Oscillation behavior for neutral delay differential equations of second-order

1.   Introduction

In this paper, we study the oscillatory behavior for neutral delay differential equations of second-order

where $\xi \geq \xi _{0}$ and

Throughout this article, we assume:

$\bf(M1)$ $\gamma$ is a quotient of odd positive integers and $\gamma \geq 1$;

$\bf(M2)$ $a\in C^{1}\left(\left[\xi _{0}, \infty \right), \left(0, \infty \right) \right),$ $p, q\in C\left(\left[\xi _{0}, \infty \right), \left[0, \infty \right) \right),$ $0\leq p\left(\xi \right) < 1$ and

$\bf(M3)$ $\tau, \sigma \in C\left(\left[\xi _{0}, \infty \right), \mathbb{R} \right),$ $\tau \left(\xi \right) \leq \xi,$ $\sigma \left(\xi \right) < \xi$ and $\lim_{\xi \rightarrow \infty }\tau \left(\xi \right) = \lim_{\xi \rightarrow \infty }\sigma \left(\xi \right) = \infty;$

$\bf(M4)$ $f\in C\left(\mathbb{R}, \mathbb{R} \right)$ and $f\left(\varkappa \right) /\varkappa ^{\gamma }\geq k$ for $\varkappa \neq 0$ and a constant $k > 0.$

By a solution of Eq (1.1), we mean a nontrivial real-valued function $\varkappa \in C\left(\left[\xi _{\varkappa }, \infty \right), \mathbb{R} \right)$ with $\xi _{\varkappa }: =$ $\min \left\{ \tau \left(\xi _{\varkappa }\right), \sigma \left(\xi _{\varkappa }\right) \right\}$ for some $\xi _{\varkappa }\geq \xi _{0},$ which has the property $a\vartheta ^{\prime }\in C^{1}\left(\left[\xi _{0}, \infty \right), \mathbb{R} \right)$ and satisfies Eq (1.1) on $\left[\xi _{0}, \infty \right)$. We will consider only those solutions of Eq (1.1) which exist on some half-line $\left[\xi _{\varkappa }, \infty \right)$ and satisfy the condition

If $y$ is either positive or negative, eventually, then $y$ is called nonoscillatory; otherwise it is called oscillatory.

Due to the many applications for differential equations of the second-order in various problems of economics, biology, and physics, there is constant interest in obtaining new sufficient conditions for the oscillation or nonoscillation of the solutions of varietal types for differential equations. The development in the study of second-order delay differential equations with a canonical operator can be followed through the works ^{[1,2,3,4,5,6]}, while the equations with a noncanonical operator ^{[7,8,9,10,11]}. The works ^{[12,13,14,15,16]} extended the results from the second-order to the higher-order delay differential equations.

For some related works, Baculikova and Dzurina ^[5] considered the oscillation of the neutral differential equation

under the condition

Grace and Lalli ^[3] studied the oscillation of the equation

Dong ^[2], Liu and Bai ^[17] and Xu and Meng ^[18,19] investigated the oscillation of equation

where $0\leq p(\xi) < 1$. Li and Han ^[20,21,22] considered the oscillation of the second-order neutral differential equation

for the case where Eq (1.2) holds. Recently, Moaaz ^[6] obtained sufficient conditions for the oscillation of neutral differential equations second order

where

The objective of this paper is to establish new oscillation results for Eq (1.1). This paper is structured as follows: Firstly, by applying the theorems of comparison that compare the second-order equations with first-order delay equations, we establish a new criterion for oscillation of Eq (1.1). Secondly, we present new results for oscillation of Eq (1.1) by using the Riccati technique. Finally, some examples are considered to illustrate the main results.

2.   Main results I

Throughout this paper, we will be employing the next notations:

and

To prove the oscillation criteria, we need the next lemmas.

Lemma 2.1. [1,Lemma 3] Let $\varkappa$ be a positive solution of Eq (1.1) on $\left[\xi _{0}, \infty \right),$ then there exists a $\xi _{1}\geq \xi _{0}$ such that

Proof. Assume that $\varkappa \left(\xi \right) > 0$ is a solution of Eq (1.1). From Eq (1.1), we get

Therefore, $\left(a\left(\xi \right) \vartheta ^{\prime }\left(\xi \right) \right) ^{\prime }$ is decreasing. Thus $\vartheta ^{\prime }\left(\xi \right) > 0$ or $\vartheta ^{\prime }\left(\xi \right) < 0$ for $\xi \geq \xi _{1}$. If $\vartheta ^{\prime }\left(\xi \right) < 0$, then there exists a constant $c$ such that

Integrating from $\xi _{1}$ to $\xi$, we have

This is a contradiction and we conclude that $\vartheta ^{\prime }\left(\xi \right) > 0.$

Theorem 2.2. If the first order delay differential equation

is oscillatory, then all solutions of Eq (1.1) are oscillatory.

Proof. Assume that Eq (1.1) has a non-oscillatory solution $\varkappa$ on $\left[\xi _{0}, \infty \right)$. Without loss of generality, we assume that there exists a $\xi _{1}\geq \xi _{0}$ such that $\varkappa (\xi) > 0$, $\varkappa (\tau (\xi)) > 0$ and $\varkappa (\sigma (\xi)) > 0$ for $\xi \geq \xi _{1}$. By the definition of $\vartheta$, using $\tau \left(\xi \right) \leq \xi$ and $\vartheta ^{\prime }\left(\xi \right) > 0,$ we obtain, for $\xi \geq \xi _{1},$

which together with Eq (1.1) implies that

From Lemma 2.1, we see that

By simple computations, we see that

Integrating Eq (2.4) from $\xi _{1}$ to $\xi$, we get

Thus, from the fact that $\left(a\left(\xi \right) \left(\vartheta ^{\prime }\left(\xi \right) \right) ^{\gamma }\right) ^{\prime }\leq 0$, we arrive at

Next, we set $w\left(\xi \right) = a\left(\xi \right) \vartheta ^{\prime }\left(\xi \right)$. Using Eqs (2.3) and (2.5), we note that $w$ be a positive solution of

Using [25,Theorem1], we have that Eq (2.2) also has a positive solution, and so, we arrive at a contradiction. This ends the proof.

Corollary 1. If

or

then all solutions of Eq (1.1) are oscillatory.

Proof. Using [23,Theorem 2.1.1], we note that the conditions Eqs (2.6) or (2.7) ensure oscillation of Eq (2.2). Thus, from Theorem 2.2, all solutions of Eq (1.1) are oscillatory.

Lemma 2.3. [4,Lemma 4] Let Eq (1.1) has an eventually positive solution $\varkappa$. Suppose that $\sigma$ is strictly increasing. Assume for some $\delta > 0$ that

Then

for every $n\geq 0$ and $\xi$ large enough, where $w\left(\xi \right) : = a\left(\xi \right) \vartheta ^{\prime }\left(\xi \right),$

Theorem 2.4. Assume that $\sigma$ is strictly increasing and Eq (2.8a) holds for some $\delta > 0$. If there exists a function $\varphi \in C^{1}\left(\left[\xi _{0}, \infty \right), \left(0, \infty \right) \right)$ such that

for sufficiently large $\xi \geq \xi _{1}$ and for some $n\geq 0,$ where $\theta _{n}\left(\delta \right)$ is defined as in Eq (2.10) and $\varphi _{+}^{\prime }\left(\xi \right) = \max \left\{ 0, \varphi ^{\prime }\left(\xi \right) \right\},$ then all solutions of Eq (1.1) are oscillatory.

Proof. Assume that there is a positive solution $\varkappa$ of Eq (1.1) on $\left[\xi _{0}, \infty \right)$. Thus, there is a $\xi _{1}\geq \xi _{0}$ such that $\varkappa \left(\xi \right) > 0$, $\varkappa (\tau (\xi)) > 0$ and $\varkappa (\sigma (\xi)) > 0$ for $\xi \geq \xi _{1}.$ It follows from Lemma 2.3 that

We define the function $\Phi \left(\xi \right)$ by

Then, $\Phi \left(\xi \right) > 0$ for $\xi \geq \xi _{1}.$ Differentiating Eq (2.13), we get

From Eqs (2.3), (2.12) and (2.11), we obtain

Integrating this inequality from $\xi _{1}$ to $\xi,$ we conclude

which contradicts with Eq (2.11). This ends the proof.

Theorem 2.5. Assume that there exists a function $\phi \in C^{1}\left(\left[\xi _{0}, \infty \right), \left(0, \infty \right) \right)$ such that

for some sufficiently large $\xi \geq \xi _{1},$ where $\phi _{+}^{\prime }\left(\xi \right) = \max \left\{ 0, \phi ^{\prime }\left(\xi \right) \right\}$. Then all solutions of Eq (1.1) are oscillatory.

Proof. Assume that there is a positive solution $\varkappa$ of Eq (1.1) on $\left[\xi _{0}, \infty \right)$. Thus, there is a $\xi _{1}\geq \xi _{0}$ such that $\varkappa \left(\xi \right) > 0$, $\varkappa (\tau (\xi)) > 0$ and $\varkappa (\sigma (\xi)) > 0$ for $\xi \geq \xi _{1}.$ From Lemma 2.1, we have that Eq (2.1) holds. As in the proof of Theorem 2.2, we arrive at Eq (2.5). From Eq (2.5), we have

Integrating this inequality from $\sigma \left(\xi \right)$ to $\xi,$ we get

Combining Eqs(2.3) and (2.15), we have

Define the function

Then $\Psi \left(\xi \right) > 0$ for $\xi > \xi _{1.}$ Differentiating Eq (2.17), we arrive at

From Eqs (2.16), (2.17) and (2.18), we deduce that

Integrating this inequality from $\xi _{1}$ to $\xi$, we find

which contradicts with Eq (2.14). This ends the proof.

Theorem 2.6. If

where

then all solutions of Eq (1.1) are oscillatory.

Proof. Proceeding as in the proof of Theorem Eq (2.5), we arrive at Eq (2.18). Using Eq (2.18) with $\phi \left(\xi \right) = 1$, we obtain

Thus, we obtain

By integrating Eq (2.20) from $\xi$ to $\rho,$ we get

Since $\Psi > 0$ and $\Psi ^{\prime } < 0$, we see that $\lim_{\rho \rightarrow \infty }\Psi \left(\rho \right) = c\geq 0.$ Thus the previous inequality becomes

Hence

Set

From Eq (2.21), $\delta \geq 1$. Taking Eqs (2.19) and (2.21) into account, we find $1+\frac{1}{4}\delta ^{2}\leq \delta$, which not possible with the permissible value $\delta \geq 1$. Thus, the proof is complete.

Example 2.7. Consider the differential equation

where $\xi > 0$. By apply Theorem 2.1 in ^[24] or Theorem 1 in ^[26], Eq (2.22) is oscillatory if $q_{0} > 1.3591$. From Theorem 2.5, Eq (2.22) is oscillatory if $q_{0} > 1.1425$. Thus, our results improves results in ^[24,26].

3.   Mian results II: improved criteria

Lemma 3.1. ^[27] Let $\alpha$ be$\ $a ratios of two odd positive integers. Then

Theorem 3.2. Assume that $\gamma = 1,$ $a^{\prime }\left(\xi \right) \geq 0,$ $\sigma ^{\prime }\left(\xi \right) > 0, \ \sigma \left(\xi \right) \leq \tau \left(\xi \right),$ $\tau ^{\prime }\geq \tau _{0} > 0$ and $\sigma \circ \tau = \tau \circ \sigma.$ Furthermore, Assume that there exists a function $\rho \left(\xi \right) \in C^{1}\left(\left[\xi _{0}, \infty \right), \left(0, \infty \right) \right)$, for all sufficiently large $\xi _{1}\geq \xi _{0}$, there is a $\xi _{2} > \xi _{1}$ such that

where $\rho _{+}^{\prime }\left(\xi \right) = \max \left\{ 0, \rho ^{\prime }\left(\xi \right) \right\}.$ Then Eq (1.1) is oscillatory.

Proof. Assume that there is a positive solution $\varkappa$ of Eq (1.1) on $\left[\xi _{0}, \infty \right)$. Thus, there is a $\xi _{1}\geq \xi _{0}$ such that $\varkappa \left(\xi \right) > 0$, $\varkappa (\tau (\xi)) > 0$ and $\varkappa (\sigma (\xi)) > 0$ for $\xi \geq \xi _{1}.$Now, from Eq (1.1), we obtain

which follows from $\sigma \circ \tau = \tau \circ \sigma$ that

Next, we define a function $\omega \left(\xi \right)$ by

then $\omega \left(\xi \right) > 0.$ Differentiating Eq (3.3) with respect to $\xi$, we have

since $\vartheta ^{\prime \prime }\left(\xi \right) \leq 0$ and $\sigma \left(\xi \right) < \xi,$ we get

It follows from Eqs (3.3) and (3.5) that

Similarly, define another function $\psi$ by

then $\psi \left(\xi \right) > 0.$ Differentiating Eq (3.7) with respect to $\xi$, we have

since $\vartheta ^{\prime \prime }\left(\xi \right) \leq 0$ and $\sigma \left(\xi \right) < \tau \left(\xi \right),$ we get

It follows from Eqs (3.7) and (3.9) that

Multiplying Eq (3.10) by $p_{0}/\tau _{0}$ and combining it with Eq (3.6), we get

From Eq (3.2), we obtain

From Lemma 3.1, Eq (3.11), becomes

integrating Eq (3.12) from $\xi _{2}$ $\left(\xi _{2}\geq \xi _{1}\right)$ to $\xi$, we get

which contradicts Eq (3.1). This ends the proof.

Example 3.3. Consider the differential equation

where $\gamma = 1$ and $q_{0} > 0$. We note that $a^{\prime }\left(\xi \right) \geq 0,$ $p\left(\xi \right) = 2,$ $\sigma ^{\prime }\left(\xi \right) = 1/ \mathrm{e} > 0, \ \sigma \left(\xi \right) = \tau \left(\xi \right) = \xi / \mathrm{e},$ $q\left(\xi \right) = q_{0}/\xi ^{2},$ $\tau _{0} = 1/\mathrm{e} > 0$ and $\sigma \circ \tau = \tau \circ \sigma = \xi /\mathrm{e}^{2}.$ It's easy to verify that

By choosing $\rho \left(\xi \right) = \xi ^{2},$ the condition Eq (3.1) is satisfied if $q_{0} > 17.496.$

Thus, from Theorem 3.2, we see that Eq (3.13) is oscillatory if $q_{0} > 17.496.$

4.   Conclusions

In this paper, by different techniques and criteria, the oscillatory behavior of a class of second-order neutral delay differential equations has been studied. The results obtained are an extension and supplement to the relevant results in the literature. It is interesting to extend the results in this paper to Emden-Fowler delay differential equations with a sublinear neutral term.

Acknowledgments

The authors present their sincere thanks to the editors and two anonymous referees.

Funding

National Natural Science Foundation of China (No. 71601072), Key Scientific Research Project of Higher Education Institutions in Henan Province of China (No. 20B110006) and the Fundamental Research Funds for the Universities of Henan Province (No. NSFRF210314).

Conflict of interest

There are no competing interests.