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Oscillation of fourthorder strongly noncanonical differential equations with delay argument
Advances in Difference Equationsvolume 2019, Article number: 388 (2019)
Abstract
The aim of this paper is to study oscillatory properties of the fourthorder strongly noncanonical equation of the form
where \(\int ^{\infty }\frac{1}{r_{i}(s)}\,\mathrm {d}{s}<\infty \), \(i=1,2,3\). Reducing possible classes of the nonoscillatory solutions, new oscillatory criteria are established.
Introduction
In the paper, we consider the fourthorder delay differential equation
where \(r_{i} \in C^{(4i)}(t_{0},\infty )\), \(r_{i}(t)>0\), \(i=1,2,3\), \(p(t)\in C(t_{0},\infty )\), \(p(t)>0\), \(\tau (t)\in C(t_{0},\infty )\), \(\tau (t)\le t\), \(\tau '(t)>0\), and \(\tau (t)\to \infty \) as \(t\to \infty \).
By a solution of Eq. (E) we mean all continuous functions \(y(t)\) for which
exist and satisfy Eq. (E) on \([T_{y},\infty )\). We consider only those solutions \(y(t)\) of (E) which satisfy \(\sup \{ \vert y(t) \vert :t \geq T\}>0\) for all \(T\geq T_{y}\). We assume that (E) possesses such a solution. A solution of (E) is called oscillatory if it has arbitrarily large zeros on \([T_{y},\infty )\) and otherwise it is called nonoscillatory. Equation (E) is said to be oscillatory if all its solutions are oscillatory.
Throughout the paper it is supposed that Eq. (E) is strongly noncanonical, that is,
Fourthorder differential equations naturally appear in models concerning physical, biological, and chemical phenomena, such as, for instance, problems of elasticity, deformation of structures, or soil settlement, see, for example, [2]. In mechanical and engineering problems, questions concerning the existence of oscillatory solutions play an important role. During the past decades, there has been a constant interest in obtaining sufficient conditions for oscillatory properties of different classes of fourthorder differential equations with deviating argument, see [2, 3, 6, 8,9,10,11,12,13,14,15,16,17,18,19,20].
In general, there are two approaches for the investigation of higherorder differential equations with noncanonical operators. One method requires to find a canonical representation of studied equation with closed form formulas for coefficients. For details, see [1, 4, 5, 7]. The second approach is to establish the conditions that reduce the number of possible classes of nonoscillatory solutions and consequently to find conditions for oscillation of (E). Our method belongs to the second one and yields easily verifiable oscillation criteria.
Preliminary results
Throughout the paper we assume that (1.1) holds, and so we can employ the notation
and
where \(i,j,k\in \{1,2,3\}\) are mutually different. To simplify the writing of quasiderivatives, we denote
where formally \(r_{4}(t)\equiv 1\). We start with the following auxiliary results which are elementary but very useful.
Lemma 1
Let (1.1) hold. Then
Proof
Since
an integration of this equality from t to ∞ yields
□
Lemma 2
Let (1.1) hold. Then
Proof
Proof of this lemma is similar to that of Lemma 1 and so it can be omitted. □
It follows from a generalization of lemma of Kiguradze [9] that the set of positive solutions of (E) has the following structure.
Lemma 3
Assume that \(y(t)\) is a positive solution of (E). Then \(y(t)\) satisfies one of the following conditions:
 \((N_{1})\)::

\(L_{1}y(t)>0\), \(L_{2}y(t)>0\), \(L_{3}y(t)>0\), \(L_{4}y(t)<0 \),
 \((N_{2})\)::

\(L_{1}y(t)>0\), \(L_{2}y(t)>0\), \(L_{3}y(t)<0\), \(L_{4}y(t)<0\),
 \((N_{3})\)::

\(L_{1}y(t)>0\), \(L_{2}y(t)<0\), \(L_{3}y(t)<0\), \(L_{4}y(t)<0\),
 \((N_{4})\)::

\(L_{1}y(t)>0\), \(L_{2}y(t)<0\), \(L_{3}y(t)>0\), \(L_{4}y(t)<0\),
 \((N_{5})\)::

\(L_{1}y(t)<0\), \(L_{2}y(t)>0\), \(L_{3}y(t)>0\), \(L_{4}y(t)<0\),
 \((N_{6})\)::

\(L_{1}y(t)<0\), \(L_{2}y(t)<0\), \(L_{3}y(t)>0\), \(L_{4}y(t)<0\),
 \((N_{A})\)::

\(L_{1}y(t)<0\), \(L_{2}y(t)>0\), \(L_{3}y(t)<0\), \(L_{4}y(t)<0\),
 \((N_{B})\)::

\(L_{1}y(t)<0\), \(L_{2}y(t)<0\), \(L_{3}y(t)<0\), \(L_{4}y(t)<0\).
The first two results are intended to reduce the number of classes that will be investigated.
Theorem 1
If
then a positive solution \(y(t)\) of (E) does not satisfy \((N_{1})\)–\((N_{4})\) of Lemma 3.
Proof
Assume on the contrary that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{1})\) or \((N_{4})\) of Lemma 3 for \(t\geq t_{1}\geq t_{0}\). Since \(y(t)\) is positive and nondecreasing, there exists a positive constant \(k>0\) such that \(y(t)\geq k\) for \(t\geq t_{1}\).
Integrating (E) from \(t_{1}\) to ∞, we get
which is a contradiction with respect to (2.1).
Now, we assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{2})\) of Lemma 3 for \(t\geq t_{1}\). Integrating (E) from \(t_{1}\) to t and using that \(y(t)\) is positive and nondecreasing, we get
Integrating the above inequality from \(t_{1}\) to ∞, we obtain
which contradicts (2.1).
Finally, we assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{3})\) of Lemma 3 for \(t\geq t_{1}\). Similarly as above, we are led to (2.2). Integrating this from \(t_{1}\) to t, we obtain
An integration from \(t_{1}\) to ∞ yields
which is a contradiction and the proof is finished. □
Theorem 2
If
then the positive solution \(y(t)\) of (E) does not satisfy \((N_{5})\), \((N_{6})\) of Lemma 3.
Proof
Assume on the contrary that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{5})\) of Lemma 3 for \(t\geq t_{1}\geq t_{0}\). Since \(L_{2}y(t)\) is a positive and increasing function there exists a positive constant \(k>0\) such that
for \(t\geq t_{1}\). Integrating the previous inequality from t to ∞, we have
After integration from \(\tau (t)\) to ∞, we get
On the other hand, in view of (2.4), an integration of (E) from \(t_{1}\) to ∞ yields
which contradicts (2.3).
Now, we assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{6})\) of Lemma 3 for \(t\geq t_{1}\geq t_{0}\). Seeing that \(L_{1}(y)\) is a negative and decreasing function, there exists a constant \(k>0\) such that
for \(t\geq t_{1}\), and integrating this inequality from \(\tau (t)\) to ∞, we have
Integrating (E) from \(t_{1}\) to ∞ and using (2.5), we obtain
which is a contradiction to (2.3). The proof is completed. □
Theorems 2.1 and 2.3 reduce the number of possible nonoscillatory solutions of (E) only to \((N_{A})\) or \((N_{B})\), which essentially simplifies examination of (E).
Main results
Now we provide useful monotonic properties of nonoscillatory solutions of (E) satisfying conditions \((N_{A})\) or \((N_{B})\) of Lemma 3. We begin with the following auxiliary result.
Lemma 4
Assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{A})\) of Lemma 3 and
Then
Proof
Assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{A})\) of Lemma 3 for \(t\geq t_{1} \geq t_{0}\).
Since \(y(t)\) is positive and decreasing, there exists \(\lim_{t\to \infty } y(t)=\ell \geq 0 \). We claim that \(\ell =0\). If not, then \(y(\tau (t))\geq \ell >0\), eventually, let us say for \(t\geq t _{1}\). An integration of (E) from \(t_{1}\) to t yields
Integrating from \(t_{1}\) to ∞, we obtain
which contradicts (3.1), and we conclude that \(y(t)\to 0\) as \(t\to \infty \).
On the other hand, since \(r_{1} y'\) is positive and decreasing, there exists
We assume on the contrary that \(\ell >0\). Then
Integrating from t to ∞, one gets
which setting into (3.3) yields
An integration from \(t_{1}\) to ∞ yields
This is a contradiction, and the proof is complete now. □
Theorem 3
Let (3.1) hold. Assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{A})\) of Lemma 3. Then
Proof
Assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{A})\) of Lemma 3 for \(t\geq t_{1} \geq t_{0}\). At first, we shall show that \(\frac{y(t)}{\mbox{$\pi$}_{12}(t)}\) is decreasing. Employing (3.2) and using that \(L_{2}y(t)\) is positive and decreasing, we have
which implies
Thus, \(\frac{r_{1}(t)y'(t)}{\mbox{$\pi$}_{2}(t)}\) is increasing, and in view of (3.2), we get
which yields
and we conclude that \(\frac{y(t)}{\mbox{$\pi$}_{12}(t)}\) is a decreasing function.
Now, we shall prove that \(\frac{y(t)}{\mbox{$\pi$}_{123}(t)}\) is an increasing function. Employing that \(L_{3}y(t)\) is a negative and decreasing function, we have
which yields
and \(\frac{L_{2}y(t)}{\pi _{3}(t)}\) is increasing. Therefore,
This inequality implies that \(\frac{r_{1}(t)y'(t) }{\pi _{23}(t)}\) is increasing. Finally,
which implies
and we conclude that \(\frac{y(t)}{\mbox{$\pi$}_{123}(t)}\) is increasing. □
Theorem 4
Assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{B})\) of Lemma 3. Then
Proof
Assume that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{B})\) of Lemma 3 for \(t\geq t_{1} \geq t_{0}\). Applying the monotonic property of \(r_{1}(t)y'(t)\), we get
which gives
and we conclude that \(\frac{y(t)}{\mbox{$\pi$}_{1}(t)}\) is increasing. □
Now, we are prepared for establishing the criteria for the essential classes \((N_{A})\) and \((N_{B})\) to be empty.
Theorem 5
Let (3.1) hold. If
where
then the class \((N_{A})\) of Lemma 3 is empty.
Proof
Assume on the contrary that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{A})\) of Lemma 3 for \(t\geq t_{1}\geq t_{0}\). Integrating (E) twice from \(t_{1}\) to t and from t to ∞, we obtain
Changing the order of integrating in the previous inequality, we see
Integrating the above inequality from t to ∞, one gets
It follows from Lemma 1 that \(\mbox{$\pi$}_{23}(s)+\mbox{$\pi$}_{32}(s)=\mbox{$\pi$}_{2}(s)\mbox{$\pi$}_{3}(s) \), and so
Integrating once more from t to ∞ and employing Lemma 2, we have
Then
Using that \(\frac{y(t)}{\mbox{$\pi$}_{12}(t)}\) is decreasing and \(\frac{y(t)}{\mbox{$\pi$}_{123}(t)}\) is increasing, the last inequality yields
Then
which contradicts (3.7). □
Theorem 6
If
where
then the class \((N_{B})\) of Lemma 3 is empty.
Proof
Assume on the contrary that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{B})\) of Lemma 3 for \(t\geq t_{1}\geq t_{0}\). Integrating (E) twice from \(t_{1}\) to t and thereafter switching the order of integration, we obtain
Integrating the above inequality again from \(t_{1}\) to t and changing the order of integration, we get
Applying Lemma 1, we can write
Integrating the previous inequality from t to ∞ and consequently switching the order of integration, we obtain
Employing the equality \(\mbox{$\pi$}_{123}(t)+\mbox{$\pi$}_{32}(t)\mbox{$\pi$}_{1}(t)\mbox{$\pi$}_{3}(t)\mbox{$\pi$}_{12}(t)=\mbox{$\pi$}_{321}(t)\), we can rewrite the above inequality into a simpler form
Using notation for \(H(s,t)\)
then
Applying the monotonic properties of \(y(t)\) (decreasing) and \(\frac{y(t)}{\mbox{$\pi$}_{1}(t)}\) (increasing), we have
which implies
This contradicts (3.8), and the proof is complete. □
The following result is intended to avoid evaluation of function \(H(s,t)\) and to simplify criterion (3.7).
Corollary 1
If
then the class \((N_{B})\) of Lemma 3 is empty.
Proof
Assume on the contrary that \(y(t)\) is an eventually positive solution of (E) satisfying condition \((N_{B})\) of Lemma 3 for \(t\geq t_{1}\geq t_{0}\). Proceeding similarly as in the proof of Theorem 6, we get (3.9). It follows from monotonic properties of \(H(s,t)\) and Lemma 2 that
Using (3.11) in (3.9), we obtain
Then
Taking into account that \(y(t)\) is decreasing and \(\frac{y(t)}{\mbox{$\pi$}_{1}(t)}\) is increasing finally, we have
which contradicts the assumption of the corollary. □
Picking up the previous results, we can establish easily verifiable oscillatory criteria.
Theorem 7
Let (2.1), (3.1), (3.7), (3.8) hold. Then (E) is oscillatory.
Theorem 8
Let (2.1), (3.1), (3.7), (3.10) hold. Then (E) is oscillatory.
We support our results with an illustrative example, in which also some comparison with existing latest ones is made.
Example 1
Let us consider noncanonical fourthorder delay differential equation in the form
where \(a>0\), \(\lambda \in (0,1)\), \(\mbox{$\pi$}_{i}(t)={1}/{t}\), \(\mbox{$\pi$}_{ij}(t)= {1}/(2t^{2})\), \(\mbox{$\pi$}_{123}(t)=\mbox{$\pi$}_{321}(t)={1}/(6t^{3})\). It is easy to verify that (2.1), (3.1) hold.
Condition (3.7) takes the form
Condition (3.8) takes the form
By Theorem 7, Eq. (\(E_{x}\)) is oscillatory provided that both (3.12) and (3.13) hold. In particular case where \(\lambda =0.8\), conditions (3.12) and (3.13) reduce to \(a>1.963\). The oscillatory criterion obtained by a different technique presented in paper [4] gives oscillation of (\(E_{x}\)) if \(a>7.913\).
On the other hand, for \(\lambda =0.9\), Theorem 7 guarantees oscillation of Eq. (\(E_{x}\)) provided that \(a>1.77\), while the best criterion from [7] requires \(a>3.50\).
So our results are more efficient than the previous ones.
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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 that helped to improve the presentation of the results and accentuate important details.
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Funding
The paper has been supported by the grant project KEGA 035TUKE4/2017.
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Correspondence to B. Baculikova.
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MSC
 34K11
 34C10
Keywords
 Noncanonical operator
 Fourth order differential equations
 Oscillation