{k+1}]$ ,
which implies that
and so
For $t\in (\tau _{k+1}, \tau _{k+2}]$ , similarly, we can show
The last inequality holds for $t=\tau _{k+2}$ . Using similar arguments and applying induction on n, it is not hard to prove that for $n\geq 1$ ,
However, from Lemma 2.2, $x(t)x^{\prime }(t)<0$ . So, using (2.20),
Now, from (2.22),
and from (2.23), this implies that there is a sufficiently large $\tau _{\ell }$ so that
for $t\geq \tau _{\ell }$ . Hence,
Since $x^{\prime }(t)<0$ ,
Integrating the last inequality on $(\tau _{\ell },t]$ yields
From (2.24), it follows that
Now, we integrate the last expression on $(\tau _{\ell },\tau _{\ell +1}]$ to obtain
Thus, using the hypothesis (2.21), it is easy to see
which leads to a contradiction because of the assumption that $x(t)$ is ultimately positive. Hence, $x(t)$ is oscillatory.
Now, we can easily establish the following oscillation criteria for (1.1).
In view of (2.1) and Lemma 2.1, it can be seen that Theorems 2.5 and 2.6 follow directly from Theorems 2.3 and 2.4, respectively.
3 Examples
In this section, we describe some examples to illustrate our results. For each example, the graphs show the discontinuities in a small interval and the oscillating behaviour on a larger interval.
Example 3.1. Consider the impulsive differential equation
Let $t_{0}=2$ . Clearly,
and so, $1-a_{k}=(k+1)/(2(k+2))>0$ and $1-c_{k}/a(\tau _{k})= 1/2$ . Thus, (H1) holds. If we choose $f(t)=3/(2t^{3}(t+1))$ , from $\omega _{k}=(k+2)/(k+1)$ ,
For $t\in (i, i+1]$ , $t\geq 2$ , we have $\overline {n}(t)=i+1$ and $\alpha (t)=i/3$ , which implies that
Thus, (H2) also holds and
If we take the limit of both sides, we see that (2.6) is satisfied. However, since $\alpha (t)<{t}/{3}$ for $t\geq 2$ , we have the estimate
Hence, from
we see that (2.13) also holds. Thus, by Theorem 2.5, (3.1) is oscillatory. This conclusion is illustrated in Figure 1.
Example 3.2. Consider the impulsive differential equation
Let $t_{0}=2$ . Clearly,
Thus, (H1) holds. Since $\omega _{k}=1$ , by choosing $f(t)=e^{-t}$ , we get
From $2/5 < e^{k}/(1+e^{k})< 1$ , we may write ${1}/{10t}<\alpha (t)<{1}/{2}$ . Then
so that
Taking the limit of both sides, it is easily seen that (2.6) holds.
Now, if we take $h(t)=e^{t}$ , we can write $2\alpha (t)f(t)h(t)<1=a(t)h^{\prime }(t)$ , and
which shows that (2.21) holds. Finally, to check the hypothesis (2.20), we write
Clearly, the last expression tends to infinity as $n\to \infty $ . Hence, by means of Theorem 2.6, (3.2) is oscillatory. The oscillation behaviour can also be seen in Figure 2.
4 Concluding remarks
The following remark demonstrates the novelty of Theorem 2.5 and hence also Theorem 2.3.
Remark 4.1. As far as we know, the only paper that deals with the oscillation of (1.1) is [Reference Akgöl and Zafer2]. If we attempt to apply the Leighton-type theorem [Reference Akgöl and Zafer2, Theorem 2.1] to (3.1), we compute
which is finite. Hence, the hypothesis of Theorem 2.1 in [Reference Akgöl and Zafer2] is not satisfied. As we have shown in Example 3.1, our result shows that this system oscillates.
Finally, the next remark shows the usefulness of Theorem 2.6 as an alternative to Theorem 2.5.
Remark 4.2. In Example 3.2, from (3.3), we have $\alpha (t)=c$ , where $1/5<c<1/2$ . Thus,
where c is a suitable positive constant. This implies that
Thus, the hypothesis (2.13) does not hold, and so Theorem 2.5 cannot be applied to the system (3.2).