Abstract
We study the oscillation of a first-order linear delay differential equation. A new technique is developed and used to obtain new oscillatory criteria for differential equation with non-monotone delay. Some of these results can improve many previous works. An example is introduced to illustrate the effectiveness and applicability of our results.
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1 Introduction
Consider the first-order linear delay differential equation
where \(p, \tau \in C([t_{0},\infty ),[0,\infty ))\), \(\tau (t)\leq t\), such that \(\lim_{t \rightarrow \infty } \tau (t)=\infty \).
By a solution of Eq. (1) we mean a continuous function \(x(t)\) on \([t_{*}, t_{0}]\), \(t_{*}=\inf_{t \geq t_{0}}\tau (t)\), continuously differentiable on \([t_{0}, \infty )\), which satisfies Eq. (1) for all \(t \in (t_{0}, \infty )\). As is customary, any solution \(x(t)\) of Eq. (1) is called oscillatory if it has arbitrarily large zeros; otherwise it is called non-oscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory; otherwise it is called non-oscillatory.
Throughout this work, we assume that the function \(\delta (t)\) is non-decreasing, continuous, and such that \(\tau (t) \leq \delta (t) \leq t\) for all \(t \geq t_{1}\) and some \(t_{1}\geq t_{0}\), and \(\delta ^{n}(t)\) stands for the n-fold composition of \(\delta (t)\). Let
and
Also, the notation \(\lambda (\xi )\) refers to the smaller real root of the transcendental equation \(\lambda = {\mathrm{e}}^{\lambda \xi }\). Finally, let
The work of Myshkis [27] can be considered as the first systematic study for the oscillation character of the class of the delay differential equations. Recently, these equations have attracted the interest of several researchers, see [1–31]. A huge number of sufficient conditions for the oscillation of Eq. (1) have been obtained. For example, the criteria
were derived respectively in [26] and [23]. In fact, the threshold \(\frac{1}{e}\) is of great importance for the oscillation problem of Eq. (1). Since, according to [23], if
then there exists a non-oscillatory solution of Eq. (1). Indeed, the oscillation problem of Eq. (1) is completely solved when the coefficient and the delay functions are constants p and τ, respectively. In this case all solutions are oscillatory if and only if \(p \tau > \frac{1}{e}\); but in the non-autonomous case, the situation is totally different. There is a gap between \(\frac{1}{e}\) and 1, when the limit \(\lim_{t \rightarrow \infty } \int _{\tau (t)}^{t}p(u)\,du\) does not exist. Many works have been done to fill this gap in the case of nondecreasing delays, \(L\leq 1\) and \(0\leq k\leq \frac{1}{e}\), see [22, 28] and the references therein. The assumption that the delay is nondecreasing plays a major role in these works. Koplatadze and Kvinikadze [24] showed that many oscillatory criteria can be generalized to equations with non-monotone delay, using a nondecreasing function \(\psi (t)\) that is defined as in (2). Since then, several mathematicians have developed and introduced many techniques to study the oscillatory behaviour of these equations. In the following, we show some of these results:
Koplatadze and Kvinikadze [24] improved condition (3) and introduced the sufficient condition
where
Braverman and Karpuz [4] improved (5) with \(n=2\), and obtained
Stavroulakis [29] improved the preceding condition and established
Infante et al. [21] improved (5) with \(n=3\), and (6), and proved that Eq. (1) is oscillatory if
or
El-Morshedy and Attia [17] showed that Eq. (1) is oscillatory, if for some \(n \in \mathbb{N}\),
where
and
In a series of papers, Chatzarakis et al. obtained many oscillatory results for Eq. (1), see [5–15]. For example, Chatzarakis [5] improved (5) with \(n=3\), and (6), and obtained the oscillatory condition
where
Bereketoglu et al. [3] improved (11), and proved that Eq. (1) oscillates if there exists \(n\in \mathbb{N}\) such that
where
Very recently, Attia, El-Morshedy and Stavroulakis [2] improved (9) and (12), and introduced the following criterion:
for some \(n,m\in \mathbb{N}\), where
and
2 Results
Let \(x(t)\) be an eventually positive solution of Eq. (1). Then
Therefore the following lemmas are applicable to \(x(t)\).
Lemma 2.1
([19, Lemma 2.1.2])
Let \(0< k\leq \frac{1}{\mathrm{e}} \). Then
Lemma 2.2
([31])
Let \(k\leq \frac{1}{\mathrm{e}} \). Then
In the sequel, we define the sequences \(\{Q_{n}(t)\}_{n=0}^{\infty }\) and \(\{\beta _{n}(t)\}_{n=1}^{\infty }\) as follows:
and
The following lemma is essential in order to obtain the main results.
Lemma 2.3
Let \(n \in \{0,1,2,\dots \}\) and \(k\leq \frac{1}{\mathrm{e}}\). Then \(\beta _{n}<1\), and
where \(x(t)\) is a positive solution of Eq. (1).
Proof
Since \(x(t)\) is a positive solution of Eq. (1), then \(x(t)\) is eventually non-increasing for all sufficiently large t. Therefore
If \(k^{*}>0\), then Lemma 2.1 implies, for sufficiently small \(\epsilon >0\), that
This inequality and the non-increasing nature of \(x(t)\) lead to
On the other hand, dividing Eq. (1) by \(x(t)\), integrating from s to t, \(s \leq t\), we get
Integrating Eq. (1) from \(\delta (t)\) to t, we obtain
Since \(\tau (s_{1})\leq \delta (t)\) for \(s_{1}\leq t\), (18) and (19) give
This equation and (17) lead to
Consequently,
Again, integrating Eq. (1) form \(\tau (s_{1})\) to \(\delta (t)\), \(s_{1} \leq t\), we obtain
Substituting into (19), we have
From this and (18), we obtain
Therefore, it follows from (21) that
Therefore
By simple induction, we get
for \(n=3,4,\dots\). Since
we get
Substituting into (22), we obtain
□
Theorem 2.1
Let \(n \in \{0,1,2,\dots \}\). If \(\beta _{i}\geq 1\), \(i=1,2,\dots,n\), or
then Eq. (1) is oscillatory.
Proof
Assume that Eq. (1) has a non-oscillatory solution \(x(t)\). Without loss of generality, let \(x(t)\) be an eventually positive solution. By using (20), we obtain
Using Lemma 2.2, it follows that
From this and Lemma 2.2, we obtain
This contradicts (23). □
Theorem 2.2
Assume that \(n \in \{0,1,2,\dots \}\) and \(\delta (t)\) is a strictly increasing for \(t \geq t_{1}\). If \(\beta _{i} \geq 1\), \(i=1,2,\dots,n\), or
then Eq. (1) is oscillatory.
Proof
Assume that there exists a positive solution \(x(t)\) of Eq. (1). From the proof of Theorem 2.1, we see that
Integrating Eq. (1) from t to \(\delta ^{-1}(t)\), we have
Since \(t \geq \tau (s_{1})\) for \(\delta ^{-1}(t) \geq s_{1}\), one has
From this, (26) and the non-increasing nature of \(x(t)\), we have
which in turn leads to
By substituting into (25), we have
Taking the upper limits of both sides as t goes to ∞, we obtain a contradiction with (24). □
Theorem 2.3
Let \(n \in \{0,1,2,\dots \}\). If \(\beta _{i}\geq 1\), \(i=1,2,\dots,n\), or
then Eq. (1) is oscillatory.
Proof
As before, let \(x(t)\) be a positive solution of Eq. (1). Then
By using (17), from the proof of Theorem 2.3, we have
Integrating Eq. (1) form \(\delta (s_{1})\) to \(\delta (t)\), \(s_{1} \leq t\), we get
Since \(\tau (s_{2}) \leq \delta (s_{1}) \) for \(s_{2} \leq s_{1}\), it follows from (17) and (30) that
that is,
From this, (29) and Lemma 2.3, we have
This, together with Lemma 2.2, implies that
This is a contradiction. □
The proof of the following result is the same as those of Theorems 2.1 and 2.2, and hence it will be omitted.
Theorem 2.4
Let \(n \in \{0,1,2,\dots \}\) and \(\delta (t)\) be strictly increasing for \(t \geq t_{1}\). If \(\beta _{i} \geq 1\), \(i=1,2,\dots,n\), or
then Eq. (1) is oscillatory.
Remark 2.1
-
The criterion (23) improves conditions (5) with \(n=2\), (6), (7), and (9) when \(k=0\).
-
Lemma 2.3 can be used to improve and generalize the oscillation results of [30, Lemma 2.1], [16, Theorem 2.6] and [18, Lemma 2.5].
Example 2.1
Consider the first order delay differential equation
where
and
where \(i, \mu _{i}\in \mathbb{N}\), \(\gamma =0.4195\), \(\xi _{1} \geq 0\), \(\mu _{i} > 1+\xi _{i}\), and \(\xi _{i+1}>\mu _{i}+9\) such that \(\lim_{i\rightarrow \infty } \xi _{i}=\infty \). Since
one has \(k=k^{*}=0\), and it follows that conditions (4), (9) and (13) fail to apply. Let \(\delta (t)=t-1\) and
Since
Then
and
where \(\mu _{i}+3 \leq v \leq \mu _{i}+5\), also
and
Then
also
Therefore \(I(\mu _{i}+8)>1.0002\). As a consequence, \(\limsup_{t\rightarrow \infty }I(t)>1\), and hence Theorem 2.2 implies that Eq. (32) is oscillatory. However, if we assume that \(\delta (t)=\psi (t)\) (which is defined as in (2)), then
Consequently, \(\int _{\tau (t)}^{t} p(s) \,ds\leq (1+\zeta )\gamma \). Then, \(\Phi _{8}(t)< 3.363136\), and it follows that
Therefore, none of conditions (5) with \(n=8\) or (6)−(8) hold. Also, since
condition (10) with \(n=2\) fails to apply. Finally,
hence, condition (12) with \(n=3\) is not satisfied.
3 Conclusion
In this work, we obtained new oscillatory criteria for Eq. (1), using improved lower bounds for the quantity \(\frac{x(\tau (u))}{x(t)}\), where \(x(t)\) is any positive solution of Eq. (1). Some of the obtained results improve many previous works. Finally, we introduced an example to demonstrate the simplicity and efficiency of some of our results, especially when \(k=0\).
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Acknowledgements
The author would like to express his deep thanks to Prof. Hassan. A. El-Morshedy for his help and suggestions . Also the author would like to express his gratitude to the anonymous referees for their help in improving the manuscript.
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This project was supported by the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University under the research project 2020/01/16630.
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Abbreviations
\(\delta ^{n}(t)\) stands for the n-fold composition of \(\delta (t)\); \(\mathbb{N}\) denotes the set of natural numbers.
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Attia, E.R. Oscillation tests for first-order linear differential equations with non-monotone delays. Adv Differ Equ 2021, 41 (2021). https://doi.org/10.1186/s13662-020-03209-4
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DOI: https://doi.org/10.1186/s13662-020-03209-4