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On oscillation of higherorder advanced trinomial differential equations
Advances in Difference Equations volume 2021, Article number: 144 (2021)
Abstract
We study the oscillatory property of the higherorder trinomial differential equation with advanced effects
Suppose that all solutions of the corresponding (\(n1\))thorder twoterm differential equation
are nonoscillatory. In order to supplement the research in the theory of oscillation proposed by (Džurina et al. in Electron. J. Differ. Equ. 2015:70, 2015), two types of clearly confirmable criteria for oscillatory behavior of the investigated equation are obtained. Some examples are offered to describe our main results.
1 Introduction
This paper focuses on the oscillatory behavior of solutions to a higherorder trinomial differential equation with advanced effects
for all \(t\geq t_{0}\). Throughout the remaining parts of this article, we need to establish some hypotheses as follows:
 (\(H_{1}\)):

\(p(t)\) and \(q(t)\in C([t_{0},\infty ))\), \(p(t)\) is nonnegative, \(q(t)\) is positive;
 (\(H_{2}\)):

\(\sigma (t)\in C([t_{0},\infty ))\), \(\sigma (t)\geq t\).
The oscillatory behavior of ordinary differential equations (ODEs) is one of the significant branching problems of differential equations. The oscillatory problems to the wings of the plane can be modeled by the oscillatory problems of ODEs. As a matter of fact, differential equations with deviating arguments have numerous applications in engineering and natural sciences (see [9, 12, 26] for more details). By a solution of Eq. (1.1), we mean a nontrivial real function \(x(t)\), \(x\in C^{1}([T_{x},\infty ),\mathbb{R})\), \(T_{x}\geq t_{0}\), which satisfies (1.1) on \([T_{x},\infty )\). We investigate only proper solutions \(x(t)\) to Eq. (1.1) with the property \(\sup \{x(t):T< t<\infty \}>0\), for any \(T\geq T_{x}\) and existing on some halfline \([T_{x},\infty )\). Tacitly, we suppose that for Eq. (1.1) there exists such a solution. As is customary, if a solution of Eq. (1.1) possesses arbitrarily many zeros on the interval \([T_{x},\infty )\), we called it an oscillatory solution. Otherwise it is said to be nonoscillatory. If all solutions to a higherorder functional and differential equation, such as Eq. (1.1), are oscillatory, it is common to call it an oscillatory equation.
With the social development and the progress of all fields of modern technology and science, such as economics, aerospace and modern physics, delay differential equations (DDEs) have received more and more consideration in the past decades. It is well known that DDEs involve the dependency of the previous state, which can help us to predict the future state with efficiency and reliability. Meanwhile, many qualitative properties such as boundedness, stability or periodicity can be explained. If we incorporate the delay effect into models, it will play a significant role when representing time taken to finish some veiled procedure. On the contrary, advanced differential equations (ADEs), different from genetic systems, can also be applied in almost all realworld areas. Population dynamics in mathematical biology, mechanical control in engineering or problems in economics are instances of areas where we can discover applications of such differential equations [15].
Regrettably, many oscillation works in the field of ODEs are considered only to finite order because several systems in engineering are naturally described by ODEs with finite order [8, 18]. Thus, the main goal of researchers in the area of ODEs is to obtain some qualitative theories as regards those equations, such as existence, uniqueness, boundedness, periodicity, and stability. Meanwhile, some criteria for the asymptotic behavior and oscillation of such equations are also important for the investigation of ODEs; see, for instance, [4, 18, 27]. We suggest the reader to consult the outstanding treatises of Elias [14], Chanturia and Kiguradze [19] and Swanson [29] for scientific research of the most important efforts made in this theory. However, several researchers, such as the authors in [2–5, 17, 25], in the field of mathematics have obtained some sufficient conditions to guarantee that all solutions of the nthorder equations
are oscillatory. In 2015 and 2017, Baculíková, Džurina and Jadlovská [11, 13] discussed the oscillatory behaviors of solutions of the two equations
and
respectively. In addition, [30, 31] were concerned with the oscillatory properties of solutions to the Swift–Hohenberg differential equation (1.3). At the end of [13], they proposed an interesting problem for further investigation: how these equations can be higherorder trinomial delay equations of the form
and
respectively. Recently, the authors in [1] have established some improved and generalized criteria for the evenorder ADE
The study of oscillation theorems to trinomial fourthorder equations without or with deviating effects is investigated in [7, 8]. Other authors were concerned with the higherorder cases; see, among others, [1, 5, 17, 21, 27]. In particular, some attempts were made for an analogue of (1.3), of the form
In 2014, Liang [27] has investigated oscillation and asymptotic properties of Eq. (1.8). He deduced some sufficient conditions to guarantee all solutions to tend to zero as \(t\rightarrow \infty \) or to oscillate by using a Philostype integral averaging technique and a generalized Riccati transformation. On the other hand, no references about the oscillation of differential equations as Eq. (1.1) can be found. The main reason of this phenomenon is the increasing of the difference between the order of the two highest derivatives. In the meantime, we must recognize the complicatedness of the structure of nonoscillatory solutions to Eq. (1.1). If we pay attention to the firstorder derivative of a nonoscillatory solution of Eq. (1.1), we can discover that it can be either negative or positive or even it may oscillate.
In this article, we focus on the asymptotic properties of the nthorder trinomial advanced linear differential equation
By deducing some novel comparison criteria together with integral criteria for Eq. (1.1), we establish some sufficient conditions to fill the holes in the oscillation theory of ODEs. We note that Eq. (1.1) with \(p(t)=0\) is exactly Eq. (1.2), Eq. (1.1) with \(n=4\) is exactly Eq. (1.3) with advanced argument. Thus, we argue that it will be useful and interesting to consider the oscillatory behavior of Eq. (1.1) because it can extend the former investigations and will offer a profitable new breakthrough for the oscillatory behavior of ODEs widely applied in the domain of economics, technology and ecology.
This article is organized into six sections. We propose some preliminary lemmas and definitions that are used in the proof of our main theorems in Sect. 2. In Sect. 3, we establish some useful preliminary results which will be applied in our main theorems. In Sect. 4, a generalized comparison criteria for the oscillation of Eq. (1.1) is deduced. Two examples are provided to check the efficiency of our main results. Several innovative integral criteria are added in Sect. 5 to provide several verifiable and calculable oscillation criteria for the system of Eq. (1.1). Finally, we propose some conclusions in Sect. 6.
2 Preliminaries
As we have proposed before, this section will introduce some notations applied in the article, discuss some preliminaries which are essential in the proofs of our main results and state the assumptions imposed on Eq. (1.1). Then we have the following three lemmas proposed by Kiguradze and Chaturia [19].
Lemma 1
([19], Corollary 2.8)
Suppose that either
or
Then the higherorder twoterm differential equation \(x^{(n)}(t)+p(t)x(t)=0\) is oscillatory, where \(m_{*n}\) and \(m_{n}^{*}\) stand for, respectively, the smallest local maxima of the two ndegree polynomials
and
Lemma 2
([19], Corollary 2.8)
Assume that \(t_{0}\geq 0\) and
for \(t\geq t_{0}\), then the higherorder twoterm differential equation \(x^{(n)}(t)+p(t)x(t)=0\) is nonoscillatory.
Lemma 3
([19])
Let \(x(t)\in C^{n}([t_{0},\infty ),\mathbb{R}^{+})\). If \(x^{(n)}(t)\) is eventually of one sign for all large t then there exist a \(t_{x}\geq t_{0}\) and an integer ℓ, \(0\leq \ell \leq n\) with \(n+\ell \) even for \(x^{(n)}(t)\geq 0\), or \(n+\ell \) odd for \(x(n)(t)\leq 0\) such that \(\ell >0\) implies that \(x(k)(t)>0\), for \(t\geq t_{x}\), \(k=0,1,\ldots ,\ell 1\), and \(\ell \leq n1\) implies that \((1)^{\ell +k}x^{(k)}(t)>0\), for \(t\geq t_{x}\), \(k=\ell ,\ell +1,\ldots ,n1\).
Definition 1
([19])
Suppose that all nontrivial solutions of Eq. (1.1) for n odd either are oscillatory or satisfy the condition
for \(t\geq t_{0}\), where \(t_{0}\) is a positive number depending on the solution, and for n even are oscillatory. Then Eq. (1.1) possesses property A.
Definition 2
([19], Definition 13.1)
Suppose that \(x(t)\), defined on \([t_{1},+\infty )\subset [t_{0},+\infty )\), represents a nontrivial solution of Eq. (1.1). If it satisfies, for \(t\geq t_{0}\) and \(i=0,1,\ldots ,n1\),
then we term it a Kneser solution.
Suppose that \(\tau (t)< t\), \(\lim_{t\rightarrow \infty }\tau (t)=+\infty \) and \(p(t)\), \(\tau (t)\in C[\mathbb{R}^{+},\mathbb{R}^{+}]\). With retarded argument, the remaining part of this section will investigate the oscillatory property of solutions to the following twoterm differential inequalities and equation:
and
It is significant to study that the oscillation of (2.1)–(2.3) holds for higherorder differential equations or not. We derive the following lemma based on Ladas [22] (see also [10, 20, 23, 28]).
Lemma 4
Suppose that
Then

(i)
the eventually positive solutions of (2.1) are nonexistent;

(ii)
the eventually negative solutions of (2.2) are nonexistent;

(iii)
the nonoscillatory solutions of (2.3) are nonexistent.
There are two important lemmas proposed by Fukagai and Kusano [16], we recall them.
Lemma 5
([16])
Suppose that \(\sigma (t)\geq t\), \(\sigma (t),p(t)\in C[\mathbb{R}^{+},\mathbb{R}^{+}]\) and
Then \(0\leq \operatorname{sgn}(x(t))x'(t)p(t)x(\sigma (t))\) is oscillatory.
Lemma 6
([16], Theorem 1)
Assume that \(\sigma (t)\geq t\) and \(p(t)\leq 0\) for \(t\geq t_{0}\). If
then the equation
is oscillatory. In addition, if,
for all large enough t, then Eq. (2.5) has an eventually nonoscillatory solution.
3 Preliminary results
We will introduce and deduce some preliminary results specific to higherorder differential equations in this section, which are important to the proof of our main theorems. At the beginning, we devote our study to the decomposition of the positive solutions of Eq. (1.1). Based on the theory of disconjugate operators, all positive solutions of Eq. (1.1) have been decomposed by supposing that \(y^{(n1)}(t)+p(t)y(t)=0\) is nonoscillatory. We argue that \(x(t)\) of Eq. (1.1), a nonoscillatory solution, is a Kneser solution if it can only obey the following condition: \(0>x'(t)x(t)\).
We recall a significant lemma proposed by Kiguradze and Chaturia [19] with a modification.
Theorem 1
Assume that all solutions of
are nonoscillatory. Then every positive solution \(x(t)\) of Eq. (1.1) (for n even) satisfies one of the following:
and every positive solution x(t) of Eq. (1.1) (for n odd) satisfies one of the following:
\(\mathcal{N}_{0}\) (Kneser solution) satisfies \(x(t)x'(t)<0\).
Proof
Suppose that \(x(t)>0\) is a nonoscillatory solution of Eq. (1.1). Without loss of generality, we can suppose that \(y(t)>0\) for \(t\geq t_{0}\). Due to \(p(t)\geq 0\), we have \(y^{(n1)}(t) \leq 0\). It is obvious that \(y(t)=x'(t)\) is a solution of
Based on Lemma 3, the situation of \(\mathcal{N}_{1},\mathcal{N}_{2},\ldots ,\mathcal{N}_{n1}\) is obvious. In what follows we only need to consider the situation of \(\mathcal{N}_{0}\). We denote
For \(t\geq t_{1}\), we set \(F(t,0,0)=0\). We derive the existence result to \(\mathcal{N}_{0}\) by applying a similar discussion to that in the proof of Theorem 13.1 of [19]. Thus, it is omitted. □
Remark 1
That all solutions of \(y^{(n1)}+py=0\) are nonoscillatory is equivalent to saying that this equation is eventually disconjugate.
We deduce the following corollary in view of the nonoscillation criterion of Eq. (3.1).
Corollary 1
Suppose that
and for Eq. (1.1) the positive solution class \(\mathcal{N}_{0}\) is an empty set. Then the nonoscillatory set \(\mathcal{N}_{\mathrm{even}}\) for Eq. (1.1) (for n even) has the following decomposition:
and the nonoscillatory set \(\mathcal{N}_{\mathrm{odd}}\) for Eq. (1.1) (for n odd) has the following decomposition:
Theorem 2
Suppose that (3.2) and
hold. If \(x(t)\in \mathcal{N}_{0}\) is a positive solution of (1.1), then
and, furthermore, \(x(t)\) is decreasing.
Proof
We suppose that Eq. (1.1) has a positive solution \(x(t)\in \mathcal{N}_{0}\). We argue that (\(P_{0}\)) means \(\lim_{t\rightarrow \infty }x'(t)=0\). If not, we suppose on the contrary that \(\lim_{t\rightarrow \infty }x'(t)=l_{0}>0\) and \(l_{0}\leq x'(t)\), and so
Setting the estimates into Eq. (1.1), we obtain
It is easy to see that
and then integrating n times, we obtain
In view of (\(P_{0}\)) of Theorem 2, we obtain a contradiction. Hence, \(\lim_{t\rightarrow \infty }x'(t)=0\). Furthermore, based on \(0< x(t)\) and \(0>x'(t)\), we can effortlessly discover that \(x(t)\) is decreasing. We check
letting us integrate Eq. (1.1) from t to ∞ to get
so we have
Integrating \(n2\) times from t to ∞, one gets
Therefore, we finish the proof of this theorem. □
Theorem 3
Suppose that (3.2) and
hold. If \(x(t)\in \mathcal{N}_{k}\) is a positive solution of (1.1), then
and, furthermore, \(\frac{x^{(k1)}(t)}{t}\) is decreasing.
Proof
We suppose that Eq. (1.1) possesses a positive solution \(x(t)\in \mathcal{N}_{k}\). We argue that (\(P_{k}\)) means \(\lim_{t\rightarrow \infty }x^{(k)}(t)=0\). If not, we suppose on the contrary that \(\lim_{t\rightarrow \infty }x^{(k)}(t)=l>0\) and \(l\leq x^{(k)}(t)\), and so \(l\frac{(tt_{1})^{k}}{k!}\leq x(t)\) and \(l\frac{(tt_{1})^{k1}}{(k1)!}\leq x'(t)\). In view of the last two estimates, we, together with Eq. (1.1), obtain
Integrating the last inequality \(nk\) times, one gets
Based on the condition (\(P_{k}\)) of Theorem 3, we obtain a contradiction. Hence, \(\lim_{t\rightarrow \infty }x^{(k)}(t)=0\). Furthermore, in view of \(0< x^{(k)}(t)\) and \(0>x^{(k+1)}(t)\), we have \(tx^{(k)}(t)t_{1}x^{(k)}(t)\leq \int _{t_{1}}^{t}x^{(k)}(s)\,ds\), \(0\leq x^{(k1)}(t_{1})t_{1}x^{(k)}(t)\) and
Replacing \(t_{1}\) by t in \(0\leq x^{(k1)}(t_{1})t_{1}x^{(k)}(t)\), one gets
Clearly, the function \(\frac{x^{(k1)}(t)}{t}\) is a decreasing function. We check
letting us integrate Eq. (1.1) from t to ∞ to get
i.e.,
which based on the monotonicity of \(x(t)\) means
Integrating the last inequality with \(nk1\) times from t to ∞, we have
And according to the monotonicity of
we have
Therefore, we finish the proof of Theorem 3. □
4 Comparison criteria of Eq. (1.1)
In this section, we will state our new technique and generalized comparison criteria, which can reduce the difficulty of the oscillation investigation of higherorder trinomial equation.
Primarily, we propose some sufficient conditions which can guarantee that all positive solutions classes \(\mathcal{N}_{k}\) are empty. For simplicity of notation, we write
Theorem 4
Suppose that all solutions of the firstorder twoterm advanced differential equation
are oscillatory for some positive constant \(\gamma \in (0,1)\) and that (3.2) holds. Then the positive solution class \(\mathcal{N}_{k}=\varnothing \) for Eq. (1.1). In addition, if
for all large enough t, then for Eq. (\(E_{k}\)) there exists an eventually nonoscillatory solution.
Proof
The second part of the theorem can be omitted due to the proof proposed by Kusano and Fukagai [16]. We have the following proof to the first part of this theorem.
By the principle of the reduction to absurdity, we can suppose on the contrary that for Eq. (1.1) there exists \(x(t)\in \mathcal{N}_{k}\), which is an eventually positive solution. Integrating Eq. (1.1) from t to ∞ leads to (3.4). In view of the estimate of \(x'(t)\):
we can deduce that
Integrating \(n2\) times from t to ∞, we derive
In addition, we can discover that \(x(t)\) satisfies the following firstorder differential inequality:
However, we can check that Lemma 3 of [6] can guarantee that for Eq. (\(E_{k}\)) there exists an eventually positive solution. This contradiction ends the proof of Theorem 4. □
Theorem 5
Suppose that all solutions of the firstorder twoterm advanced differential equation
are oscillatory for some \(\gamma \in (0,1)\) and that (3.2) holds. Then the positive solution class \(\mathcal{N}_{0}=\varnothing \) for Eq. (1.1). In addition, if
for all large enough t, then for Eq. (\(E_{0}\)) there exists an eventually nonoscillatory solution (Kneser solution).
Proof
The second part of the theorem can be omitted due to the proof proposed by Kusano and Fukagai [16]. We have the following proof to the first part of this theorem.
By the principle of the reduction to absurdity, we can suppose on the contrary that for Eq. (1.1) there exists \(x(t)>0\), which belongs to \(\mathcal{N}_{0}\). Integrating Eq. (1.1) \(n1\) times from t to ∞, one gets
In addition, we can discover that \(x(t)\) fulfills the inequality
However, we can check that Theorem 1 of [16] can guarantee that for Eq. (\(E_{0}\)) there exists a positive solution. This contradiction ends the proof of Theorem 5. □
Thanks to the above theorems, we can easily derive the following oscillation theorem for Eq. (1.1).
Theorem 6
Suppose that all solutions of the firstorder differential equations (\(E_{0}\)) and (\(E_{k}\)) are oscillatory for some \(\gamma \in (0,1)\) and that (3.2) holds, then all solutions of Eq. (1.1) are oscillatory.
We get an effortlessly confirmable theorem for the oscillatory properties of the investigated trinomial differential equations by applying some sufficient conditions to the oscillation of the firstorder advanced equations.
Theorem 7
Suppose that
where \(i=k\), \(k=0,1,2,3,\ldots ,n1\) and that (3.2) holds. Then Eq. (1.1) is oscillatory. In addition, if
for all large enough t, then for Eq. (\(E_{k}\)) there exists an eventually nonoscillatory solution.
Proof
The second part of the theorem can be omitted due to the proof proposed by Kusano and Fukagai [16]. We have the following proof to the first part of this theorem.
By the principle of the reduction to absurdity, we can suppose on the contrary that for Eq. (1.1) there exists \(x(t)\), which is an eventually positive solution. Theorem 1 guarantees that \(x(t)\in \mathcal{N}_{k}\). It follows from (\(C_{k}\)) that there is some \(\gamma \in (0,1)\) which can guarantee that
which by Theorem 2.4.1 of [24] implies that all solutions of Eq. (\(E_{k}\)) are oscillatory, which based on Theorem 4 guarantees \(\mathcal{N}_{k}=\varnothing \). This contradiction ends the proof of Theorem 7. □
Example 1
Based on Example 3.5 in [11], we investigate the following linear fourthorder trinomial ADE:
For the investigated Eq. (\(E_{1}\)), the corresponding Eq. (3.1) has the following form:
with a solution \(y(t)=t^{0.1}>0\). By direct calculation, we can effortlessly see that \(0< a=0.231<\frac{2\sqrt{3}}{9}\approx 0.3849\). Via a direct computation with Eq. (\(E_{1}\)) we see that
where \(t_{1}\) is large enough. Thanks to Theorem 7, all solutions of Eq. (\(E_{1}\)) are oscillatory under the following conditions:
For example, with \(\lambda =e\) it happens provided that \(b>2.1254\).
Example 2
We study the fifthorder ADE of Euler type
For the investigated Eq. (\(E_{2}\)), the corresponding Eq. (3.1) has the form
with an eventually positive solution \(y(t)=t^{0.1}\), where \(0< a=0.4959<\frac{9}{16}=0.5625\). It is seen via a direct computation with Eq. (\(E_{2}\)) that
Based on Theorem 7, all solutions of Eq. (\(E_{2}\)) are oscillatory under the following conditions:
For e.g. \(\lambda =e\) it happens provided that \(b>8.8291\).
5 Integral criteria of Eq. (1.1)
We will propose some new integral conditions for oscillatory properties to investigate advanced differential equations in this section which will provide a more precise result than the comparison criteria. At first, we need to use the preliminary results in Sect. 3.
Theorem 8
Suppose that (3.2) and
hold. Then \(\mathcal{N}_{n1}\) is an empty set for Eq. (1.1).
Proof
By the principle of the reduction to absurdity, we suppose on the contrary that Eq. (1.1) has a positive solution \(x(t)\) and this solution belongs to the positive solution class \(\mathcal{N}_{n1}\). Applying the estimation of the inequality offered by Theorem 3 which is a previous result proposed by Kiguradze [19], we have
for some \(\gamma \in (0,1)\). Thus,
from which one can conclude to a contradiction with the hypotheses of Theorem 8. This ends the proof. □
Theorem 9
Assume that (3.2) and
hold. Then the positive solution class \(\mathcal{N}_{k}\) is empty for Eq. (1.1).
Proof
By the principle of the reduction to absurdity, we suppose on the contrary that Eq. (1.1) has a positive solution \(x(t)\) and this solution belongs to the positive solution class \(\mathcal{N}_{k}\). After that, we integrate (4.2) from \(t_{1}\) to t to get
or
Denoting \(\mathcal{A}=st_{1}\), \(\mathcal{B}=s\sigma (t)\) and applying the Lagrangian middlevalue theorem yields \(\mathcal{A}^{n1}\mathcal{B}^{n1}\geq (n1)(\mathcal{A} \mathcal{B})\mathcal{B}^{n2}\) for \(\mathcal{A}\geq \mathcal{B}\geq 0\). Therefore, the last inequality leads to
for any \(\gamma \in (0,1)\), from which one concludes to a contradiction with the hypotheses of this theorem. This ends the proof. □
Theorem 10
Assume that (3.2) and
hold. Then the positive solution class \(\mathcal{N}_{0}=\varnothing \) for Eq. (1.1).
Proof
For the sake of proof by contradiction, we suppose that for Eq. (1.1) there exists an eventually positive solution \(x(t)\) and this solution belongs to the positive solution class \(\mathcal{N}_{0}\). Next, we integrate (4.2) from \(t_{1}\) to t to obtain
or
Denoting \(\mathcal{A}=st_{1}\), \(\mathcal{B}=s\sigma (t)\) and applying the Lagrangian middlevalue theorem yield \(\mathcal{A}^{n1}\mathcal{B}^{n1}\leq (n1)(\mathcal{A} \mathcal{B})\mathcal{B}^{n2}\) for \(\mathcal{A}\geq \mathcal{B}\geq 0\), the above estimate guarantees
It is obvious that
for any \(\gamma \in (0,1)\), from which one can conclude to a contradiction with the hypotheses of this theorem. This ends the proof. □
According to the above theorems, we can deduce the sufficient conditions to guarantee the oscillatory behavior of Eq. (1.1).
Theorem 11
If all hypotheses of Theorems 8–10are satisfied, then we can conclude that all solutions of the nthorder trinomial differential equation Eq. (1.1) are oscillatory.
Example 3
We discuss once more the fourthorder trinomial ADE
Based on the conditions of Theorems 8 and 9, we have
which guarantee that the positive solution classes \(\mathcal{N}_{k}\) and \(\mathcal{N}_{n1}\) are empty and all solutions of Eq. (\(E_{1}\)) are oscillatory. For instance, for \(\lambda =e\) it happens provided that \(b>1.9259\).
By the same principle, we can also apply integral criteria to study the fifthorder trinomial ADE of Euler type. Hence, it is omitted.
6 Conclusion
In this article, two methods, comparison criteria and integral criteria, have been applied to obtain some sufficient conditions of asymptotic and oscillatory behavior of a higherorder advanced trinomial differential equation under the substantial difficulty derived from the middle positive or negative term \(p(t)x'(t)\). In 2015, Džurina, Baculíková and Jadlovská [11] have obtained the oscillatory behavior of the following equation:
The previous fourthorder differential equation can be generalized to higher order by the two general types
or
The second equation has been addressed by Liang [27] in 2014. However, so far no researcher addressed the investigation of the first type advanced equations in the theory of oscillatory and asymptotic behaviors. Therefore, this gap in the theory of oscillation has been filled.
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It is a pleasure to express my sincere thanks to the anonymous referee for his (or her) helpful comments and valuable suggestions which led to improvements of this paper.
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This research was supported by the National Natural Science Foundation of China (Nos. 11671406 and 12071491).
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The main idea of this paper was proposed by DL. DL prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Luo, D. On oscillation of higherorder advanced trinomial differential equations. Adv Differ Equ 2021, 144 (2021). https://doi.org/10.1186/s13662021032921
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DOI: https://doi.org/10.1186/s13662021032921