Oscillation theorems of solution of second-order neutral differential equations

1.   Introduction

In this work, we are concerned with the oscillation of solutions of the second-order neutral differential equation

Throughout this work, we assume:

$\left(S_1\right)$ $p, \ r\in C^{1}\left(\left[ l_{0}, \infty\right) \right), r\left(l\right) > 0, r^{\prime }\left(l\right) \geq 0, 0\leq p\left(l\right) < 1$ and $\inf_{l\geq l_{0}}p\left(l\right)\neq 0, p, q$ do not vanish identically on any half-line of form $\left[l_{0}, \infty \right)$;

$\left(S_2\right)$ $q\in C\left(\left[ l_{0}, \infty \right)\times \left(a, b\right), \left[ 0, \infty \right) \right), q\left(l, s\right) \geq 0$ and

$\left(S_3\right)$ $f\in C\left(\mathbb{R}, \mathbb{R}\right),$ and there exists a positive constant $k$ such that $f\left(l, u\right) /u^{\alpha }\geq k$ for $u\neq 0, \ \alpha$ is a quotient of odd positive integers and $\alpha > 1$;

$\left(S_4\right)$ $\tau \in C\left(\left[ l_{0}, \infty \right), \left(0, \infty \right) \right), \ \tau \left(l\right) \leq l$ and $\lim_{l\rightarrow \infty }\tau \left(l\right) = \infty;$

$\left(S_5\right)$ $\eta \in C\left(\left[ l_{0}, \infty \right)\times \left(a, b\right), \left(0, \infty \right) \right), \eta \left(l, s\right) \geq l, \eta$ has nonnegative partial derivatives and $\lim_{l\rightarrow \infty }\eta \left(l, s\right) = \infty.$

By a solution of (1.1) we mean a function $u\in C^{1}[l_{u}, \infty), \ l_{u}\geq l_{0},$ which has the property $r\left(l\right) \left(z^{\prime }\left(l\right) \right) \in C^{1}[l_{u}, \infty),$ and satisfies (1.1) on $[l_{u}, \infty)$. We consider only those solutions $u$ of (1.1) which satisfy $\alpha \sup \{\left\vert u\left(l\right) \right\vert : l\geq L\} > 0$ for all $L\geq l_{u}$. A solution $u$ of (1.1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory. Equation (1.1) itself is called oscillatory if all its solutions are oscillatory.

The neutral differential equations find a wide range of applications in certain high-tech fields, such as control theory, mechanical engineering, physics, population dynamics, economics, and so on, see ^[1,2]. Thus, we can see that investigating the oscillatory and asymptotic behavior of solutions of neutral differential equations is of great importance. During the past period, many papers appeared on the oscillatory behavior of differential equations of neutral and delay type, see ^{[3,4,5,6,7,8,9,10,11,12,13,14,15,16]}, and the references mentioned therein.

Second-order differential equations are derived from nuclear physics, fluid mechanics, gas dynamics, and astrophysics applications. Wong ^[17] established the oscillation criteria for equation

in the super-linear case. After that, many researchers developed several criteria for oscillation for second-order differential equations, see for example ^{[18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]} and the references mentioned therein.

Wang et al. ^[21] considered the oscillation behavior of solutions of the second-order neutral differential equations of the form

where $\alpha = 1$.

Baculikova and Dzurina ^[34] studied the asymptotic and oscillation behavior of the solutions of second-order delay differential equation

where

The main purpose of this work is to establish new criteria for oscillation of (1.1). New criteria ensure that all solutions are oscillatory, which is an extension and expansion of previous results. An example was provided to illustrate the results.

2.   Main results

In this section, we will mention our main results.

To prove our results, we will need the following notation:

Lemma 2.1. Assume that (1.2) holds and

Moreover, assume that (1.1) has a positive solution $u(l)$ on $\left[l_{1}, \infty \right)$, where $l_{1}\in$ $\left[ l_{0}, \infty \right)$ is sufficiently large. Then,

Furthermore,

Proof. Suppose that $u(l)$ is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that $u(l) > 0, u\left(\tau\left(l\right) \right) > 0$ and $u\left(\eta \left(l, s\right) \right) > 0$ for $l\geq l_{1}\geq l_{0}$. From (1.1), we have

Therefore, $z^{\prime }\left(l\right)$ is either eventually negative or eventually positive. Assume on the contrary that there exists $l_{2}\geq l_{1}$ such that $z^{\prime }\left(l\right) > 0$. Then,

and so,

which together with (2.4) implies that

Since $\eta \left(l, s\right)$ is nondecreasing with respect to $s,$ we get $\eta \left(l, s\right) \geq \eta \left(l, a\right)$ for $s\in \left(a, b\right)$, and so

Define the Riccati function by

Differentiating and using (1.1) and (2.5), we arrive at

Integrating (2.7) from $l_{2}$ to $l$, we obtain

From (2.1), we find that (2.8) comes to contradiction with the positivity of $\omega \left(l\right)$. Hence, the case $z^{\prime }\left(l\right) > 0$ is impossible. Thus, $z\left(l\right)$ satisfies (2.2) for $l\geq l_{1}$. Now, it follows from the monotonicity of $r\left(l\right) z^{\prime }\left(l\right)$ that

that is

Now

Using (2.10) and (2.11), we conclude that

This completes the proof.

Theorem 2.1. Assume that (1.2) holds. If

and

then, (1.1) is oscillatory.

Proof. Suppose that $u(l)$ is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that $u(l) > 0, u\left(\tau \left(l\right) \right) > 0$ and $u\left(\eta \left(l, s\right) \right) > 0$ for $l\geq l_{1}\geq l_{0}$. By comparing (2.1) with (2.13), we find that (2.1) is necessary for (2.13) to be correct. In fact, since the function

is unbounded due to (1.2) and $\pi ^{\prime }\left(l\right) < 0$, (2.1) must hold. Thus, by Lemma 2.1, $z\left(l\right)$ satisfies (2.2). It follows from (2.3) that there is $c > 0$ such that

and

Using (2.14) and (2.15) in (1.1), we have

Integrating (2.17) from $l_{1}$ to $l$, we have

that is,

Integrating (2.18) from $l_{1}$ to $l$ and taking (2.12) and (2.13) into account, we get

which is a contradiction. This completes the proof.

Theorem 2.2. Assume that (1.2) and (2.1) hold. If

and

is oscillatory, where

then, (1.1) is oscillatory.

Proof. Suppose that $u(l)$ is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that $u(l) > 0, u\left(\tau \left(l\right) \right) > 0$ and $u\left(\eta \left(l, s\right) \right) > 0$ for $l\geq l_{1}\geq l_{0}$. Since (2.1) holds, we can conclude that $z\left(l\right)$ satisfies (2.2) for $l\geq l_{1}$.

Considering the following

from (1.1) and (2.15), (2.21) becomes

Thus, we find $\Theta \left(l\right) = z\left(l\right) +r\left(l\right)z^{\prime }\left(l\right) \pi \left(l\right) \geq 0$ is nonincreasing. Integrating (2.22) from $l$ to $\infty$ and using (2.10), we have

since $r\left(l\right) z^{\prime }\left(l\right) \pi \left(l\right) < 0,$ we get

Substituting (2.23) in (2.16), we see that $W\left(l\right) = -r\left(l\right) z^{\prime }\left(l\right)$ is a positive solution of the following inequality

Using increasing property of $W\left(l\right)$, we get

that is,

which is a contradiction. This completes the proof.

3.   Example

In this section, we will present example to illustrate the results.

Example 3.1. Consider the second-order neutral differential equation

where $\alpha > 1, \ l_{0} = 0$. Note that$\ r\left(l\right) = l^{4},$ $p\left(l\right) = l^{-3}/3\left(l-1\right) ^{-3},$ $q\left(l, s\right) = 3^{\alpha }4\left(l+s+1\right) ^{3\alpha }l^{3},$ $\tau \left(l\right) = l-1, \ \eta\left(l, s\right) = l+s+1, \ a = 1/2, \ b = 1$ and $f\left(u\right) = u^{\alpha}.$

Now, we find

By using Theorem 2.1, we see that

and the condition (2.13) is satisfied, where

Then (3.1) is oscillatory.

4.   Conclusions

This article presents an interesting outcome by studying a class of second-order neutral functional differential equations. By explicitly taking advantage of proposing new criteria, we ensure that all solutions are oscillatory. The obtained results can provide theoretical support and empower the oscillation study for a class of second-order neutral differential equations. An illustrated example is presented to verify our results. For researchers interested in this field, and as part of our future research, we seek to find new results for the oscillation of the differential equation

under condition

Conflict of interest

The author declares that there is no competing interest.