Abstract
This work is concerned with the oscillatory behavior of solutions of fourth-order neutral differential equations. By using the Riccati transformation and integral averaging techniques we obtain some new Kamenev-type and Philos-type oscillation criteria. Our results extend and improve some known results in the literature. An example is given to illustrate our main results.
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1 Introduction
In this paper, we establish some oscillation criteria for the fourth-order neutral differential equation of the form
where \(L_{y}=r ( t ) ( z^{\prime\prime\prime} ( t ) ) ^{\gamma}\) and \(z ( t ) :=y ( t ) +p ( t ) y ( \tau ( t ) ) \). We suppose that:
- \(( S_{1} ) \):
γ and β are quotients of odd positive integers,
- \(( S_{2} ) \):
\(r,p,q\in C[t_{0},\infty)\), \(r ( t ) >0\), \(r^{\prime} ( t ) \geq0\), \(q ( t ) >0\), \(0\leq p ( t ) < p_{0}<1\), \(\tau,\delta\in C[t_{0},\infty)\), \(\tau ( t ) \leq t\), \(\lim_{t\rightarrow\infty}\tau ( t ) =\lim_{t\rightarrow\infty}\delta ( t ) =\infty\). and
$$ \int_{t_{0}}^{\infty}\frac{1}{r^{1/\gamma} ( s ) } \,\mathrm{d}s=\infty. $$(2)
By a solution of (1) we mean a function \(y\in C^{3}[t_{y},\infty )\), \(t_{y}\geq t_{0}\), satisfying (1) on \([t_{y},\infty)\) and such that \(r ( t ) ( z^{\prime\prime\prime} ( t ) ) ^{\gamma}\in C^{1}[t_{y},\infty)\). We consider only those solutions y of (1) that satisfy \(\sup \{ \vert y ( t ) \vert :t\geq T\}>0 \) for all \(T\geq t_{y}\).
A solution y of (1) is said to be nonoscillatory if it is ultimately positive or negative; otherwise, it is said to be oscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.
Delay differential equations play an important role in applications of real-world life. One area of active research in recent years is studying the sufficient conditions for oscillation of delay differential equations, see [1–23] and the references therein.
In particular, the Emden–Fowler delay differential equations have numerous applications in mathematical, theoretical, and chemical physics; see, for instance, [24–27].
Let us briefly comment on a number of related results, which motivated our study. The authors in [28, 29] were concerned with oscillatory behavior of solutions of fourth-order neutral differential equations and established some new oscillation criteria.
In [30, 31] the authors considered the equation
and established the criteria for the solutions to be oscillatory when \(0\leq p ( t ) <1\).
Xing et al. [32] proved that the equation
is oscillatory if
and
where n is even, and \(\widehat{q} ( t ) :=\min \{ q ( \delta^{-1} ( t ) ) ,q ( \delta^{-1} ( \tau ( t ) ) ) \} \).
Moaaz et al. [33] proved that if there exist positive functions \(\eta , \zeta\in C^{1} ( [ t_{0},\infty ) ,{R} ) \) such that the equations
and
are oscillatory, where
and
then (1) is oscillatory.
Our aim in the present paper is employing the Riccati technique to establish some new Kamenev-type and Philos-type conditions for the oscillation of all solutions of equation (1) under condition (2).
The paper is organized as follows. In Sect. 2, we give four lemmas to prove the main results. In Sect. 3, we establish new oscillation results for (1) by using Riccati transformation. In Sect. 4, we establish some new Kamenev-type oscillation criteria for (1). In Sect. 5, we use the integral averaging technique to establish some new Philos-type conditions for the oscillation of all solutions of equation (1). Finally, we present an example and some conclusions to illustrate the main results.
Remark 1.1
All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all t large enough.
Remark 1.2
Without loss of generality, we can deal only with the positive solutions of (1).
Notation
For convenience, we use the following notation:
and
2 Some auxiliary lemmas
We will employ the following lemmas:
Lemma 2.1
([34], Lemma 2.1)
Let\(\gamma\geq1\)be the ratio of two odd numbers, and let\(V>0\)andUbe constants. Then
Lemma 2.2
([1, Lemma 2.2.3])
Let\(y\in C^{n} ( [ t_{0},\infty ) , ( 0,\infty ) ) \). Assume that\(y^{ ( n ) } ( t ) \)is of fixed sign and not identically zero on\([ t_{0},\infty ) \)and that there exists\(t_{1}\geq t_{0}\)such that\(y^{ ( n-1 ) } ( t ) y^{ ( n ) } ( t ) \leq0\)for all\(t\geq t_{1}\). If\(\lim_{t\rightarrow \infty}y ( t ) \neq0\), then for every\(\mu\in ( 0,1 ) \), there exists\(t_{\mu}\geq t_{1}\)such that
Lemma 2.3
([35])
Let\(y ( t ) \)be a positive andn-times differentiable function on an interval\([ T,\infty ) \)with itsnth derivative\(y^{ ( n ) } ( t ) \)nonpositive on\([ T,\infty ) \), not identically zero on any interval of the form\([ T^{\prime},\infty ) \), \(T^{\prime}\geq T\), and such that\(y^{ ( n-1 ) } ( t ) y^{ ( n ) } ( t ) \leq0\), \(t\geq t_{y}\). Then there exist constants\(0<\theta<1 \)and\(N>0 \)such that
for all sufficient larget.
Lemma 2.4
Assume thatyis an eventually positive solution of (1). Then
Proof
Let y be an eventually positive solution of (1). Then there exists \(t_{1}\geq t_{0}\) such that \(y ( t ) >0\), \(y ( \tau ( t ) ) >0\) and \(y ( \delta ( t ) ) >0\) for \(t\geq t_{1}\). Since \(r^{\prime} ( t ) >0\), we have
for \(t\geq t_{1}\). From the definition of z we get
which, together with (1), gives
The proof is complete. □
3 Oscillation criteria
In this section, we establish new oscillation results for (1) by using the Riccati transformation.
Lemma 3.1
Letybe an eventually positive solution of (1). If there exist constants\(\varepsilon\in ( 0,1 ) \)and\(\zeta>0 \)such that
then
Proof
Let y be an eventually positive solution of (1). Using Lemma 2.4, we obtain that (14) holds. From (17) we see that \(\varphi ( t ) >0 \) for \(t\geq t_{1}\), and using (14), we obtain
From Lemma 2.3 we have
which is
Using (17) we have
Since \(z^{\prime} ( t ) >0\), there exist \(t_{2}\geq t_{1}\) and a constant \(M>0\) such that
Then (22) turns into
that is,
The proof is complete. □
Theorem 3.1
Assume that (2) holds. If
then (1) is oscillatory.
Proof
Let y be an eventually positive solution of (1). Then there exists \(t_{1}\geq t_{0}\) such that \(y ( t ) >0\), \(y ( \tau ( t ) ) >0\), and \(y ( \delta ( t ) ) >0\) for \(t\geq t_{1}\). By Lemma 3.1 we get that (18) holds.
Integrating (18) from t to l, we get
Letting \(l\rightarrow\infty\) and using \(\varphi>0\) and \(\varphi ^{\prime }<0\), we have
This implies
Let \(\lambda=\inf_{t\geq T}\varphi ( t ) /\tilde{A}_{1} ( t ) \). Then obviously \(\lambda\geq1\). Thus from (26) and (29) we see that
or
which contradicts the admissible values of \(\lambda\geq1 \) and \(\gamma >0\). Therefore the proof is complete. □
4 Kamenev-type criteria
In this section, we establish new Kamenev-type oscillation criteria for (1).
Lemma 4.1
Letybe an eventually positive solution of (1), and suppose that (15) holds. If there exist a function\(\pi\in C^{1} ( [ t_{0},\infty ) ,R ^{+} ) \)and constants\(\varepsilon\in ( 0,1 ) \)and\(\zeta>0\)such that
then
Proof
Let y be an eventually positive solution of (1). Using Lemma 2.4, we obtain that (14) holds. From (32) we see that \(\varpi ( t ) >0 \) for \(t\geq t_{1}\), and using (14), we obtain
From Lemma 2.3 we have
which is
By (32) we have
Since \(z^{\prime} ( t ) >0\), there exist \(t_{2}\geq t_{1}\) and \(M>0\) such that
Hence we obtain
that is,
The proof is complete. □
Theorem 4.1
Assume that (2) holds. If there exist a function\(\pi \in C^{1} ( [ t_{0},\infty ) ,R ^{+} ) \)such that
then (1) is oscillatory.
Proof
Let y be a nonoscillatory solution of (1) on \([ t_{0},\infty ) \). Without loss of generality, we can assume that u is eventually positive. Using Lemma 4.1, we get that (33) holds. From Lemma 2.1 we set
Thus we have
and
Since
we get
Hence
and so
which contradicts (36), and this completes the proof. □
5 Philos-type oscillation result
In the section, we employ the integral averaging technique to establish a Philos-type oscillation criterion for (1).
Definition
Let
A kernel function \(H\in C ( D,R ) \) is said to belong to the function class ℑ, written as \(H\in \Im\), if
- (i)
\(H ( t,s ) =0\) for \(t\geq t_{0}\), \(H ( t,s ) >0\), \(( t,s ) \in D_{0}\);
- (ii)
\(H ( t,s ) \) has a continuous and nonpositive partial derivative \(\partial H/\partial s\) on \(D_{0}\), and there exist functions \(\pi \in C^{1} ( [ t_{0},\infty ) , ( 0,\infty ) ) \) and \(h\in C ( D_{0},R ) \) such that
$$ \frac{\partial}{\partial s}H ( t,s ) +\frac{\pi^{\prime } ( s ) }{\pi ( s ) }H ( t,s ) =h ( t,s ) H^{\gamma/ ( \gamma+1 ) } ( t,s ) . $$(44)
Theorem 5.1
Assume that (2) holds. If there exist a positive function\(\pi\in C^{1} ( [ t_{0},\infty ) ,R ) \)such that
then (1) is oscillatory.
Proof
Let y is a nonoscillatory solution of (1) on \([ t_{0},\infty ) \). Without loss of generality, we can assume that u is eventually positive. From Lemma 4.1 we get that (33) holds. Multiplying (33) by \(H ( t,s ) \) and integrating the resulting inequality from \(t_{1}\) to t, we find that
From (44) we get
Using Lemma 2.1 with \(V=B_{3} ( s ) H ( t,s ) \), \(U=h ( t,s ) H^{\gamma/ ( \gamma+1 ) } ( t,s ) \), and \(y=\varpi ( s ) \), we get
which implies that
a contradiction to (45).
Theorem 5.1 is proved. □
Corollary 5.1
If condition (45) in Theorem5.1is replaced by the conditions
and
then (1) is oscillatory.
Example
Consider the differential equation
where \(q_{0}>0\) is a constant. Note that \(\gamma=\beta=1\), \(r ( t ) =t\), \(p_{0} ( t ) =1/2\), \(q ( t ) =q_{0}/t^{4}\), \(\delta ( t ) =t/2\), and \(\tau ( t ) =t/3\). If we set \(\pi ( t ) =t^{2}\), then
and
Thus we get
Therefore by Theorem 4.1 all solutions of (49) are oscillatory if \(q_{0}>32\).
Remark 5.1
We can easily see that the results obtained in [32, 33] cannot be applied to (36), so our results are new.
Remark 5.2
We can generalize our results by studying the equation
For this, we leave the results to interested researchers.
Remark 5.3
For interested researchers, there is a good problem of finding new results for (1) where
6 Conclusions
The aim of this paper was to provide a study of asymptotic nature for a class of fourth-order neutral delay differential equations. We used a Riccati substitution and the integral averaging technique to ensure that every solution of the studied equation is oscillatory. The results presented complement some of the known results reported in the literature.
A further extension of this paper is using our results to study a class of systems of higher-order neutral differential equations, including those of fractional order. Some research in this area is in progress.
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Bazighifan, O. Kamenev and Philos-types oscillation criteria for fourth-order neutral differential equations. Adv Differ Equ 2020, 201 (2020). https://doi.org/10.1186/s13662-020-02661-6
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DOI: https://doi.org/10.1186/s13662-020-02661-6