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Sharp oscillation theorem for fourthorder linear delay differential equations
Journal of Inequalities and Applications volume 2022, Article number: 122 (2022)
Abstract
In this paper, we present a singlecondition sharp criterion for the oscillation of the fourthorder linear delay differential equation
by employing a novel method of iteratively improved monotonicity properties of nonoscillatory solutions. The result obtained improves a large number of existing ones in the literature.
1 Introduction
Consider the fourthorder linear delay differential equation
where \(p\in C([t_{0},\infty ))\) is positive, and the delay function \(\tau \in C([t_{0},\infty ))\) satisfies \(\tau (t)\le t\) and \(\tau (t) \to \infty \) as \(t\to \infty \).
By a solution of (1) we understand a four times differentiable realvalued function x that satisfies (1) for all t large enough. Our attention is restricted to those solutions of (1) that satisfy the condition \(\sup \{ x(t): T\le t<\infty \}>0\) for any large \(T\ge t_{0}\). We make a standing hypothesis that equation (1) possesses such solutions. A nontrivial solution of (1) is said to be oscillatory if it has infinitely many zeros and nonoscillatory otherwise. Equation (1) is called oscillatory if all its solutions are oscillatory.
The study of fourthorder differential equations originated with the vibrating rod problem in the first half of the 18th century and is generally of great practical importance. Such equations naturally arise in the modeling of physical and biological phenomena, such as, for instance, elasticity problems, deformation of structures, or oscillations of neuromuscular systems; see, e.g., [1, 2] for more detail.
With regard to their practical importance and the number of mathematical problems involved, the subject of the qualitative theory for such equations has undergone rapid development. In particular, oscillation theory of fourthorder differential equations involving (1) as a particular case has attracted a lot of attention over the last decades, which is evidenced by extensive research in the area, and we refer the reader to the recent related works [3–6] and the references therein. On the other hand, equation (1) can be understood as a prototype of evenorder binomial differential equations, investigated in detail in the monographs of Elias [7], Kiguradze and Chanturia [8], Koplatadze [9], and Swanson [10].
The aim of this paper is to obtain an unimprovable result for (1) to be oscillatory, depending on whether the limit inferior
is finite or not. To start, we briefly explain where the motivation behind this research comes from. As a particular case of a more complex work for halflinear delay differential equations, Jadlovská and Džurina [11] showed, via a novel method of iteratively improved monotonicity properties of nonoscillatory solutions, that the secondorder delay differential equation
is oscillatory if
where
We recall that the main purpose of the method is to find for any nonoscillatory, say positive, solution x of the studied binomial equation optimal values of positive constants a and b such that
which correspond to the monotonicities
respectively. The oscillation criterion is just an immediate consequence of these monotonicities; see [12] for a detailed description of the method.
Very recently, Graef, Jadlovská, and Tunç [13] extended the approach from [11] and showed that any nonoscillatory solution of the thirdorder delay differential equation
tends to zero asymptotically if
where
Three facts are important to notice about the above results. Firstly, no restriction is posed on the monotonicity or boundedness of the delay function \(\tau (t)\). Secondly, the result applies also in the ordinary case \(\tau (t) = t\). Thirdly, the oscillation constant \(M_{1}\) (\(M_{2}\)) is optimal for equation (2) (equation (5)) in the sense that the strict inequality in condition (3) (condition (6)) cannot be replaced by a nonstrict one without affecting validity of the result.
A natural question that arises is whether the method of iteratively improved monotonicity properties employed in the abovementioned works can be extended to obtain sharp results for the fourthorder delay differential equation (1). In this paper, we give a positive answer to this question. Our arguments essentially uses a classical result of Kiguradze [8, Lemma 1.1], by which the set \(\mathcal{S}\) of all positive nonoscillatory solutions of (1) has the decomposition
where
For each of these classes of nonoscillatory solutions, it is possible to initiate an iterative process that converges to the optimal monotonicity values a and b. As a side product of this finding, we formulate a singlecondition oscillation criterion with an unimprovable oscillation constant. To the best of our knowledge, there is no qualitatively the same result for (1) in the literature for general \(\tau (t)\); see Remarks 2 and 3 for more details.
The organization of the paper is as follows. In Sect. 2, we introduce the basic notations and the core of the method developed. In Sects. 3 and 4, we provide a series of lemmas, iteratively improving monotonicity properties of nonoscillatory solutions belonging to the classes \(\mathcal{S}_{3}\) and \(\mathcal{S}_{1}\), respectively. In Sect. 5, we present our main result, an oscillation criterion for (1). As usual, the improvement made over the existing results from the literature is illustrated via Eulertype differential equations. Finally, we propose several open problems for further research.
2 Notation
In our proofs, we will use the constants
All our results require, directly or indirectly, that \(\beta _{*}\) and \(\gamma _{*}\) are positive. Obviously, for arbitrary but fixed \(\beta \in (0,\beta _{*})\), \(\gamma \in (0,\gamma _{*})\), \(\delta \in (1, \delta _{*})\) for \(\delta _{*}>1\), and \(\delta = \delta _{*}\) for \(\delta _{*} = 1\), there is \(t_{1}\ge t_{0}\) large enough such that
In view of the above, let us define (as far as they exist) the sequences \(\{\beta _{n}\}\) and \(\{\gamma _{n}\}\) by
where \([k]\) stands for either β or γ. By induction it is easy to show that if \([k]_{i}<1\) for \(i = 1,2,\ldots ,n\), then \([k]_{n+1}\) exists, and
where
For arbitrary but fixed \([k]\in (0,[k]_{*})\) and \([k]_{i}<1\), \(i = 1,2,\ldots , n\), we also define the sequence \(\{\varepsilon _{[k]_{n}}\}\):
Again, by induction it follows that \(0<\varepsilon _{[k]_{n}}<\varepsilon _{[k]_{n+1}}<1\) for n\in {\mathbb{N}}_{0} and
We will use these facts in our proofs. As usual, and without loss of generality, we can suppose from now on that nonoscillatory solutions of (1) are eventually positive.
3 Nonexistence of \(\mathcal{S}_{3}\)type solutions
In view of (8), from (1) we see that
We begin with a simple lemma, which gives information on the behavior of possible nonoscillatory solutions belonging to the class \(\mathcal{S}_{3}\).
Lemma 1
Let \(\beta _{*}>0\) and assume that x is a solution of (1) belonging to the class \(\mathcal{S}_{3}\). Then for t sufficiently large:

(i)
\(\lim_{t\to \infty}x'''(t) = \lim_{t\to \infty}x''(t) /t =\lim_{t \to \infty}x'(t) /t^{2} = \lim_{t\to \infty}x(t) /t^{3} =0 \);

(ii)
\(x''(t)/t\) is decreasing, and \(x''(t)>tx'''(t)\);

(iii)
\(x'(t)/t^{2}\) is decreasing, and \(x'(t)>tx''(t)/2\);

(iv)
\(x(t)/t^{3}\) is decreasing, and \(x(t)>tx'(t)/3\).
Proof
Let \(x\in \mathcal{S}_{3}\) and choose \(t_{1}\ge t_{0}\) such that \(x(\tau (t))>0\) for \(t\ge t_{1}\).
(i) Since \(x'''(t)\) is a nonincreasing positive function, there exists a finite limit
If \(\ell >0\), then \(x'''(t)\ge \ell >0\), and so \(x(t)\ge \ell (tt_{1})^{3}/3!\) eventually, say for \(t\ge t_{2}\ge t_{1}\). Using this in (12) gives
Clearly, there exists \(t_{3}> t_{2}\) such that
which implies
Integrating the above inequality from \(t_{3}\) to t gives
which is a contradiction. Hence \(\ell = 0\).
Note that if \(x \in \mathcal{S}_{3}\), then \(x(t) \to \infty \) and \(x'(t) \to \infty \) as \(t\to \infty \). Also, if \(x''(t) \not \to \infty \), then \(\frac{x''(t)}{t} \to 0\) as \(t\to \infty \) since \(x''(t) >0\) is increasing. Hence we can apply l’Hôpital’s rule to see that (i) holds.
(ii) Again using the fact that \(x'''(t)\) is positive and nonincreasing, we find that
In view of (i), there is \(t_{4}>t_{1}\) such that \(x''(t_{1})>t_{1}x'''(t)\) for \(t\ge t_{4}\). Therefore
and so
which proves (ii).
(iii) Since \(x''(t)/t\) is a decreasing function tending to zero (see (i) and (ii)), we have
for some \(t_{5}>t_{4}\). Therefore
which proves (iii).
(iv) Similarly, since \(x'(t)/t^{2}\) is a decreasing function tending to zero (see (i) and (iii)), we get
for some \(t_{6}>t_{5}\). Thus
which proves (iv) and completes the proof of the lemma. □
Remark 1
When investigating the asymptotic properties of solutions of higherorder differential equations, authors often refer to the famous monograph of Kiguradze and Chanturia [8] and use the following result: for the function h satisfying \(h^{(i)}(t)>0\), \(i = 0,1,\ldots , m\), and \(h^{(m+1)}(t)\le 0\) eventually, \(h(t)/h'(t) \ge t/m\). However, as remarked in [14], only
holds eventually for every \(\ell \in (0,1)\). The necessity of the constant \(\ell \in (0,1)\) in inequality (13) has been shown by means of counterexamples; see [14] for details. Obviously, the application of (13) to the solution \(x(t)\in \mathcal{S}_{3}\) would lead to
which is a weaker result than Lemma 1 provides. The omission of the constant ℓ was made possible by the requirement of having \(\beta _{*}\) positive.
The next lemma provides some additional properties of solutions in the class \(\mathcal{S}_{3}\).
Lemma 2
Let \(\beta _{*}>0\) and assume that x is a solution of (1) belonging to the class \(\mathcal{S}_{3}\). Then for any \(\beta \in (0,\beta _{*})\) and t sufficiently large:

(v)
\(x''(t)/t^{1\beta}\) is decreasing, and \((1\beta )x''(t)>tx'''(t)\);

(vi)
\(\beta <1\);

(vii)
\(\lim_{t\to \infty}x''(t)/t^{1\beta} = \lim_{t\to \infty}x'(t)/t^{2 \beta} =\lim_{t\to \infty}x(t)/t^{3\beta} =0\);

(viii)
\(x'(t)/t^{2\beta}\) is decreasing, and \(x'(t)>tx''(t)/(2\beta )\);

(ix)
\(x(t)/t^{3\beta}\) is decreasing, and \(x(t)>tx'(t)/(3\beta )\).
Proof
Let \(x\in \mathcal{S}_{3}\) and choose \(t_{1}\ge t_{0}\) such that \(x(\tau (t))>0\) and parts (i)–(iv) of Lemma 1 hold for \(t\ge t_{1}\).
(v) Define the function
which is positive in view of (ii). Differentiating z and using (12) and the monotonicity of \(x(t)/t^{3}\) (see (iv)), we obtain
Using the estimates from (iv) and (iii), respectively, in the above inequality, we find
Integrating (16) from \(t_{1}\) to t and using the fact that \(x''(t)/t\) is decreasing and tends to zero (see (i) and (ii)), we obtain
for some \(t_{2} > t_{1}\), that is,
and so
Hence part (v) holds.
(vi) This clearly follows from (v) and the fact that \(x''(t)\) is increasing.
(vii) To show
it suffices to prove that there is \(\varepsilon >1\) such that, for sufficiently large t,
Indeed, if
then
which is a contradiction. Using (17), we see that for any \(k\in ((2\beta )/2,1)\), there is \(t_{3}\ge t_{2}\) sufficiently large such that
Using this in (16), we have
Integrating from \(t_{3}\) to t and using (i), this becomes
for some \(t_{4}>t_{3}\). In view of the definition of z (see (14)), the above inequality implies
Then it is easy to see that (19) holds with
The remaining limits in (vii) follow from (18) and an application of l’Hôpital’s rule.
(viii) Using (18) in (20), we have
for some \(t_{5}>t_{4}\). Thus
(ix) Finally, since \(x'(t)/t^{2\beta}\) is a decreasing function tending to zero (see (vii) and (viii)), we have
for some \(t_{6}>t_{5}\). Therefore
The proof is complete. □
The next result is a simple consequence of (ix).
Lemma 3
Assume that \(\beta _{*}>0\) and \(\delta _{*} = \infty \). Then \(\mathcal{S}_{3} = \emptyset \).
Proof
Suppose to the contrary that \(x \in \mathcal{S}_{3}\) and let \(t_{1}\ge t_{0}\) be such that \(x(\tau (t))>0\) for \(t\ge t_{1}\). Using (ix) and (8) in (15), we find
Also, from (ix) and (viii) it follows, respectively, that
eventually, say for \(t\ge t_{2}\) for some \(t_{2}\ge t_{1}\). Proceeding as in the proof of (v), we arrive at
Since δ can be arbitrarily large, we choose it so that
This implies that \(x''(t)>tx'''(t)\), which is a contradiction. This proves the lemma. □
In view of Lemma 2(vi) and Lemma 3, it is reasonable to assume that \(\delta _{*}<\infty \), so that \(\mathcal{S}_{3}\neq \emptyset \). The following lemma can be seen as an iterative version of Lemma 2.
Lemma 4
Let \(\beta _{*}>0\) and assume that x is a solution of (1) belonging to the class \(\mathcal{S}_{3}\). Then for any \(\varepsilon _{\beta _{n}}\in (0,1)\) and sufficiently large t,
 \((\mathrm{I})_{n}\):

\(x''(t)/t^{1\tilde{\beta}_{n}}\) is decreasing, and \((1\tilde{\beta}_{n})x''(t)>tx'''(t)\);
 \((\mathrm{II})_{n}\):

\(\tilde{\beta}_{n}<1\);
 \((\mathrm{III})_{n}\):

\(\lim_{t\to \infty}x''(t)/t^{1\tilde{\beta}_{n}} = \lim_{t\to \infty}x'(t)/t^{2\tilde{\beta}_{n}} =\lim_{t\to \infty}x(t)/t^{3 \tilde{\beta}_{n}} =0\);
 \((\mathrm{IV})_{n}\):

\(x'(t)/t^{2\tilde{\beta}_{n}}\) is decreasing, and \(x'(t)>tx''(t)/(2\tilde{\beta}_{n})\);
 \((\mathrm{V})_{n}\):

\(x(t)/t^{3\tilde{\beta}_{n}}\) is decreasing, and \(x(t)>tx'(t)/(3\tilde{\beta}_{n})\);
where \(\tilde{\beta}_{n} =\varepsilon _{\beta _{n}}\beta _{n}\).
Proof
Let \(x \in \mathcal{S}_{3}\) with \(x(\tau (t))>0\) for \(t\ge t_{1}\) for some \(t_{1}\ge t_{0}\). We will proceed by induction on n. For \(n = 0\), the conclusion clearly follows from Lemma 2 with \(\beta = \tilde{\beta}_{0}\). Next, assume that \((\mathrm{I})_{n}\)–\((\mathrm{IV})_{n}\) hold for \(n\ge 1\) and \(t\ge t_{n}\ge t_{1}\). We need to show that they all hold for \(n+1\).
(I)\(_{n+1}\) Using (viii) and (8) in (15), we have
Then by \((\mathrm{V})_{n}\) and \((\mathrm{IV})_{n}\) we get
Integrating (21) from \(t_{n}\) to t and using \((\mathrm{I})_{n}\) and \((\mathrm{III})_{n}\) in the resulting inequality, we see that there exists \(t_{n}'>t_{n}\) such that
i.e.,
and
which proves (I)\(_{n+1}\).
(II)\(_{n+1}\) This follows immediately from (I)\(_{n+1}\) and the fact that \(x''(t)\) is increasing.
(III)\(_{n+1}\) As for the case \(n = 0\), it suffices to show that there is \(\varepsilon >1\) such that
Using (23), we see that for any \(k\in (0,1)\), there is \(t_{n}''\ge t_{n}'\) sufficiently large such that
Combining this with (21), we obtain
which after integrating from \(t_{n}''\) to t and using \((\mathrm{III})_{n}\) yields
Since \(\tilde{\beta}_{n}<\tilde{\beta}_{n+1}\), we can choose k such that \(\varepsilon >1\), and hence (19) holds. The rest of proof is the same as that for \(n = 0\).
(IV)\(_{n+1}\) Using (III)\(_{n+1}\) in (25) gives
and so (IV)\(_{n+1}\) holds.
(V)\(_{n+1}\) As in the case \(n = 0\), using the fact that \(x'/t^{2\tilde{\beta}_{n+1}}\) is decreasing and tends to zero, we get
which implies (V)\(_{n+1}\) and completes the proof of the lemma. □
From the above arguments we can immediately obtain the following lemma.
Lemma 5
Assume that \(\delta _{*}<\infty \) and
where
Then \(\mathcal{S}_{3} = \emptyset \).
Proof
Suppose to the contrary that \(x \in \mathcal{S}_{3}\) and let \(t_{1}\ge t_{0}\) be such that \(x(\tau (t))>0\) for \(t\ge t_{1}\). We claim that
By \((\mathrm{II})_{n}\), \(\tilde{\beta}_{n} <1\). Since \(\varepsilon _{\beta _{n}} \in (0,1)\) can be chosen arbitrarily, set \(\varepsilon _{\beta _{n}} > 1/\ell _{\beta _{n}}\), where \(\ell _{\beta _{n}}\) is defined by (10). Then
which proves the claim. In view of (29), we conclude that the sequence \(\{\beta _{n}\}_{n = 0}^{\infty}\) defined by (9) is increasing and bounded from above, that is, there exists a finite limit
where \(c \in (0,1)\) is a root of the equation
However, condition (27) implies that (30) does not possess positive solutions. Hence \(\mathcal{S}_{3} = \emptyset \), and the proof is complete. □
4 Nonexistence of \(\mathcal{S}_{1}\)type solutions
In this section, we prove similar results to those in Sect. 3 for solutions in the class \(\mathcal{S}_{1}\). In view of (8), equation (1) becomes
Lemma 6
Let \(\gamma _{*}>0\) and assume that x is a solution of (1) belonging to the class \(\mathcal{S}_{1}\). Then for t sufficiently large:

(i)
\(\lim_{t\to \infty}x'(t) = \lim_{t\to \infty}x(t) /t =0 \);

(ii)
\(x(t)/t\) is decreasing.
Proof
Let \(x \in \mathcal{S}_{1}\) and choose \(t_{1}\ge t_{0}\) such that \(x(\tau (t))>0\) for \(t\ge t_{1}\).
(i) Since \(x'(t)\) is a decreasing positive function, there exists a finite limit
If \(\ell >0\), then \(x'(t)\ge \ell >0\), and so \(x(t)\ge \ell (tt_{1})>\ell t/3\) for \(t\ge t_{2}\) for some \(t_{2}\ge t_{1}\). Using this in (31) gives
Integrating twice from t to ∞, we have
and after integrating from \(t_{2}\) to t,
which is a contradiction. Hence \(\ell = 0\). Applying l’Hôpital’s rule, we see that (i) holds.
(ii) Again using the monotonicity of \(x'\) and (i), we see that
for \(t \ge t_{3}>t_{1}\), where \(t_{3}\) is sufficiently large such that \(x(t_{1})  x'(t)t_{1}>0\) for \(t\ge t_{3}\). Thus
and the proof is complete. □
Lemma 7
Let \(\gamma _{*}>0\) and assume that x is a solution of (1) belonging to the class \(\mathcal{S}_{1}\). Then for any \(\gamma \in (0,\gamma _{*})\) and t sufficiently large:

(iii)
\(x(t)/t^{1\gamma}\) is decreasing;

(iv)
\(\gamma <1\);

(v)
\(\lim_{t\to \infty}x(t)/t^{1\gamma} = 0\);

(vi)
\(x(t)/t^{\gamma}\) is nondecreasing.
Proof
Let \(x \in \mathcal{S}_{1}\) and choose \(t_{1}\ge t_{0}\) such that \(x(\tau (t))>0\) for \(t\ge t_{1}\).
(iii) Since \(x(t)/t\) is decreasing (see (ii)) in (31), we obtain
Integrating this inequality twice from t to ∞ and using at each step that x is increasing yields
Define the function
which is clearly positive in view of (ii). Differentiating w and using (32), we see that
Integrating from \(t_{1}\) to t and using again that \(x(t)/t\) is decreasing and tends to zero, we have
where \(t_{2}>t_{1}\) is sufficiently large such that \(w(t_{1})  t_{1}x(t)/t>0\) for \(t\ge t_{2}\). Hence
and
so (iii) holds.
(iv) This clearly follows from (iii) and the fact that x is increasing.
(v) To prove this, similarly to the proof for the class \(\mathcal{S}_{3}\), it suffices to show that
for some \(\varepsilon >1\). Using (35) in (34), we see that for any \(k\in (1\gamma ,1)\), there is \(t_{3}\ge t_{2}\) such that
from which it follows that
Now it is clear that (36) holds with \(\varepsilon = k/(1\gamma )>1\).
(vi) Integrating (32) from t to ∞ and using the monotonicity of x, we get
and so
The proof of the lemma is now complete. □
Lemma 8
Assume that \(\gamma _{*}>0\) and \(\delta _{*} = \infty \). Then \(\mathcal{S}_{1} = \emptyset \).
Proof
Suppose to the contrary that \(x \in \mathcal{S}_{1}\) and let \(t_{1}\ge t_{0}\) be such that \(x(\tau (t))>0\) for \(t\ge t_{1}\). Using (iii) and (8) in (31), we see that
Integrating twice from t to ∞ and using repeatedly that x is increasing, we obtain
Then using (38) in (33) yields
Integrating as in (34) with γ replaced by \(\gamma \delta ^{\gamma}\) leads to
Since δ can be arbitrarily large, we can choose it so that
which implies that \(x(t)>tx'(t)\), a contradiction. This proves the lemma. □
Next, we obtain an iterative form of Lemma 7.
Lemma 9
Let \(\gamma _{*}>0\) and assume that x is a solution of (1) belonging to the class \(\mathcal{S}_{1}\). Then for any \(\varepsilon _{\gamma _{n}}\in (0,1)\) and sufficiently large t:
 \((\mathrm{I})_{n}\):

\(x(t)/t^{1\tilde{\gamma}_{n}}\) is decreasing;
 \((\mathrm{II})_{n}\):

\(\tilde{\gamma}_{n}<1\);
 \((\mathrm{III})_{n}\):

\(\lim_{t\to \infty}x(t)/t^{1\tilde{\gamma}_{n}} = 0\);
 \((\mathrm{IV})_{n}\):

\(x(t)/t^{\tilde{\gamma}_{n}}\) is nondecreasing;
where \(\tilde{\gamma}_{n} = \varepsilon _{\gamma _{n}}\gamma _{n}\).
Proof
Let \(x \in \mathcal{S}_{1}\) with \(x(\tau (t))>0\) for \(t\ge t_{1}\) for some \(t_{1}\ge t_{0}\). We will proceed by induction on n. For \(n = 0\), the conclusion follows from Lemma 7. Next, assume that \((\mathrm{I})_{n}\)–\((\mathrm{IV})_{n}\) hold for \(n\ge 1\) and \(t\ge t_{n}\ge t_{1}\) and let us show that (I)\(_{n+1}\) holds. Using \((\mathrm{I})_{n}\) in (31) gives
Integrating the above inequality from t to ∞ and using the fact that \(x(t)/t^{\tilde{\gamma}_{n}}\) is increasing (see \((\mathrm{IV})_{n}\)), we obtain
Repeating this step, we get
and using this in (33), we have
Integrating from \(t_{n}\) to t and using \((\mathrm{I})_{n}\) and \((\mathrm{III})_{n}\), we obtain
where \(t_{n}'>t_{n}\) is sufficiently large such that \(w(t_{n})  t_{n}^{1\tilde{\gamma}_{n}}x(t)/t^{1\tilde{\gamma}_{n}}>0\) for \(t\ge t_{n}'\). By the definition of w we see that
and
which proves (I)\(_{n+1}\). Since the proofs of the other parts are similar to those in the case \(n = 0\), we omit the details. □
Lemma 10
Assume that \(\delta _{*}<\infty \) and
where M is defined by (28). Then \(\mathcal{S}_{1} = \emptyset \).
Proof
The proof is similar to that of Lemma 5 and hence is omitted. □
5 Main result and discussion
Combining the results from the previous two sections, we now present the main result in this paper.
Theorem 1
Assume that
where
Then equation (1) is oscillatory.
Proof
Notice that condition (43) implies \(\beta _{*} >0\), and since
we see that \(\gamma _{*} >0\). Now if \(\delta _{*} = \infty \), then Lemmas 3 and 8 imply that \(\mathcal{S}_{1} = \mathcal{S}_{3} = \emptyset \). For \(\delta _{*} < \infty \), the same conclusion follows from Lemmas 5 and 10. This proves the theorem. □
Corollary 1
Let \(\tau (t) = \alpha t\) with \(0<\alpha \le 1\). If
then (1) is oscillatory.
Remark 2
As an important related result, we recall a particular case of integraltype criterion due to Koplatadze [9, Corollary 6.4] obtained by a different technique for evenorder differential equations with deviating arguments. He proved that (1) is oscillatory if \(\tau (t)\ge \alpha t\), \(0<\alpha \le 1\), and
Clearly, if \(\tau (t) = \alpha t\), then (44) and (45) are qualitatively the same for (1). Hence, for \(n = 4\), Theorem 1 can be regarded as a generalization of [9, Corollary 6.4], removing the restrictive condition \(\tau (t)\ge \alpha t\). For similar related comparison results, we refer the reader to [15].
We demonstrate the sharpness of the newly obtained oscillation criterion on Eulertype differential equations.
Example 1
Consider the fourthorder Euler delay differential equation
The associated characteristic equation is
which is obtained by setting
Denoting the local maxima of \(f(\alpha )\) by
and noting that for any \(\delta _{*}\),
it is easy to verify that (46) has the nonoscillatory solution (48) if
By Theorem 1, equation (46) is oscillatory if
which shows that our oscillation constant cannot be improved. As a result, we see that this paper provides an optimal method for the study of oscillatory properties of fourthorder delay differential equations.
In the particular case \(\delta _{*} = 2\), we see that equation (46) is oscillatory if
Remark 3
For completeness, we review a few known oscillation constants for equation (46) with \(\delta _{*} = 2\) given in the previous works.

1.
$$ p_{0}>1728; $$

2.
Zafer [18]:
$$ p_{0}>\frac{192}{\mathrm{e}\ln 2} \simeq 101.902; $$ 
3.
Grace [19]:
$$ p_{0}>\frac{6}{1 + \ln 2} \simeq 3.545; $$ 
4.
Karpuz et al. [20], Zhang and Yan [21]:
$$ p_{0}>\frac{6}{\mathrm{e}\ln 2} \simeq 3.184; $$ 
5.
Koplatadze [22]:
$$ p_{0}>\frac{3!}{3 + \ln 2} \simeq 1.625; $$ 
6.
Baculíková and Džurina [23]:
$$ p_{0}>1. $$
It is worth noting that cases (2)–(5) above require \(\tau (t)< t\) and \(\tau '(t)\ge 0\), which is not needed in Theorem 1. Hence Theorem 1 significantly improves many existing results in the literature, even without the usual restrictive assumptions on the deviating argument.
Remark 4
The results presented in this paper open many fruitful problems for further research, and we state at least the most obvious ones.
The first problem consists in extending the sharp results from this paper to the more general fourthorder equation
with \(r_{i} \in \mathcal{C}([t_{0},\infty ),(0,\infty ))\) in the socalled canonical form, i.e., where
For a similar extension in the case of thirdorder equations, we refer the reader to the recent paper [24].
It is also an open question how to obtain a sharp singlecondition oscillation criterion for (51) in the noncanonical case where
Note that the nonexistence of eight possible classes of nonoscillatory solutions must be shown for (51) to be oscillatory (see [2, 25] for more detail).
It would be also interesting to establish the corresponding results for equation (51) with an advanced argument \(\tau (t) \geq t\) in either the canonical or noncanonical case. We refer the reader to [26] for a similar sharp oscillation criteria for secondorder advanced differential equations. It is also worth mentioning that the method of iteratively improved monotonicities of nonoscillatory solutions has not as yet been applied to equations with damping. In view of the increasing interest in the study of third and fourthorder difference equations with deviating arguments, another possible direction for future research is to extend the approach used in this paper to the discrete case, similar to what was done for secondorder difference equations with deviating arguments in [27, 28].
Finally, another possibility is developing a unified approach by investigating the oscillatory and asymptotic properties of solutions of fourthorder dynamic equations with deviating arguments on time scales via the method of iteratively improved monotonicities.
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References
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Acknowledgements
The authors express their sincere gratitude to the editors and three anonymous referees for the careful reading of the original manuscript and useful comments, which helped to improve the presentation of the results and accentuate important details.
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The work on this research has been supported by the Slovak Research and Development Agency under Contract Nos. APVV190590 and APVV180373 and Slovak Grant Agency VEGA No. 2/0127/20.
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IJ made the major analysis and the original draft preparation. JD, JG, and SG analyzed all the results, gave many comments, and made necessary improvements. All authors read and approved the final manuscript.
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Jadlovská, I., Džurina, J., Graef, J.R. et al. Sharp oscillation theorem for fourthorder linear delay differential equations. J Inequal Appl 2022, 122 (2022). https://doi.org/10.1186/s13660022028590
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DOI: https://doi.org/10.1186/s13660022028590