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Oscillation of thirdorder neutral differential equations with damping and distributed delay
Advances in Difference Equations volume 2019, Article number: 426 (2019)
Abstract
The present paper focuses on the oscillation of the thirdorder nonlinear neutral differential equations with damping and distributed delay. The oscillation of the thirdorder damped equations is often discussed by reducing the equations to the secondorder ones. However, by applying the Riccati transformation and the integral averaging technique, we give an analytical method for the estimation of Riccati dynamic inequality to establish several oscillation criteria for the discussed equation, which show that any solution either oscillates or converges to zero. The results make significant improvement and extend the earlier works such as (Zhang et al. in Appl. Math. Lett. 25:1514–1519 2012). Finally, some examples are given to demonstrate the effectiveness of the obtained oscillation results.
Introduction
Differential equations arise in modeling situations to describe population growth, biology, economics, chemical reactions, neural networks, and so forth; see, e.g., [2,3,4,5,6,7,8]. In the present paper, we investigate the oscillatory behavior of a thirdorder neutral differential equation with damping and distributed delay. The equation is given as follows:
Throughout this article, we always make the hypotheses as follows:
 (H_{1}):

\(r(t)\in C^{1}([t_{0},\infty ),(0,\infty ))\), \(m(t) \in C([t_{0},\infty ),(0,\infty ))\), \({\int _{t_{0}}^{\infty }}\frac{1}{r(t)}\exp(\int _{t_{0}}^{t} \frac{m(s)}{r(s)}\,\mathrm {d}s)\,\mathrm {d}t=\infty \);
 (H_{2}):

\(\alpha (t)\in C^{1}([t_{0},\infty ),(0,\infty ))\), \({\int _{t_{0}}^{\infty }}\frac{1}{\alpha (t)}\,\mathrm {d}t= \infty \);
 (H_{3}):

\(p(t,\mu )\in C([t_{0},\infty )\times [a,b],(0,\infty ))\), \(0\le p(t)= {\int _{a}^{b}} p(t,\mu )\,\mathrm {d}\mu \le p<1\);
 (H_{4}):

\(\tau (t,\mu )\in C([t_{0},\infty )\times [a,b],(0, \infty ))\) is not a decreasing function with respect to μ and satisfies \(\tau (t,\mu )\le t\) and \(\lim_{t\rightarrow \infty }\inf_{\mu \in [a,b]}\tau (t,\mu )=\infty \);
 (H_{5}):

\(g(t,\zeta )\in C([t_{0},\infty )\times [c,d],[\delta ,\infty ))\) for \(\delta >0\) is not a decreasing function with respect to ζ and satisfies \(g(t,\zeta )\le t\) and \(\lim_{t\rightarrow \infty }\inf_{\zeta \in [c,d]}g(t,\zeta )=\infty \);
 (H_{6}):

\(F(t,\zeta ,w)\in C([t_{0},\infty )\times [c,d]\times (0,\infty ),(0,\infty ))\), \(q(t,\zeta )\in C([t_{0},\infty )\times [c,d],(0, \infty ))\), \(\frac{F(t,\zeta ,w)}{w}\ge q(t,\zeta )\).
Letting
a function \(x(t)\) is the solution of equation (1.1) if \(x(t)\) satisfies (1.1) on \([T_{x},\infty )\) for every \(t\ge T_{x}\ge t _{0}\) with \(x(t)\), \(\alpha (t)y'(t)\) and \(r(t)(\alpha (t)y'(t))'\in C ^{1}[T_{x},\infty )\). We focus on the solutions satisfying \(\sup \{x(t):T\le t< \infty \}>0, T\ge T_{x}\). The solution with arbitrarily large zeros on \([T_{x},\infty )\) is treated as an oscillatory solution.
More and more scholars pay attention to the oscillatory solution of functional differential equations, especially for the firstorder or secondorder equations. With the development of the oscillation for the secondorder equations, researchers began to study the oscillation for the thirdorder equations, such as [9,10,11,12,13,14,15,16,17,18] for the delay equations, [19,20,21,22,23,24,25,26,27] for the equations on time scales, [28,29,30,31,32,33,34,35,36,37] for the damping equations. For the neutral delay equation
the oscillation was discussed in [38], and its general cases were discussed in [39,40,41,42]. For the distributed neutral delay equations
the Philostype oscillation criteria were studied by Zhang et al. [1], and the further investigation for the oscillation was given in [43,44,45] by Riccati transformation and integral averaging technique.
However, our focus is on the oscillation for thirdorder neutral differential equations with distributed delay and damping term, such as [46]. The research on the damped differential equations of thirdorder has been developed in recent years. Furthermore, the methods discussed are relatively limited. A general means used in the above mentioned papers [28,29,30,31,32,33,34,35,36,37] is reducing the thirdorder equations to the secondorder ones. We notice that in the discussion of oscillation for the differential equations, the key is the inequality estimation techniques. In [46], by the Riccati transformation we give a method for the estimation of Riccati dynamic inequality to get some oscillation criteria. Moreover, the main contribution in this paper is that we provide another method for the inequality estimation to discuss the oscillation of differential equations with damping and distributed delay on the basis of the Riccati transformation and the integral averaging technique. The results obtained continue and extend the analytic works in [1], where the methods using Lemmas 2.3 and 2.4 for the inequality estimation cannot be applied for (1.1).
Preliminaries
For the oscillatory solutions of (1.1), we usually talk about the eventually positive solutions. In this section, the following results may play an important role in establishing new oscillation criteria for (1.1).
Lemma 2.1
Assume that \(x(t)\) is the positive solution of (1.1). Then there are two cases as follows.

(I)
\(y(t)>0\), \(y'(t)>0\), \((\alpha (t)y'(t))'>0\);

(II)
\(y(t)>0\), \(y'(t)<0\), \((\alpha (t)y'(t))'>0\)
for \(t\ge t_{1}\ge t_{0}\) with sufficiently large \(t_{1}\).
Proof
We set that \(x(t)\) is the positive solution of (1.1) for \([t_{0},\infty )\). Then it follows from \((\mbox{H}_{4})\) and \((\mbox{H}_{5})\) that \(x(\tau (t,\mu ))>0\) and \(x(g(t,\zeta ))>0\) for \(t\ge t_{1}\) with sufficiently large \(t_{1}\), respectively. It is easy to get \(y(t)>x(t)>0\).
From (1.1) and \((\mbox{H}_{6})\), we get
It follows that \(\frac{\mathrm{d} }{\mathrm{d} t} [\exp(\int _{t_{1}}^{t} \frac{m(s)}{r(s)}\,\mathrm {d}s)r(t) (\alpha (t)y'(t) )' ]<0\). Then \(\exp( {\int _{t_{1}}^{t}}\frac{m(s)}{r(s)}\,\mathrm {d}s)r(t) (\alpha (t)y'(t) )'\) is a decreasing function with one sign eventually. Thus, from \((\mbox{H}_{1})\),
for \(t\ge t_{2}\ge t_{1}\).
We claim that \((\alpha (t)y'(t) )'>0\). Suppose \((\alpha (t)y'(t) )'\le 0\). According to the monotonicity of \(\exp( {\int _{t_{1}}^{t}}\frac{m(s)}{r(s)}\,\mathrm {d}s)r(t) (\alpha (t)y'(t) )'\), we have
for \(M>0\). Integrate the above inequality on \([t_{2},t]\) to get
Letting \(t\rightarrow \infty \), we have \(\alpha (t)y'(t)\rightarrow  \infty \) by \((\mbox{H}_{1})\). Then it follows from \((\alpha (t)y'(t) )'\le 0\) that \(\alpha (t)y'(t)\le \alpha (t_{3})y'(t_{3})<0\) for \(t\ge t_{3}\ge t_{2}\). Dividing by \(\alpha (t)\) and integrating on \([t_{3},t]\), we have that
From condition \((\mbox{H}_{2})\), we have \(y(t)\rightarrow \infty \) as \(t\rightarrow \infty \). This contradicts \(y(t)>0\), which implies \((\alpha (t)y'(t) )'>0\). We complete the proof. □
Lemma 2.2
Assume that \(x(t)\) is the positive solution of (1.1) and \(y(t)\) satisfies case (II). Suppose
where \(q(t)= {\int _{c}^{d}}q(t,\zeta )\,\mathrm {d}\zeta \). Then \(\lim_{t\rightarrow \infty }x(t)=0\).
Proof
We set that \(x(t)\) is the positive solution of (1.1) for \([t_{0},\infty )\). Due to the fact that case (II) is valid for \(y(t)\), we obtain \(\lim_{t\rightarrow \infty }y(t)=l\ge 0\). And then we use proof by contradiction to prove \(l=0\). Suppose \(l>0\). Then it follows that \(l+\varepsilon >y(t)>l\) for \(\varepsilon >0\) with \(t\ge t_{1}\ge t_{0}\). Taking ε such that \(p\varepsilon < l(1p)\), from \((\mbox{H}_{3})\), \((\mbox{H}_{4})\), and property (II), we have
where \(K=\frac{l(1p)p\varepsilon }{l+\varepsilon }>0\). It follows from \((\mbox{H}_{5})\), \((\mbox{H}_{6})\), (2.2), and property (II) that
Taking \(z(t)=\exp( {\int _{t_{1}}^{t}}\frac{m(s)}{r(s)}\,\mathrm {d}s)\), we get
Integrate on \([t,\infty )\) to obtain
By virtue of \(y(g(t,d))>l\) and \(z'(t)>0\), we conclude
This yields
by the integration from t to ∞. Further integrate on \([t_{1},\infty )\) to get
This contradicts (2.1), which leads to \(l=0\), and then \(\lim_{t\rightarrow \infty }x(t)=0\) from \(y(t)>x(t)>0\). We complete the proof. □
Oscillation results
Based on the lemmas in Section 2, some new oscillation criteria for (1.1) are obtained by applying Riccati transformation, inequality estimation, and integral averaging technique due to Philos [47]. Putting
a function \(H\in C(D,\mathbb{R})\) is said to belong to X class \((H\in X)\) if it satisfies

(i)
\(H(t,t)=0\), \(t\ge t_{0}\) and \(H(t,s)>0\), \((t,s)\in D_{0}\);

(ii)
\(H(t,s)\) has a continuous and nonpositive partial derivative on D with respect to the second variable;

(iii)
There exists \(h(t,s)\in C(D,\mathbb{R})\) such that
$$ \frac{\partial H(t,s)}{\partial s}=h(t,s)\sqrt{H(t,s)} \quad \text{for all } (t,s)\in D. $$
Theorem 3.1
Assume that (2.1) holds and there exist \(H\in X\) and \(\phi \in C([t _{0},\infty ),\mathbb{R})\) such that
and
for \(t\ge T\ge t_{0}, k>1\), \(\theta >0\), where
Then any solution \(x(t)\) of (1.1) either oscillates or converges to zero.
Proof
Let \(x(t)\) be a nonoscillatory solution of (1.1). Without loss of generality, we assume that \(x(t)>0\) on \([t_{1},\infty )\). From \((\mbox{H}_{4})\) and \((\mbox{H}_{5})\), we have \(x(\tau (t,\mu ))>0\), \((t, \mu )\in [t_{1},\infty )\times [a,b]\), \(x(g(t,\zeta ))>0\), \((t,\zeta ) \in [t_{1},\infty )\times [c,d]\) for sufficiently large \(t_{1}\). From Lemma 2.1, \(y(t)\) is one case of (I) and (II).
If \(y(t)\) satisfies case (I), then
from \((\mbox{H}_{3})\) and \((\mbox{H}_{4})\). By \((\mbox{H}_{5})\), \((\mbox{H}_{6})\) and the above inequality, it is obvious that
Putting that \(w(t)=\rho (t)\frac{r(t) (\alpha (t)y'(t) )'}{ \alpha (t)y'(t)}, t\ge t_{1}\) with \(\rho (t)\) given in (3.4), we know
By property (I), there exists a limit of \(\frac{1}{y'(t)}\) as \(t\rightarrow \infty \), which is denoted by \(\lim_{t\rightarrow \infty }\frac{1}{y'(t)}=\eta \). Choosing \(\varepsilon =\frac{\eta }{2}\), we obtain \(\frac{1}{y'(t)}>\frac{ \eta }{2}\) for \(t\ge t_{2}\ge t_{1}\). Letting \(\theta =\frac{y( \delta )\eta }{2}\), from \(g(t,c)\ge \delta \) in \((\mbox{H}_{5})\) we have
where \(P(t)\) is defined in (3.4). Multiply the above inequality by \(H(t,s)\) and integrate the inequality from \(t_{2}\) to t to get
from the integral averaging technique. Then
Thus, it follows from (3.3) that
and
Next we claim
Suppose \({\int _{t_{2}}^{\infty }}\frac{w^{2}(t)}{\rho (t)r(t)}\,\mathrm {d}t=\infty \). It follows from (3.1) that
for \(\mu >0\), and then \(\frac{H(t,t_{3})}{H(t,t_{0})}>\mu \) for \(t\ge t_{3}\ge t_{2}\). Thus we have
for \(M_{1}>0\). Then, when \(t\ge t_{3}\), we conclude
This implies
This leads to a contradiction with (3.5). Then we conclude \({\int _{t_{2}}^{\infty }}\frac{w^{2}(t)}{\rho (t)r(t)}\,\mathrm {d}t<\infty \), which contradicts (3.2).
If \(y(t)\) satisfies case (II), then \(\lim_{t\rightarrow \infty }x(t)=0\) from (2.1) and Lemma 2.2. The proof is complete. □
Theorem 3.2
Assume that (2.1) holds and there exist \(H\in X\) and \(R(t)\in C([t _{0},\infty ),\mathbb{R})\) such that
for \(k>1\), \(\theta >0\), and \(t\ge T\ge t_{0}\), where
Then any solution \(x(t)\) of (1.1) either oscillates or converges to zero.
Proof
Let \(x(t)\) be a nonoscillatory solution of (1.1). Without loss of generality, we assert that \(x(t)>0\) on \([t_{1},\infty )\). It follows from \((\mbox{H}_{4})\) and \((\mbox{H}_{5})\) that \(x(\tau (t,\mu ))>0\), \((t, \mu )\in [t_{1},\infty )\times [a,b]\), \(x(g(t,\zeta ))>0\), \((t,\zeta ) \in [t_{1},\infty )\times [c,d]\) for sufficiently large \(t_{1}\). From Lemma 2.1, \(y(t)\) is one case of (I) and (II).
If \(y(t)\) satisfies case (I), then by letting
we conclude that
In the same way as Theorem 3.1, taking \(\lim_{t\rightarrow \infty }\frac{1}{y'(t)}=\eta \), \(\varepsilon =\frac{ \eta }{2}\), and \(\theta =\frac{y(\delta )\eta }{2}\), we have
where \(Q(t)\) is defined by (3.8). Based on \(BuAu^{2}\le \frac{B ^{2}}{4A}\) for \(A>0\), \(u\in \mathbb{R}\), we get
where \(D(t)\) is given by (3.7). Multiplying the inequality by \(H(t,s)\) and integrating on \([T,t]\), we obtain
Then
This contradicts (3.6).
If \(y(t)\) satisfies case (II), then \(\lim_{t\rightarrow \infty }x(t)=0\) from (2.1) and Lemma 2.2. The proof is complete. □
Remark 3.3
The proofs of Theorems 3.1 and 3.2 provide a method for the estimation of Riccati dynamic inequality, which is different from [46] and useful for the oscillation criteria.
Examples
Example 4.1
Consider the equation
By (4.1), we note that \(r(t)=\frac{1}{t}\), \(\alpha (t)=e^{t}\), \(p(t,\mu )=\frac{\mu }{6t}\), \(\tau (t,\mu )=t\mu \), \(m(t)=\frac{2}{t ^{2}}\), \(g(t,\zeta )=(t3\sqrt{t})\zeta \), \(a=1\), \(b=2\), \(c= \frac{1}{2}\), \(d=1\), which satisfy conditions \((\mbox{H}_{1})\)–\(( \mbox{H}_{6})\). Furthermore, we choose \(H(t,s)=(ts)^{2}\), \(k=2\), \(\theta =2\), \(p=\frac{1}{2}\), \(q(t,\zeta )=t\zeta \), \(\phi (t)=t\), \(t _{0}=1\). By Theorem 3.1, we obtain \(h(t,s)=2\), \(\rho (t)=t^{2}1\), \(q(t)=\frac{3}{8}t\), \(P(t)=\frac{3}{8}t(t^{2}1) e^{t}\) and
Clearly, it is obvious that other conditions of Theorem 3.1 are valid. Thus, it follows from Theorem 3.1 that any solution of (4.1) is oscillatory or converges to zero as \(t\rightarrow \infty \).
Example 4.2
We consider the equation
From (4.2), we find that \(r(t)=\frac{2}{t}\), \(\alpha (t)= \sqrt{t}\), \(p(t,\mu )=e^{t}\mu ^{2}\), \(\tau (t,\mu )=\frac{1}{3}t \mu \), \(m(t)=\frac{1}{t^{2}}\), \(g(t,\zeta )=t\zeta \), \(a=0\), \(b=1\), \(c= \frac{1}{2}\), \(d=1\), which satisfy conditions \((\mbox{H}_{1})\)–\(( \mbox{H}_{6})\). Furthermore, we choose \(H(t,s)=(ts)^{2}\), \(k=2\), \(\theta =2\), \(p=\frac{1}{2}\), \(q(t,\zeta )=\frac{\zeta }{\sqrt{t}}\), \(R(t)=\frac{1}{t}\), \(t_{0}=1\). By Theorem 3.2, we have \(h(t,s)=2\), \(\rho (t)=\sqrt{t}1\), \(q(t)=\frac{3}{8}t^{\frac{1}{2}}\), \(D(t)=(\sqrt{t}1)(\frac{3}{8}t^{1}+t^{3})\) and
Then by Theorem 3.2 we know that any solution of (4.2) is oscillatory or converges to zero as \(t\rightarrow \infty \).
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Acknowledgements
The authors express their sincere gratitude to the editors and reviewers for the careful reading of the original manuscript and useful comments.
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The work is supported by the National Natural Science Foundation of China (Grant Nos. 11501496 and 61503171), the Doctor Startup Research Fund of Yulin University (Grant No. 13GK04), Key Research and Development Program of Shandong Province (Grant Nos. 2019GNC106027 and 2019GGX101003), Scientific Research Plan of Universities in Shandong Province (Grant No. J18KA352), and Doctoral Scientific Research Foundation of Qilu University of Technology (Shandong Academy of Sciences) (Grant No. 81110240).
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Wei, M., Jiang, C. & Li, T. Oscillation of thirdorder neutral differential equations with damping and distributed delay. Adv Differ Equ 2019, 426 (2019). https://doi.org/10.1186/s1366201923632
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Keywords
 Oscillation
 Thirdorder
 Distributed delay
 Damping term
 Riccati transformation