On the oscillation of solutions of third-order differential equations with non-positive neutral coefficients

1.   Introduction

In this article, we are concerned with the oscillatory behavior of solutions of a general class of third-order differential equations with non-positive neutral coefficients of the type

where $w(l) = y\left(l\right) -\sum_{j = 1}^{m}a_{j}\left(l\right) y\left(\delta _{j}\left(l\right) \right)$, $\alpha$ and $\beta _{i}$ are quotients of odd positive integers, and $m, n$ are positive integers. We shall consider the following two cases: the canonical case

and the non-canonical case

Throughout the paper, we assume that

$\left(H_{1}\right)$ $d\left(l\right) \in C\left([l_{0}, \infty), \left(0, \infty \right) \right), a_{j}\left(l\right) \in C\left([l_{0}, \infty)\right), 0\leq a_{j}\left(l\right) \leq a_{0j}, \sum_{j = 1}^{m}a_{0j} < 1, j = 1, 2, ..., m;$

$\left(H_{2}\right)$ $\delta _{j}\left(l\right), \tau _{i}\left(l\right), q_{i}\left(l\right) \in C\left([l_{0}, \infty)\right), \delta _{j}\left(l\right) \leq l, \lim_{l\rightarrow \infty }\delta _{j}\left(l\right) = \lim_{l\rightarrow \infty }\tau _{i}\left(l\right) = \infty,$ $q_{i}\left(l\right) > 0,$ $i = 1, 2, ..., n, j = 1, 2, ..., m.$

Any nontrivial function $y\left(l\right) \in C\left([l_{y}, \infty)\right), l_{y}\geq l_{0},$ such that $w\in C^{2}\left([l_{y}, \infty)\right),$ $d\left(l\right) \left(w^{^{\prime \prime }}\left(l\right) \right) ^{\alpha }\in C^{1}\left([l_{y}, \infty)\right)$ and $y\left(l\right)$ satisfies (1.1) on $[l_{y}, \infty)$ is called a solution of (1.1). Our attention is restricted to those solutions $y\left(l\right)$ of (1.1) that satisfy $\sup \left\{ \left\vert y\left(l\right) \right\vert :l\geq T\right\} > 0$ for $T\geq l_{y}.$ We tacitly suppose that (1.1) possesses such a solution. A solution $y\left(l\right)$ of (1.1) is termed oscillatory if it has arbitrarily large zeros on $[l_{y}, \infty);$ otherwise, it is said to be non-oscillatory. The equation itself is termed oscillatory if all its solutions oscillate.

In dynamical models, delay and oscillation effects are often formulated by means of external sources and/or nonlinear diffusion, perturbing the natural evolution of related systems (see, e.g., ^[1,2,3,4]). Recently, there has been considerable interest in studying the qualitative properties of solutions of various types of differential equations, see, e.g., ^[5,6,7], for the oscillation of second-order differential equations, while ^[8,9], for fourth-order differential equations, and the references ^[10,11,12], for the oscillation of nth-order differential equations. In fact, it is notable that the analysis of differential equations with non-positive neutral coefficients is more difficult in comparison with that of non-negative neutral coefficients. The same thing can be said also for the non-canonical case compared to the canonical case. Moreover, although there has been a lot of interest in the oscillatory behavior of solutions of third-order equations with non-negative neutral coefficients (see, e.g., ^{[13,14,15,16]}), for equations with non-positive neutral coefficients there are relatively fewer published results and so they are not very prevalent in the literature (see, e.g., ^{[17,18,19,20]}). For instance, we mention here some of the related works that motivate our work. In ^[20], Qiu established new oscillation criteria for the third-order nonlinear dynamic equation on time scales of the type,

with $\int_{l_{0}}^{\infty }\frac{1}{d_{1}^{\frac{1}{\gamma _{1}}}\left(l\right) }\Delta l = \infty, \int_{l_{0}}^{\infty }\frac{1}{d_{2}^{\frac{1}{ \gamma _{2}}}\left(l\right) }\Delta l = \infty$. The authors in ^[17] were concerned with the D.E.,

and established numerous criteria for the so-called Hile and Nehari type under the assumptions $0\leq b\left(l\right) \leq 1$ and $d^{\prime }\left(l\right) \geq 0$ in the canonical case (1.2). Meanwhile, Jiang et al. ^[19] were motivated by the work of Baculucova and Duzirina ^[17], and Li et al. ^[21], to establish an affirmative answer to the question about the possibility of obtaining asymptotic criteria for the D.E.,

where $\alpha > 0$ is a quotient of odd positive integers, $0\leq \int_{a}^{b}p\left(l, \zeta \right) d\zeta \leq p_{0} < 1,$ without the need for the restrictive condition $d^{\prime }\left(l\right) \geq 0$. More recently Garce ^[22] studied the oscillatory behavior of solutions of the nonlinear differential equation

with $\tau (l)\leq l, \sigma (l)\leq l, \tau ^{^{\prime }}(l) > 0,$ and $\sigma ^{^{\prime }}(l) > 0.$ Meanwhile, Garce et al. ^[18] were concerned with the oscillatory behavior of solutions of nonlinear differential equations of the type

using comparison methods and integral conditions, with $\gamma \geq \beta, \tau (l)\leq l, \sigma \left(l\right) \leq l, \tau ^{^{\prime }}(l) > 0$, and $\sigma ^{^{\prime }}(l) > 0$ in the canonical case (1.2).

The principal goal of this paper is to study the oscillatory behavior of solutions of the nonlinear third-order differential equation with non-positive neutral coefficients (1.1) in the two cases canonical (1.2) and non-canonical (1.3) by using Riccati transformation without the need for the restrictive condition $d^{\prime }\left(l\right) \geq 0.$ Moreover, we do not need specific restrictions on the functions $\tau _{i}(l),$ that is, $\tau _{i}(l)$ may be delayed or advanced; furthermore, we considered the two cases $\beta _{i}\geq \alpha$ and $\beta _{i}\leq \alpha.$

2.   Preliminaries

This section is devoted to present some notations and lemmas needed for our results.

Define

We first start with the following two lemmas, which are very similar to Lemmas 2.1 and 2.2 of ^[19].

Lemma 2.1. Assume that $y\left(l\right)$ is an eventually positive solution of (1.1), such that (1.2) be satisfied. Then there exists $l_{1}\geq l_{0},$ such that for all $l\geq l_{1}$ the corresponding function $w$ satisfies one of the following four cases:

Lemma 2.2. If for any eventually positive solution $y\left(l\right)$ of (1.1), the corresponding $w\left(l\right)$ satisfies case (i) of Lemma 2.1, then for any $l_{2} > l_{1}\geq l_{0},$

and $\frac{w^{\prime }\left(l\right) }{\int_{l_{1}}^{l}d^{\frac{-1}{\alpha } }\left(h\right) dh}$ is nonincreasing eventually.

Now, we introduce the following preliminary result:

Lemma 2.3. If for any eventually positive solution $y\left(l\right)$ of (1.1), the corresponding $w\left(l\right)$ satisfies case (ii) of Lemma 2.1, and

then $\lim_{l\rightarrow \infty }y\left(l\right) = 0.$

Proof. Since $w \left(l\right)$ satisfies property $\left(ii\right)$, then there exists a finite constant $M\geq 0$ such that $\lim_{l\rightarrow \infty }w\left(l\right) = M.$ We claim that $M = 0.$ Otherwise, assume that $M > 0.$ By the definition of $w,$ $y\left(l\right) \geq$ $w\left(l\right) > M.$ Consequently, by (1.1), we have

where $\kappa = { \{}_{\max \beta _{i}\, \, \, M < 1 }^{\min \beta _{i}\, \, \, M\geq 1}.$ Integrating (2.2) from $l$ to $\infty,$ we obtain

Therefore, by integrating from $l$ to $\infty$ and then integrating the result from $l_{1}\ $ to $\infty,$ it follows that

This contradicts (2.1). Hence, $M = 0$ and $\lim_{l\rightarrow \infty }w\left(l\right) = 0.$ Next, we claim that $y(l)$ is bounded. If this is false, then there exists a sequence $\left \{ l_{m}\right \}$ such that $\lim_{l\rightarrow \infty }l_{m} = \infty$ and $\lim_{m\rightarrow \infty }y\left(l_{m}\right) = \infty,$ where $y\left(l_{m}\right) = \max \left \{ y\left(s\right) :l_{0}\leq s\leq l_{m}\right \}.$ Since $\lim_{l\rightarrow \infty }\delta _{j}\left(l\right) = \infty, \delta _{j}\left(l_{m}\right) > l_{0}$ for sufficiently large $m$. By $\delta _{j}\left(l\right)\leq l$, we conclude that

and so

which yields $\lim_{l\rightarrow \infty }w\left(l_{m}\right) = \infty.$ This contradicts $\lim_{l\rightarrow \infty }w\left(l\right) = 0,$ therefore $y\left(l\right)$ is bounded, and hence we may suppose that $\lim \sup_{l\rightarrow \infty }$ $y\left(l\right) = b_{0},$ where $0\leq b_{0} < \infty.$ Then there exists a sequence $\left \{ l_{k}\right \}$ such that $\lim_{l\rightarrow \infty }l_{k} = \infty$ and $\lim_{l\rightarrow \infty }y\left(l_{k}\right) = b_{0}.$ Now assuming that $b_{0} > 0$, and letting

we have $y\left(\delta _{j}\left(l_{k}\right) \right) < b_{0}+\epsilon$ eventually, and thus

which is a contradiction. Thus $b_{0} = 0$ and $\lim_{l\rightarrow \infty }y\left(l\right) = 0$. The proof is complete. □

3.   Main results

Theorem 3.1. Assume that $\beta _{i}\geq \alpha$, $i = 1, ..., n,$ (1.2) and (2.1) hold. Suppose that there exists $I\left(l\right) \in C\left([l_{0}, \infty)\right)$ such that

If there exist a function $v\left(l\right) \in C^{1}\left([l_{0}, \infty), \left(0, \infty \right) \right),$ and a constant $C_{1} > 0$ such that, for all sufficiently large $l_{1}$ $\geq l_{0}$ and for some $l_{3} > l_{2} > l_{1}$,

where

then Eq (1.1) is almost oscillatory.

Proof. Assume that $y\left(l\right)$ is an eventually positive solution of (1.1). Then there exists a $l_{1}\geq l_{0}$ such that $y\left(l\right) > 0, y\left(\delta _{j}\left(l\right) \right) > 0$ and $y\left(\tau _{i}\left(l\right) \right) > 0$ for $l\geq l_{1}$. It is clear by Lemma 2.1 that the function $w\left(l\right)$ obeys one of four possible cases $\left(i\right)$, $\left(ii\right)$, $\left(iii\right)$, or $\left(iv\right)$. Assume first that case $\left(i\right)$ is satisfied for $l\geq l_{1}.$ Define the Riccati transformation $\phi \left(l\right)$ by

then $\phi \left(l\right) > 0$ for $l\geq l_{1}$, and

But since from (1.1) and the definition of $w$, we have

Hence, since $w^{\prime }\left(l\right) > 0$ and $\tau _{i}\left(l\right) \geq I\left(l\right),$ then

Then from (3.4), we have

Now since $w\left(l\right)$ is positive and increasing, then there exist a $l_{2}\geq l_{1}$ and $C_{1} > 0$ such that

This, with (3.6), leads to

Since $I\left(l\right) \leq l$, then by using the nonincreasing property of $\frac{w^{\prime }\left(l\right) }{\int_{l_{1}}^{l}d^{\frac{-1}{\alpha } }\left(h\right) dh}$ (see Lemma 2.2), we obtain

Now by using Lemma 2.2, we have

and so, by substituting from (3.10) into (3.8), we obtain

Applying the inequality

with $V = \phi \left(l\right)$, $R = \frac{\alpha }{\left(d\left(l\right) v\left(l\right) \right) ^{\frac{1}{\alpha }}}$ and $T =$ $\frac{v^{\prime }\left(l\right) }{v\left(l\right) },$ we obtain

By integrating from $l_{3}$ $\left(l_{3} > l_{2}\right)$ to $l,$ we arrive at

this contradicts (3.2). Now consider the case $\left(ii\right)$, then by Lemma 2.3, $\lim_{l\rightarrow \infty }y\left(l\right) = 0.$ In both cases $\left(iii\right)$ and $\left(iv\right)$, similar analysis to that in [19, Theorem 3.1], case $\left(iii\right)$, and case $\left(iv\right)$ can be used to arrive at the conclusion $\lim_{l\rightarrow \infty }y\left(l\right) = 0.$ This completes the proof. □

Theorem 3.2. Assume that $\beta _{i}\leq \alpha$, $i = 1, ..., n,$ (1.2) and (2.1) hold. Suppose that there exists $I\left(l\right) \in C\left([l_{0}, \infty)\right)$ satisfies (3.1). Suppose further that there exist a function $k\left(l\right) \in C^{1}\left([l_{0}, \infty), \left(0, \infty \right) \right),$ and a constant $C_{2} > 0,$ for sufficiently large $l_{1}$ $\geq l_{0}$ and some $l_{3} > l_{2} > l_{1}$. If

then Eq (1.1) is almost oscillatory.

Proof. For the sake of contradiction, suppose that (1.1) has an eventually positive solution $y\left(l\right)$. Then for any, $l_{1}\geq l_{0}$, we have $y\left(l\right) > 0, y\left(\delta _{j}\left(l\right) \right) > 0$ and $y\left(\tau _{i}\left(l\right) \right) > 0, i = 1, 2, ..., n, j = 1, 2, ..., m$. It is clear by Lemma 2.1, that the function $w\left(l\right)$ obeys one of the four possible cases $\left(i\right), \left(ii\right), \left(iii\right)$, or $\left(iv\right)$. Assume first that case $\left(i\right)$ is satisfied for $l\geq l_{1}.$ Define

then $\Omega \left(l\right) > 0$, and by using (3.5), we obtain

But since $w^{\prime }\left(l\right) > 0$ and $d\left(l\right) \left(w^{\prime \prime }\left(l\right) \right) ^{\alpha }$ is nonincreasing, we obtain

Substituting into (3.13), we obtain

By the definition of $\Omega,$ we have

By applying the inequality (3.11), with

we obtain

But since from (3.14), we have

which leads to

Hence

This with $I\left(l\right) \leq l,$ yields

Substituting from (3.17) into (3.15), we obtain

Now since by (3.16), $\frac{w\left(l\right) }{D_{2}\left(l\, ,l_{2}\right) }$is decreasing, there exists a constant $C_{2} > 0$ such that for $l_{3} > l_{2},$ we have

Substituting into (3.18), we obtain

Integrating (3.19) from $l_{3}$ to $l,$ we obtain

This contradicts (3.12). The proofs of the cases $\left(ii\right)$–$\left(iv\right)$ are as in the proof of Theorem 3.1. □

Now, we discuss the oscillatory behavior of Eq (1.1) in the non-canonical case (1.3).

Theorem 3.3. Assume that $\beta _{i}\geq \alpha$, $i = 1, ..., n,$ (1.3) and (2.1) hold. Assume that $I\left(l\right)$ be as in Theorem 3.1, for sufficiently large $l_{1}$ $\geq l_{0}$ and for some $l_{3} > l_{2} > l_{1}$, (3.2) is satisfied. Suppose further that there exist constants $C_{3}$ $> 0\ $ and $0 < L < 1$, such that

then Eq (1.1) is almost oscillatory.

Proof. For the sake of contradiction suppose that (1.1) has an eventually positive solution $y\left(l\right)$. In view of (1.3), there exist six possible cases including $\left(i\right)$–$\left(iv\right)$ (as in Lemma 2.1), and the two extra cases:

The proofs of the four cases $\left(i\right)$–$\left(iv\right)$ follow the same arguments of Theorem 3.1. Now consider the case$\left(v\right)$. Since $d\left(w^{\prime \prime }\right) ^{\alpha }$ is decreasing, then

Integrating from $l$ to $g$ and letting $g\rightarrow \infty,$ we have

In view of case $\left(v\right),$ since $w\left(l\right) > 0, w^{\prime }\left(l\right) > 0$ and $w^{\prime \prime }\left(l\right) < 0$ on $\left[ l_{1}, \infty \right],$ for any constant $L\in \left(0, 1\right)$, we have

Now define

then $\Phi \left(l\right) < 0$ for $l\geq l_{2\text{ }},$ and

Thus, from (3.5), we have

Therefore, in view of (3.23), we obtain

But since $I\left(l\right) \leq l$, then

This with (3.23), leads to

Consequently, by substituting in (3.25), we have

Now since, from the positivity and increasing properties of $w\left(l\right),$ there exists a constant $C_{3} > 0$ such that $w\left(l\right) \geq C_{3},$ then we have

It is clear by (3.22) and (3.24) that

Multiplying (3.29) by $D^{\alpha }\left(l\right)$ and integrating from $l_{3}$ to $l,$ we have

Set $R = \frac{D^{\alpha }\left(s\right) }{d^{\frac{1}{\alpha }}\left(s\right) }, T = d^{-\frac{1}{\alpha }}\left(s\right) D^{\alpha -1}\left(s\right)$ and $v = -\Phi \left(s\right),$ then using the inequality (3.11), we have

This is a contradiction with (3.20). Assume case $\left(vi\right)$ holds. Now, by using an argument similar to that used in Lemma 2.3, we arrive at the conclusion that $\lim_{l\rightarrow \infty }$ $y\left(l\right) = 0.$ The proof is complete. □

Theorem 3.4. Assume that $\beta _{i}\leq \alpha$, $i = 1, ..., n,$ (1.3) and (2.1) hold. Assume further that $\ I\left(l\right)$ be as in Theorem 3.1, for sufficiently large $l_{1}$ $\geq l_{0}$ and for some $l_{3} > l_{2} > l_{1}$, (3.12) is satisfied. If there exists a constant $C_{4}$ $> 0$, such that

then Eq (1.1) is almost oscillatory.

Proof. Let $y\left(l\right)$ be a non-oscillatory solution of (1.1) such that $y\left(l\right) > 0,$ in view of (1.3), there exist six possible cases $\left(i\right) -\left(vi\right)$ (as in Theorem 3.3). The proofs in the four cases $\left(i\right) -\left(iv\right)$ are as in Theorem 3.2. Now assume that case $\left(v\right)$ holds. Then following the same lines of the proof of Theorem 3.3, we arrive at (3.28), but since by (3.26), $\frac{w\left(l\right) }{l^{\frac{1}{L}} }$ is nonincreasing, then there exists a positive constant $C_{4}$ such that

This with (3.28), leads to

Multiplying both sides of (3.32) by $D^{\alpha }\left(l\right)$ and integrating from $l_{4}\left(>l_{3}\right)$ to $l,$ and then applying the inequality (3.11), we obtain

This contradicts (3.31). Assume that case $\left(vi\right)$ holds. By a similar argument to that used in Lemma 2.3, we arrive at the conclusion that $\lim_{l\rightarrow \infty }$ $y\left(l\right) = 0.$ The proof is complete. □

4.   Examples

Example 4.1. Consider the differential equation

Here $a_{1} = \frac{1}{l^{2}}, a_{2} = \frac{1}{l^{4}}, d\left(l\right) = 1,$ $q_{1}\left(l\right) = \frac{1}{l^{3}}, q_{2}\left(l\right) = \frac{1}{l^{4}}, \delta _{1}\left(l\right) = \frac{l}{5}, \delta _{2}\left(l\right) = \frac{l }{2}, \tau _{1}\left(l\right) = l, \tau _{2}\left(l\right) = 2l,$ $\alpha = \beta _{1} \ = \beta _{2} = 1.$ Note that $\int_{l_{0}}^{\infty }d^{\frac{1}{ \alpha }}\left(s\right) ds = \int_{2}^{\infty }ds = \infty,$

Choosing $\ v\left(l\right) = l$ and $\ I\left(l\right) = l,$ we have

and

Thus, by Theorem 3.1, Eq (4.1) is almost oscillatory.

Example 4.2. Consider the differential equation

Here $a_{1} = \frac{1}{2l}, a_{2} = \frac{1}{7l}, d\left(l\right) = l^{\frac{4}{3} },$ $q_{1}\left(l\right) = \frac{1}{l^{\frac{5}{3}}}$, $q_{2}\left(l\right) = \frac{1}{l^{4}}, \delta _{1}\left(l\right) = \frac{l}{2}$, $\delta _{2}\left(l\right) = \frac{l}{3}, \tau _{1}\left(l\right) = \frac{l}{2}$, $\tau _{2}\left(l\right) = \frac{l}{3},$ $\alpha = 1, \beta _{1} = 3, \beta _{2} = 5.$ Note that $\ \int_{l_{0}}^{\infty }d^{\frac{1}{\alpha }}\left(s\right) ds = \int_{1}^{\infty }l^{-\frac{4}{3}}ds < \infty,$

Taking $I\left(l\right) = \frac{l}{3},$ we have

choosing $l_{1} = 1, l_{2} = (1, 2),$ $l_{3} > 6$ and $v\left(l\right) = \frac{1}{l^{ \frac{1}{3}}}$, then

for $C_{1}^{2} > \frac{1}{12}$, and

for $C_{3}^{2}L\left(\frac{1}{3}\right) ^{\frac{1}{L}} > \frac{1}{36}.$ Thus, by Theorem 3.3, Eq (4.2) is almost oscillatory.

5.   Conclusions

In this article, we discussed a general class of third-order differential equations with non-positive neutral coefficients (1.1) in the two cases of canonical and non-canonical conditions. Our criteria do not need to determine whether the functions $\tau _{i}\left(l\right)$ are delayed or advanced. Moreover, our new criteria do not need the restrictive condition $d^{\prime }\left(l\right) \geq 0$.

Author contributions

A. A. El-Gaber: Investigation, Software, Supervision, Writing-original draft; M. M. A. El-Sheikh: Investigation, Software, Supervision, Writing-original draft; M. Zakarya: Writing-review editing and Funding; A.A. I Al-Thaqfan: Writing-review editing and Funding; H. M. Rezk: Investigation, Softwar, Writing-original draft. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through large Research Project under grant number RGP 2/190/45.

Conflict of interest

The authors declare that there are no conflicts of interest in this paper.