Abstract
In this paper, we discuss the oscillatory behavior of solutions of a class of Super-linear fourth-order differential equations with several sub-linear neutral terms using the Riccati and generalized Riccati transformations. Some Kamenev–Philos-type oscillation criteria are established. New oscillation criteria are deduced in both canonical and non-canonical cases. An illustrative example is given.
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1 Introduction
The aim of this paper is to discuss the oscillatory behavior of solutions of a class of super-linear fourth-order neutral differential equations of the type,
where \(z(t)=x ( t ) +\sum_{j=1}^{n}a_{j} ( t ) x^{ \alpha _{j}} ( \sigma _{j} ( t ) )\), m, n are positive integers, and \(\alpha _{j}\), γ are ratios of odd positive integers and \(0<\alpha _{j}\leq 1 \), \(\gamma\geq 1\), under the conditions
and
Throughout the paper, we assume the following assumptions
- \(( A_{1} ) \):
-
\(r ( t ) \in C^{1} ( [t_{0},\infty ), ( 0, \infty ) )\), \(r^{\prime} ( t ) \geq 0\);
- \(( A_{2} )\):
-
\(a_{j} ( t ),\sigma _{j} ( t ) ,\tau _{i} ( t ) \in C[t_{0},\infty ))\), \(\sigma _{j} ( t ) \leq t\), \(\lim_{t\rightarrow \infty}\sigma _{j} ( t )=\infty \);
- \(( A_{3} ) \):
-
there exists a function \(\tau \in C^{1} ( [t_{0},\infty ), R )\) such that \(\tau ( t ) \leq \tau _{i} ( t ) \) for \(i=1,2,\ldots,m\), \(\tau ( t ) \leq t\), \(\tau ^{\prime} ( t ) >0\) and \(\lim_{t\rightarrow \infty}\tau ( t ) =\infty \);
- \(( A_{4} ) \):
-
\(0\leq a_{j} ( t ) \leq a_{0j} ( t )\), \(\sum_{j=1}^{n}a_{0j} ( t ) <1\), \(f_{i} ( t,x ) \in C ( [t_{0},\infty )\times R,R ) \) satisfy \(xf_{i} ( t,x )>0\) for all \(x\neq 0\), and there exist positive continuous functions \(q_{i} ( t ) \) defined on \([t_{0},\infty )\) such that \(\vert f_{i} ( t,x ) \vert \geq q_{i} ( t ) \vert x \vert ^{\gamma }\).
By a solution of (1.1), we mean a nontrivial real function \(x ( t ) \) such that \(r ( t ) ( [ x ( t ) +\sum_{j=1}^{n}a_{j} ( t ) x^{\alpha _{j}} ( \sigma _{j} ( t ) ) ] ^{\prime \prime \prime} ) ^{\gamma}\) is continuously differentiable satisfying (1.1) for any \(t_{1}\geq t_{0}\).
A solution of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
Oscillation phenomena take part in different models from real-world applications; see, e.g., paper [8] for more details. In the last three decades, there has been considerable interest in studying the oscillation of solutions of several kinds of differential equations [1–5, 7, 8, 10–20, 22–24, 26–39]. The half-linear equations have numerous applications in the study of p-Laplace equations, non-Newtonian fluid theory, porous medium, etc.; see, e.g., papers [6, 21, 25] for more details. In particular, papers [11, 24] were concerned with the oscillation of various classes of half-linear differential equations, whereas the papers [3–5, 7, 10, 20, 26, 38] were concerned with the oscillatory behavior of the fourth-order differential equation (1.1) and its special cases. In what follows, we briefly comment on a number of closely related results which motivated our work. The authors in [3, 4, 26] discussed in their recent papers, the special case of (1.1) of the form,
Under the condition (1.2), Dassios and Bazighifan in [10] discussed the oscillation of the same equation under condition (1.3). In [20], Li et al. studied the oscillatory behavior of a class of fourth-order differential equations with the p-Laplacian-like operator of the type,
where \(z(t)=x ( t ) +a ( t ) x ( \sigma ( t ) )\). Under the condition \(\int _{t_{0}}^{\infty } \frac{1}{r^{\frac{1}{p-2}} ( t ) }\,dt<\infty \), they used the Riccati transformation and integral averaging technique and presented a Kamenev-type oscillation criterion.
More recently, Bazighifan et al. [5] studied the asymptotic behavior of solutions of the fourth-order neutral differential equation with the continuously distributed delay of the form
where α, β are quotients of odd positive integers, and \(\beta \geq \alpha \) under the condition (1.2).
2 Preliminaries
The following preliminary results will be needed for our proofs.
Lemma 1
([9])
Let \(h>0\). Then
Lemma 2
([28])
Let \(z ( t )\) be a positive and n-times differentiable function on an interval \([T,\infty )\) with non-positive nth derivative \(z^{ ( n ) } ( t ) \) on \([T,\infty )\), which is not identically zero on any interval of the form \([T^{\prime},\infty )\), \(T^{\prime}\geq T\) and such that \(z^{ ( n-1 ) } ( t ) z^{ ( n ) } ( t ) \leq 0\). Then, there exist constants \(0<\theta <1\) and \(N>0\) such that \(z^{\prime} ( \theta t ) \geq Nt^{n-2}z^{ ( n-1 ) } ( t ) \) for all sufficient large t.
Lemma 3
([26])
Let \(z^{ ( n ) } ( t )\) be of fixed sign and \(z^{ ( n-1 ) } ( t ) z^{ ( n ) } ( t ) \leq 0\) for all \(t\geq t_{1}\). If \(\lim_{t \rightarrow \infty}z ( t ) \neq 0\), then for every \(\lambda \in ( 0,1 ) \), there exists \(t_{\lambda}\) ≥t such that \(z ( t ) \geq \frac{\lambda}{ ( n-1 ) !}t^{n-1} \vert z^{ ( n-1 ) } ( t ) \vert \) for \(t\geq t_{\lambda}\).
Lemma 4
([2])
Let α is a ratio of two odd numbers. Suppose that U, V are constants with \(V>0\). Then, \(UY-VY^{\frac{ ( \gamma +1 ) }{\gamma}}\leq \frac{\gamma ^{\gamma}}{ ( \gamma +1 ) ^{\gamma +1}} \frac{U^{\gamma +1}}{V^{\gamma}}\).
Lemma 5
Assume that \(x ( t )\) is an eventually positive solution of (1.1), \(z^{\prime} ( t ) >0\), and there exists a positive decreasing function \(\delta ( t ) \in C ( [t_{0},\infty ) ) \) tending to zero such that \(\theta ( \tau _{i} ( t ) ) >0\) for \(t\geq t_{0}\) where \(\theta ( t ) =1-\sum_{j=1}^{n}\alpha _{j}a_{j} ( t ) -\frac{1}{\delta ( t ) }\sum_{j=1}^{n} ( 1- \alpha _{j} ) a_{j} ( t )\). Then,
Proof
Let x be an eventually positive solution of Eq. (1.1). Then, there exists a \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\), \(x ( \sigma _{j} ( t ) ) >0\) and \(x ( \tau _{i} ( t ) )>0\) for \(t\geq t_{1}\). Now from the definition of z, we have
Then, by Lemma 1, we have
Now since \(z ( t ) \) is positive and increasing, and \(\delta ( t ) \) is a positive decreasing function tending to zero, then there exists a \(t_{2}\geq t_{1}\) such that \(z ( t ) \geq \delta ( t ) \), and
That is \(x ( t ) \geq \theta ( t ) z ( t )\). Therefore, from (1.1), it follows that
Thus, the proof is completed. □
The following two auxiliary results are very similar to those reported in [3] and [10].
Lemma 6
Let \(x ( t ) \) be a positive solution of (1.1). If (1.2) is satisfied, then there exists \(t\geq t_{1}\) such that
Lemma 7
Let \(x ( t ) \) be a positive solution of (1.1). If (1.3) is satisfied, then there exist three possible cases for sufficiently large \(t\geq t_{1}\)
- \(( S_{1} ) \):
-
\(z ( t ) >0\), \(z^{\prime} ( t ) >0\), \(z^{\prime \prime \prime} ( t ) >0\), \(z^{ ( 4 ) } ( t ) \leq 0\);
- \(( S_{2} ) \):
-
\(z ( t ) >0\), \(z^{\prime} ( t ) >0\), \(z^{\prime \prime} ( t )>0\), \(z^{\prime \prime \prime} ( t )<0\);
- \(( S_{3} ) \):
-
\(z ( t ) >0\), \(z^{\prime} ( t ) <0\), \(z^{\prime \prime} ( t ) >0\), \(z^{\prime \prime \prime} ( t )<0\).
3 Main results
We first consider the case \(R ( t_{0} ) =\infty \).
Theorem 8
If there exist \(\eta ( t ) \in C^{1} ( [t_{0},\infty ), ( 0, \infty ) ) \), \(b ( t ) \in C^{1} ( [t_{0},\infty ),[0,\infty ) )\), \(\zeta \in ( 0,1 ) \) and \(\epsilon >0\) such that
then (1.1) is oscillatory, where \(Q ( t ) =\eta ( t ) \sum_{i=1}^{m}q_{i} ( t ) \theta ^{\gamma } ( \tau _{i} ( t ) ) -\eta ( t ) [ r ( t ) b ( t ) ] ^{\prime}+\zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) r ( t ) \eta ( t ) b^{1+\frac{1}{\gamma}} ( t )\).
Proof
Suppose for the contrary that x is an eventually positive solution of (1.1). Then there exists a \(t_{1}\geq t_{0}\) such that \(x ( t )>0\), \(x ( \sigma _{j} ( t ) )>0\) and \(x ( \tau _{i} ( t ) )>0\) for \(t\geq t_{1}\). Using Lemma 5, we obtain (2.1). Define
It is clear that \(\psi ( t ) >0\) for \(t\geq t_{1}\), and
Thus, by (2.1), it follows that
By Lemma 2, we have
However, since \(z ( t )\) is increasing, then \(z ( \tau ( t ) ) \geq z ( \zeta \tau ( t ) ) \). Therefore,
Moreover, since from (3.2), we have
then
As in [35], we use the inequality
with
to get
Using inequalities (3.3) and (3.4), for \(t\geq T\), we have
Then,
i.e.
Now let
Then, by Lemma 4, we obtain
Thus, we have
Integrating (3.6) from T to t, we get
which contradicts (3.1), and this completes the proof. □
The following result deals with the Kamenev-type oscillation for Eq. (1.1) under the condition (1.2).
Theorem 9
If
then (1.1) is oscillatory.
Proof
Let x be a nonoscillatory solution of (1.1) on \([t_{0},\infty )\). Without loss of generality, we may assume that x is an eventually positive solution. Define \(\psi ( t )\) as in (3.2). Then, following the same steps as in the proof of Theorem 8, we arrive at (3.6). Multiplying (3.6) by \(( t-s ) ^{n}\) and integrating the resulting inequality from \(t_{0}\) to t, we have
However, since
then from (3.8), we get
Hence,
and so
which contradicts (3.7), and this completes the proof. □
Now we are going to discuss the so called Philos-type oscillation criteria for Eq. (1.1) under condition (1.2), but we first outline the following definition.
Definition 10
Let \(D= \{ ( t,s ) \in R^{2}:t\geq s\geq t_{0} \} \) and \(D_{0}= \{ ( t,s ) \in R^{2}:t>s \geq t_{0} \}\). The functions \(K_{i} ( t,s ) \in C ( D,R ) \), \(i=1,2\) are said to belong to the class X (written \(K_{i}\in X\)) if they satisfy
-
(I)
\(K_{i} ( t,t ) =0\) for \(t\geq t_{0}\), \(K_{i} ( t,s ) >0\), \(( t,s ) \in D_{0}\)
-
(II)
\(\frac{\partial K_{i} ( t,s ) }{\partial s}\leq 0\), and there exist \(\rho ( t ) \in C^{1} ( [t_{0},\infty ), ( 0,\infty ) ) \) and \(L_{i} ( t,s ) \in C ( D,R ) \) such that
$$ -\frac{\partial K_{1} ( t,s ) }{\partial s}=K_{1} ( t,s ) \biggl[ \frac{\eta ^{\prime} ( t ) }{\eta ( t ) }+ ( \gamma +1 ) \zeta \epsilon \tau ^{\prime} ( t ) \tau ^{2} ( t ) b^{\frac{1}{\gamma}} ( t ) \biggr] +L_{1} ( t,s ), $$and
$$ \frac{\partial K_{2} ( t,s ) }{\partial s}+ \frac{\rho ^{\prime } ( t ) }{\rho ( t ) }K_{2} ( t,s ) = \frac{L_{2} ( t,s ) }{\rho ( t ) } \bigl[ K_{2} ( t,s ) \bigr] ^{\frac{\gamma}{\gamma +1}}. $$
Theorem 11
Assume that there exists a function \(K_{1}\in X\) such that
Then, Eq. (1.1) is oscillatory.
Proof
Let x be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x is an eventually positive solution of (1.1). Now define \(\psi ( t ) \) as in (3.2). Following the same steps as in the proof of Theorem 8, we arrive at (3.5). Multiplying (3.5) by \(K_{1} ( t,s ) \) and integrating the resulting inequality from T to t, we have
where
Then, we have
Putting \(U= \vert L_{1} ( t,s ) \vert \), \(V=K_{1} ( t,s ) B ( s )\) and then using Lemma 4, we obtain
Then,
Hence,
for all sufficiently large t, which contradicts (3.9). □
Theorem 12
Assume that
where
Then, (1.1) is oscillatory.
Proof
Assume that \(x ( t ) \) is an eventually positive solution of (1.1). Then, there exists a \(t_{1}\geq t_{0}\) such that \(x ( t ) >0\), \(x ( \sigma _{j} ( t ) ) >0\) and \(x ( \tau _{i} ( t ) ) >0\) for \(t\geq t_{1}\). Using Lemma 5, we arrive at (2.1). Define
Then, it is clear by (2.1) that
Since, by Lemma 2, we have
then
i.e.
Integrating the above inequality from t to l, we get
Letting \(l\rightarrow \infty \) and using the fact that \(\omega ( t ) \) is positive and decreasing, we get
Let \(\delta =\inf_{t\geq T} \frac{\omega ( t ) }{\phi _{1}^{\ast } ( t ) }\). Then obviously \(\delta \geq 1\), and by (3.10) and (3.11), it follows that
which contradicts the admissible values of \(\delta \geq 1\) and \(\gamma \geq 1\). Therefore, the proof is completed. □
4 The case \(R ( t_{0} ) <\infty \)
Now we are going to discuss the oscillatory behavior of Eq. (1.1) under the condition (1.3). First we need the following lemma.
Lemma 13
Assume that x is an eventually positive solution of Eq. (1.1) and \(( S_{2} ) \) holds. If
then
Proof
Since x is an eventually positive solution of Eq. (1.1) and \(( S_{2} ) \) holds, then using Lemma 5, we obtain (2.1). Now from Eq. (4.1), we see that \(\vartheta ( t ) <0\) for \(t\geq t_{1}\), and
This with (2.1) and (4.1) leads to
i.e.
Now since \(z^{\prime \prime} ( t )\) is decreasing, then it follows that \(- \frac{z^{\prime \prime} ( \tau ( t ) ) }{z^{\prime \prime} ( t ) }\leq -1\). Consequently, by Lemma 3, we have \(z ( \tau ( t ) ) \geq \frac{\lambda}{2}\tau ^{2} ( t ) z^{\prime \prime} ( \tau ( t ) ) \). Then
The proof is completed. □
Theorem 14
Suppose that (3.9) holds, and
If
or
then Eq. (1.1) is oscillatory.
Proof
Suppose for the contrary that there exists a nonoscillatory solution \(x ( t ) >0\) of (1.1). Then, we have one of the three possible cases of Lemma 7. We first assume that \(( S_{1} ) \) holds. Then by Theorem 11, if (3.9) holds, Eq. (1.1) is oscillatory. Secondly, if \(( S_{2} ) \) holds, then by Lemma 13, we get (4.2). Multiplying (4.2) by \(K_{2} ( t,s ) \) and integrating from \(t_{1}\) to t, we obtain
Setting
Then, by Lemma 4, we have
Hence,
This contradicts (4.3). Finally, assume the case \(( S_{3} ) \). Hence, since \(r ( t ) ( z^{{\prime \prime \prime}} ( t ) ) ^{\gamma}\) is nonincreasing, then for \(s\geq t\geq t_{1,}\) we have
Going through as in the proof of Theorem 2.3 case 1 in [20], we get a contradiction with (4.4) and (4.5), and so the proof is completed. □
Remark 15
Theorem 14 remains true if we used (3.1), or (3.7), or (3.10) instead of (3.9).
5 Example
Example 16
Consider the fourth-order differential equation
Here \(\gamma =1\), \(r ( t ) =t\), \(a_{1}=\frac{1}{t^{3}}\), \(a_{2}= \frac{1}{t^{4}}\), \(\alpha _{1}=\frac{1}{3}\), \(\alpha _{2}=\frac{1}{5}\), \(q_{1}= \frac{3}{t}\), \(q_{2}=\frac{1}{t^{3}}\), \(\tau _{1} ( t ) =t\), \(\tau _{2} ( t ) =2t\). Let \(\tau ( t ) =\frac{t}{2}\rightarrow \) \(\tau ( t ) \leq \) \(\tau _{i} ( t ) \), \(\lim_{t\rightarrow \infty }\) \(\tau ( t ) =\infty \), \(\tau ^{\prime } ( t ) =\frac{1}{2}>0\). Therefore, the conditions \(( A_{1} ) - ( A_{5} ) \) and (1.2) are satisfied. Choosing \(\delta ( t ) =\frac{1}{t}\). Then \(\delta ( t ) \rightarrow 0\) for \(t\rightarrow \infty \). Moreover, \(\theta ( \tau _{1} ( t ) ) =\theta ( t ) = [ 1-\frac{2}{3t^{2}}-\frac{17}{15t^{3}}-\frac{1}{5t^{4}} ] >0\) for \(t\geq 2\), and \(\theta ( \tau _{2} ( t ) ) =\theta ( 2t ) = [ 1-\frac{1}{6t^{2}}-\frac{17}{120t^{3}}- \frac{1}{80t^{4}} ] >0\) for \(t\geq 2\). Choosing \(\eta ( t ) =1\), \(b ( t ) =\frac{1}{t^{2}}\), we have
Therefore, by Theorem 8, every solution of (5.1) is oscillatory.
6 Conclusions
In this paper, we consider a general class of super-linear fourth-order differential equations with several sub-linear neutral terms of the type (1.1). Using the Riccati and generalized Riccati transformations, we establish new oscillation criteria in both cases of canonical case \(\int _{t_{0}}^{\infty } \frac{1}{r^{\frac{1}{\alpha }} ( t ) }\,dt=\infty \) and non-canonical case \(\int _{t_{0}}^{\infty } \frac{1}{r^{\frac{1}{\alpha }} ( t ) }\,dt<\infty \). With the help of the methods given in this paper, we derive some the Kamenev–Philos-type oscillation criteria for (1.1). An illustrative example is given. For interested researchers, there is a good deal of finding new results for (1.1) when \(z(t)=x ( t ) -\sum_{j=1}^{n}a_{j} ( t ) x^{ \alpha _{j}} ( \sigma _{j} ( t ) )\).
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References
Agarwal, R.P., Bohner, M., Li, T., Zhang, C.: Oscillation of third-order nonlinear delay differential equations. Taiwan. J. Math. 17(2), 545–558 (2013)
Agarwal, R.P., Zhang, C., Li, T.: Some remarks on oscillation of second order neutral differential equations. Appl. Math. Comput. 274, 178–181 (2016)
Bazighifan, O.: Kamenev and Philos-types oscillation criteria for fourth-order neutral differential equations. Adv. Differ. Equ. 201, 1–12 (2020)
Bazighifan, O., Cesarano, C.: A Philos-type oscillation criteria for fourth-order neutral differential equations. Symmetry 379(12), 1–10 (2020)
Bazighifan, O., Minhos, F., Moaaz, O.: Sufficient conditions for oscillation of fourth-order neutral differential equations with distributed deviating arguments. Axioms 39(9), 1–11 (2020)
Bohner, M., Li, T.: Kamenev-type criteria for nonlinear damped dynamic equations. Sci. China Math. 58(7), 1445–1452 (2015)
Chatzarakis, G.E., Elabbasy, E.M., Bazighifan, O.: An oscillation criterion in 4th-order neutral differential equations with a continuously distributed delay. Adv. Differ. Equ. 3366, 1 (2019)
Chiu, K.S., Li, T.: Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments. Math. Nachr. 292(10), 2153–2164 (2019)
Cloud, M.J., Drachman, B.C.: Inequalities with Applications to Enginering. Springer, New York (1998)
Dassios, I., Bazighifan, O.: Oscillation conditions for certain fourth-order non-linear neutral differential equation. Symmetry 1096(12), 1–9 (2020)
Džuurina, J., Grace, S.R., Jadlovská, I., Li, T.: Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term. Math. Nachr. 293(5), 910–922 (2020)
El-Sheikh, M.M.A.: Oscillation and nonoscillation criteria for second order nonlinear differential equations. J. Math. Anal. Appl. 179(1), 14–27 (1993)
El-Sheikh, M.M.A., Sallam, R.A., Elimy, D.: Oscillation criteria for second order nonlinear equations with damping. Adv. Differ. Equ. Control Process. 8(2), 127–142 (2011)
El-Sheikh, M.M.A., Sallam, R.A., Salem, S.: Oscillation of nonlinear third-order differential equations with several sublinear neutral terms. Math. Slovaca 71(6), 1411–1426 (2021)
Fu, Y., Tian, Y., Jiang, C., Li, T.: On the asymptotic properties of nonlinear third-order neutral delay differential equations with distributed deviating arguments. J. Funct. Spaces 2016, 1–5 (2016)
Jiang, C., Jiang, Y., Li, T.: Asymptotic behavior of third-order differential equations with nonpositive neutral coefficients and distributed deviating arguments. Adv. Differ. Equ. 105, 1–14 (2016)
Jiang, C., Li, T.: Oscillation criteria for third-order nonlinear neutral differential equations with distributed deviating arguments. J. Nonlinear Sci. Appl. 9, 6170–6182 (2016)
Jiang, C., Tian, Y., Jiang, Y., Li, T.: Some oscillation results for nonlinear second-order differential equations with damping. Adv. Differ. Equ. 2015, 354 (2015)
Jiang, Y., Jianng, C., Li, T.: Oscillatory behavior of third-order nonlinear neutral delay differential equations. Adv. Differ. Equ. 2016, 171 (2016)
Li, T., Baculikova, B., Dzurina, J., Zhang, C.: Oscillation of fourth-order neutral differential equations with p-Laplacian like operators. Bound. Value Probl. 56, 1–9 (2014)
Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70(3), 86 (2019)
Li, T., Rogovchenko, Y.V.: Asymptotic behavior of an odd-order delay differential equation. Bound. Value Probl. 2014, 1 (2014)
Li, T., Rogovchenko, Y.V.: On asymptotic behavior of solutions to higher order sublinear Emden-Fowler delay differential equations. Appl. Math. Lett. 667, 53–59 (2017)
Li, T., Rogovchenko, Y.V.: On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appl. Math. Lett. 105, 106293 (2020)
Li, T., Viglialoro, G.: Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime. Differ. Integral Equ. 34(5–6), 315–336 (2021)
Moaaz, O., Cesarano, C., Muhib, A.: Some new oscillation results for fourth-order neutral differential equations. Eur. J. Appl. Math. 13(2), 185–199 (2020)
Moaaz, O., Metwally, E., Elabbasy, M., Shaaban, E.: Oscillation criteria for a class of third order damped differential. Arab J. Math. Sci. 24(1), 16–30 (2018)
Philos, C.: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations will positive delays. Arch. Math. 36, 168–178 (1981)
Qin, G., Huang, C., Xie, Y., Wen, F.: Asymptotic behavior for third-order quasi-linear differential equations. Adv. Differ. Equ. 305, 1–8 (2013)
Qiu, Y.-C., Jadlovska, I., Chiu, K.-S., Li, T.: Existence of nonoscillatory solutions tending to zero of third-order neutral dynamic equations on time scales. Adv. Differ. Equ. 231, 1–9 (2020)
Qiu, Y.C., Zada, A., Qin, H., Li, T.: Oscillation criteria for nonlinear third-order neutral dynamic equations with damping on time scales. J. Funct. Spaces 2017, 1–18 (2017)
Sallam, R.A., El-Sheikh, M.M.A., El-Saedy, E.I.: On the oscillation of second order nonlinear neutral delay differential equations. Math. Slovaca 71(4), 859–870 (2021)
Sallam, R.A., Salem, S., El-Sheikh, M.M.A.: Oscillation of solutions of third order nonlinear neutral differential equations. Adv. Differ. Equ. 314, 1–25 (2020)
Thandapani, E., El-Sheikh, M.M.A., Sallam, R., Salem, S.: On the oscillatory behavior of third order differential equations with a sublinear neutral term. Math. Slovaca 70(1), 95–106 (2020)
Tian, Y., Cail, Y., Fu, Y., Li, T.: Oscillation and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments. Adv. Differ. Equ. 267, 1–14 (2015)
Tiryaki, A., Aktas, M.F.: Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping. J. Math. Anal. Appl. 325, 54–68 (2007)
Wang, H., Chen, G., Jiang, Y., Jiang, C., Li, T.: Asymptotic behavior of third-order neutral differential equations with distributed deviating arguments. J. Math. Comput. Sci. 17, 194–199 (2017)
Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of fourth-order delay dynamic equations. Sci. China Math. 58(1), 143–160 (2015)
Zhang, Q., Gao, L., Yu, Y.: Oscillation criteria for third-order neutral differential equations with continuously distributed delay. Appl. Math. Lett. 25, 1514–1519 (2012)
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El-Gaber, A.A., El-Sheikh, M.M.A. & El-Saedy, E.I. Oscillation of super-linear fourth-order differential equations with several sub-linear neutral terms. Bound Value Probl 2022, 41 (2022). https://doi.org/10.1186/s13661-022-01620-2
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DOI: https://doi.org/10.1186/s13661-022-01620-2