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On the oscillation of fourthorder delay differential equations
Advances in Difference Equations volume 2019, Article number: 118 (2019)
Abstract
In the paper, fourthorder delay differential equations of the form
under the assumption
are investigated. Our newly proposed approach allows us to greatly reduce a number of conditions ensuring that all solutions of the studied equation oscillate. An example is also presented to test the strength and applicability of the results obtained.
Introduction
Consider the fourthorder linear delay differential equation
where
In the sequel, we will assume that:
 \((\mathrm {H}_{1})\) :

\(r_{i}\in \mathcal{C}([t_{0},\infty ), \mathbb {R})\), \(i = 1,2,3\) are positive and satisfy
$$ \pi _{i}(t_{0}): = \int _{t_{0}}^{\infty }\frac{\mathrm {d}t}{r_{i}(t)} < \infty ; $$  \((\mathrm {H}_{2})\) :

\(q\in \mathcal{C}([t_{0},\infty ),\mathbb {R})\) is nonnegative and does not vanish eventually;
 \((\mathrm {H}_{3})\) :

\(\tau \in \mathcal{C}^{1}([t_{0},\infty ),\mathbb {R})\), \(\tau (t)\le t\), and \(\lim_{t\to \infty } \tau (t) = \infty \).
A solution y of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all its solutions oscillate.
The foundations of vibration theory for continuous media established in the first half of the 18th century by the two close collaborators Daniel Bernoulli and Leonard Euler have generated the investigation of linear fourthorder differential equations [23]. Since then, the Euler–Bernoulli beam theory has shown to be of great practical importance due to its wide applications in civil, mechanical and aeronautical engineering and has been outlined in the literature over the years.
Being aware of the continuous interest in the study of selfexcited oscillation phenomena which occur in bridges, it is worth mentioning that an oversimplified model concerns traveling waves in a suspension bridges [8, 18, 19]. Here, beams are used as the basis of supporting of the bridge or as the mainframe foundation in axles. The governing equation reads
where x is the coordinate along the beam axis, t is the time, \(u = u(x,t)\) is the lateral displacement, δ is the viscous damping coefficient and γ is the stiffness coefficient per unit length. To find the traveling wave solutions of this partial differential equation, we may use the substitution of the form
with period c and one has to solve the nonlinear fourthorder differential equation of the form
As another important example on use of fourthorder equations, we mention the famous Swift–Hohenberg equation
which serves as a model of pattern formation in many physical, chemical or biological systems [15].
It is also worth to mention the oscillatory muscle movement model represented by a fourthorder delay differential equation, which can arise due to the interaction of a muscle with its inertial load [20]. An unexpected area where fourthorder differential equations have occurred is in the context of number theory [4].
Because of the above motivating factors for the study of fourthorder differential equations, as well as because of the theoretical interest in generalizing and extending some known results from those given for lowerorder equations, the study of oscillation of such equations has received considerable portion of attention. For a systematic summary of the most significant efforts made as regards this theory, the reader is referred to the monographs of Elias [7], Kiguradze and Chanturia [13], and Swanson [21].
As far as the oscillation theory of fourthorder differential equations is concerned, the problem of investigating ways of factoring disconjugate operator \(L_{4}y\) has been of special interest.
Motivated by the famous work of George Pólya, Trench [22] showed that we can always write the operator \(L_{4} y\) in an equivalent canonical form
such that the functions \(p_{i}\in \mathcal{C}([t_{0},\infty ), \mathbb {R})\), \(i = 0,1,2,3,4\) are positive,
and uniquely determined up to positive multiplicative constants with the product 1. The explicit forms of the functions \(p_{i}\) generally depend on whether the integrals \(\pi _{i}\) (\(i = 1,2,3\)), which we defined in \((\mathrm {H}_{1})\), are convergent or divergent. Consequently, the investigation of the qualitative behavior of canonical fourthorder functional differential equations of the form
its generalizations or particular cases, especially with regard to their oscillatory properties, has become the subject of intensive research; see, for instance, [1,2,3, 5, 11, 12, 16, 17, 24,25,26] and the references cited therein.
The main advantage of studying (1.1) in canonical form (1.2) essentially lies in the direct application of the wellknown Kiguradze lemma [13, Lemma 1], which allows one to classify the set of possible nonoscillatory solutions. In particular, if y is a positive solution of the canonical equation (1.2), then there are only two possible cases for y:
for t large enough.
However, the formulas for the corresponding functions \(p_{i}\) resulting from Trench’s theory of canonical operators are in general too complicated to allow the practical application of existing results obtained for canonical equations. Another possible approach elaborated by several authors is to investigate the original equation, at the cost of the existence of additional classes of possible nonoscillatory solutions. In particular, the authors in [9, 10] established oscillation results for (1.1) under assumptions
and
respectively. Although the technique used in these papers is different, their results have in common that the oscillation of the studied equation was ensured via four independent conditions, eliminating nonoscillatory solutions pertaining to particular classes.
Very recently, Džurina and Jadlovská [6] investigated the oscillatory behavior of thirdorder differential equations of the form
under the condition
By careful observation, they pointed out that various conditions, traditionally imposed in the existing results are redundant. This observation led to the gain of various twocondition oscillation criteria for (1.3).
To the best of our knowledge, there is nothing known about the oscillation of (1.1) under the assumption \((\mathrm {H}_{1})\). Inspired by the ideas adopted in [6], our primary goal is to fill this gap by presenting simple criteria for the oscillation of all solutions of (1.1). Most importantly, we stress that the nonexistence of eight possible classes of nonoscillatory solutions (see Lemma 1) is shown only through two conditions. Our newly proposed approach could hopefully serve as a reference in the lessdeveloped theory for noncanonical equations. Finally, we illustrate the importance of the results obtained via Eulertype equations.
Main results
For the sake of clarity, we list the functions to be used in the paper. That is, for \(t\ge t_{*}\ge t_{0}\), we put
As usual, all functional inequalities considered in this paper are supposed to be satisfied for all t large enough. In what follows, we need only to consider eventually positive solutions of (1.1), since if y satisfies (1.1), then so does −y.
Lemma 1
Assume that \((\mathrm {H}_{1})\)–\((\mathrm {H}_{3})\) hold and y is an eventually positive solution to (1.1). Then there exists a \(t_{1} \in [t_{0}, \infty )\) such that y satisfies one of the following cases:
for \(t\ge t_{1}\).
Proof
The proof is obvious and so we omit it. □
We start with a simple condition ensuring the nonexistence of solutions of types (1)–(4). As will be shown later, this condition is already included in those eliminating positive decreasing solutions.
Lemma 2
Assume that \((\mathrm {H}_{1})\)–\((\mathrm {H}_{3})\) hold. Let y be an eventually positive solution of (1.1). If
then cases (1)–(4) from Lemma 1 are impossible.
Proof
First of all, it is important to note that if both \((\mathrm {H}_{1})\) and (2.1) hold, then
Now, assume on the contrary that \(y(t)\) is an eventually positive solution of (1.1) satisfying one of the cases (1)–(4) from Lemma 1 and pick a \(t_{1}\in [t_{0},\infty )\) such that \(y(\tau (t))>0\) for \(t\ge t_{1}\). Since y is increasing, there exist a constant \(c>0\) and a \(t_{2}\ge t_{1}\) such that \(y(\tau (t)) \ge c\) for \(t\ge t_{2}\). Using this inequality in (1.1), we get
Integrating (2.3) from \(t_{2}\) to t, we find
If we assume that y belongs either to case (1) or case (3), then from (2.2) and (2.4), we obtain
which contradicts the fact that \(L_{3}y\) is nonincreasing.
Next, assume that case (2) holds. Then (2.4) becomes
that is,
Integrating (2.6) from \(t_{2}\) to t, we have
which, by virtue of (2.2), gives
which clearly contradicts the fact that \(L_{2}y\) is decreasing.
Finally, let us assume that case (4) holds. In the same way as in the previous case, we arrive at (2.7), namely,
Integrating this inequality from \(t_{2}\) to t, we get
which in view of (2.1) yields
which contradicts the fact that \(L_{1}y\) is decreasing.
The proof is complete. □
In the following result, a simple condition ensuring that any nonoscillatory solution converges to zero as \(t\to \infty \) is established.
Theorem 1
Assume that \((\mathrm {H}_{1})\)–\((\mathrm {H}_{3})\) hold. If
then any solution y of (1.1) is oscillatory or \(\lim_{t\to \infty }y(t) = 0\).
Proof
Assume that y is a nonoscillatory solution of (1.1) on \([t_{0},\infty )\). Without loss of generality, we may take a \(t_{1} \ge t_{0}\) such that \(y(t)>0\) and \(y(\tau (t))>0\) for \(t\ge t_{1}\). By Lemma 1, eight possible cases may occur for \(t\ge t_{1}\).
Since (2.10) together with \((\mathrm {H}_{1})\) implies that \(\int _{t_{0}} ^{\infty }Q(t,t_{0})\,\mathrm {d}t\) cannot be bounded, by Lemma 2, cases (1)–(4) are impossible.
Let one of the cases (5)–(8) hold. Since y is decreasing, there exists a finite nonnegative limit \(y(\infty ) = \lim_{t\to \infty }y(t) = c\). Assume on the contrary that \(c> 0\). Then there exists a \(t_{2}\ge t_{1}\) such that \(y(\tau (t)) \ge c\) for \(t\ge t_{2}\) and inequality (2.3) is satisfied. Then one can arrive at contradiction (2.5) in cases (5) and (7), and contradiction (2.8) in case (6). Thus, we conclude that \(c = 0\).
If we assume that case (8) holds, then we get (2.9), that is,
or
Integrating the above inequality from \(t_{2}\) to t, we obtain
However, the integral on the righthand side of the above inequality tends to ∞ as \(t\to \infty \) due to (2.10), which contradicts the fact that y is decreasing.
The proof is complete. □
In the sequel, we present various twocondition oscillation criteria for (1.1).
Theorem 2
Assume that \((\mathrm {H}_{1})\)–\((\mathrm {H}_{3})\) hold and τ is nondecreasing. If
for any \(t_{1}\ge t_{0}\), where
then (1.1) is oscillatory.
Proof
Assume that y is a nonoscillatory solution of (1.1) on \([t_{0},\infty )\). Without loss of generality, we may take a \(t_{1} \ge t_{0}\) such that \(y(t)>0\) and \(y(\tau (t))>0\) for \(t\ge t_{1}\). By Lemma 1, eight possible cases may occur for \(t\ge t_{1}\).
At first, it is useful to note that, in view of \((\mathrm {H}_{1})\), it is necessary for the validity of (2.11) that
From Lemma 2, the above condition ensures that cases (1)–(4) from Lemma 1 are impossible. We shall consider the remaining possible cases (5)–(8) separately.
Assume that case (5) holds. From the monotonicity of \(L_{2}y\), we deduce that
that is,
Integrating the above inequality from t to ∞, we get
Using (2.13) and the increasing property of \(L_{2}y\) in (1.1), there exist a constant \(c>0\) and a \(t_{2}\ge t_{1}\) such that
Integrating the above inequality from \(t_{2}\) to t, we have
Taking \((\mathrm {H}_{1})\) and (2.12) into account, it is easy to see that
Using (2.15) in (2.14), we arrive at a contradiction with the fact that \(L_{3}y\) is nonincreasing.
Assume that case (6) holds. From the monotonicity of \(L_{3}y\), we have
Therefore,
which implies that \(L_{2}y/\pi _{3}\) is nondecreasing. Using further this property, we obtain
Hence,
and so \(L_{1}y/\pi _{23}\) is nondecreasing. Finally, we arrive at
Using (2.17) in the above inequality, we get
Therefore,
Integrating this inequality from \(t_{1}\) to t and using the monotonicity of \(L_{2}y\), we find
From (2.16) and (2.19), we obtain
Dividing the above inequality by \(L_{3}y\) and taking the lim sup on both sides of the resulting inequality, one arrives at a contradiction with (2.11).
Assume that case (7) holds. From the decreasing property of \(L_{1}y\), we get
Thus,
which means that \(y/\pi _{1}\) is nondecreasing. Integrating (1.1) from \(t_{1}\) to t and using the monotonicity of y, we conclude that
On the other hand, using \((\mathrm {H}_{1})\) and (2.15), it is easy to see that, for any constant \(k>0\),
This in view of inequality (2.20) contradicts the fact that \(L_{3}y\) is nonincreasing.
Assume that case (8) holds. Integrating (1.1) from \(t_{1}\) to t, we have
Dividing both sides of the above inequality by \(r_{3}(t)\) and integrating the resulting inequality again from \(t_{1}\) to t, we get
Similarly, we obtain
that is,
which clearly contradicts (2.11).
The proof is complete. □
Theorem 3
Assume that \((\mathrm {H}_{1})\)–\((\mathrm {H}_{3})\) hold and τ is nondecreasing. If
for any \(t_{1}\ge t_{0}\), where
then (1.1) is oscillatory.
Proof
Assume that y is a nonoscillatory solution of (1.1) on \([t_{0},\infty )\). Without loss of generality, we may take a \(t_{1} \ge t_{0}\) such that \(y(t)>0\) and \(y(\tau (t))>0\) for \(t\ge t_{1}\). By Lemma 1, eight possible cases may occur for \(t\ge t_{1}\).
First, note that it is necessary for the validity of (2.23) that
which in view of \((\mathrm {H}_{1})\) implies that (2.12) holds. From Lemma 2, we see that the above condition ensures that cases (1)–(4) from Lemma 1 are impossible. We will consider the remaining possible cases (5)–(8) separately.
Since cases (5) and (7) can be treated exactly as in the proof of Theorem 2, we omit this part of the proof. Assume that case (6) holds. Proceeding as in the proof of Theorem 2 case (6), we arrive at (2.19), i.e.,
that is,
where we set \(x(t) = L_{2}y(t)>0\). It follows from (2.23) that
however, by [14, Theorem 2.1.1], this condition ensures that inequality (2.24) does not possess a positive solution, which is a contradiction with our initial assumption.
Assume that case (8) holds. Proceeding as in the proof of Theorem 2 case (8), we arrive at (2.22), i.e.,
or
Similar to case (7), we arrive at a contradiction.
The proof is complete. □
The last criterion is obtained by employing the classical Riccati transformation technique.
Theorem 4
Assume that \((\mathrm {H}_{1})\)–\((\mathrm {H}_{3})\) hold. If, for all sufficiently large \(t_{1}\ge t_{0}\),
and
then (1.1) is oscillatory.
Proof
Suppose for the sake of contradiction that y is a nonoscillatory solution of (1.1) on \([t_{0},\infty )\). Without loss of generality, we may take a \(t_{1}\ge t_{0}\) such that \(y(t)>0\) and \(y(\tau (t))>0\) for \(t\ge t_{1}\). By Lemma 1, eight possible cases may occur for \(t\ge t_{1}\). From (2.27), we see that
which in view of \((\mathrm {H}_{1})\) implies that \(Q(\infty ,t_{0}) = \infty \). Thus, by Lemma 2, cases (1)–(4) from Lemma 1 are impossible. Therefore, it is enough to consider cases (5)–(8).
Assume that case (5) holds. From (2.26), we have
Then, proceeding as in the proof of Theorem 2 case (5), we arrive at the contradiction.
Assume that case (6) holds. Let us define the function
Combining (2.16) and (2.18), we obtain
which yields
Also, proceeding as in the proof of Theorem 2 case (6), we derive from (2.16) and (2.17) that
By (1.1), (2.29), and the monotonicity of y, we conclude that
Multiplying the above inequality by \(\pi _{123}\) and integrating the resulting inequality from \(t_{1}\) to t, we get
Therefore, by virtue of (2.28),
which contradicts (2.26).
Assume that case (7) holds. Note that
is necessary for (2.26). Then, for any \(k>0\), we have
Proceeding as in the proof of Theorem 2 case (7), we arrive at the contradiction.
Assume that case (8) holds. Let us define the function
From (2.21), we obtain
On the other hand, from (2.25), we see that
Then
Now, multiplying both sides of the above inequality by \(\pi _{1}(t)\) and integrating the resulting inequality from \(t_{1}\) to t, we get
Hence, in view of (2.30),
which contradicts (2.27).
The proof is complete. □
We conclude the paper by providing an example that illustrates the applicability and strength of the results obtained.
Example 1
Let us consider the fourthorder differential equation of Euler type
where \(q_{0} >0\) and \(\lambda \in (0,1]\). It is easy to verify that condition (2.10) is satisfied and by Theorem 1, we conclude that any nonoscillatory solution of (\(E_{x}\)) converges to zero as t approaches infinity.
By Theorem 2, we see that (\(E_{x}\)) is oscillatory if
The same conclusion follows from Theorem 3 if \(\lambda <1\) and
and from Theorem 4, if
Thus, Theorem 4 provides a stronger result than Theorem 2. Both theorems, however, do not depend on the value λ. In fact, Theorem 3 is more efficient for almost all values of λ, namely for \(\lambda \in (0,0.9215)\).
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Acknowledgements
The authors express their sincere gratitude to the editors 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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This research is supported by NNSF of P.R. China (Grant No. 61503171), CPSF (Grant No. 2015M582091), and NSF of Shandong Province (Grant No. ZR2016JL021). The research of the second and third authors is supported by the grant project KEGA 035TUKE4/2017.
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Grace, S.R., Džurina, J., Jadlovská, I. et al. On the oscillation of fourthorder delay differential equations. Adv Differ Equ 2019, 118 (2019). https://doi.org/10.1186/s1366201920601
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MSC
 34C10
 34K11
Keywords
 Linear differential equation
 Delay
 Fourthorder
 Noncanonical operator
 Oscillation