Nonlinear Hybrid Second-Order Neutral Delay Differential Equation with Mixed Sign Terms:Oscillation via Symmetric Transform

1. Introduction

Second-order neutral differential equations with positive and negative terms are important for modeling complex systems in engineering and science, particularly where delays occur in the system dynamics, such as in high-speed electrical networks, biological population models, and mechanical systems. The inclusion of both positive and negative coefficients allows for more realistic and intricate descriptions of these phenomena, leading to diverse behaviors such as oscillations or stability, which are crucial for understanding and predicting the system’s long-term behavior. The applicability of these equations extends to diverse fields, including control theory, population dynamics, neuroscience, and, more recently, the study of complex networks and cyber-physical systems. Their ability to capture intricate temporal dynamics makes them a powerful tool for both theoretical analysis and practical applications (see [1,2]).

Motivated by these facts, in this paper we focus on the oscillatory properties of solutions of the second-order nonlinear hybrid-type neutral delay differential equation of the form

$${\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }-p\left(t\right)x\left(t\right)+q\left(t\right)x^{\alpha }\left(\sigma \left(t\right)\right)=0,t\ge t_0,$$

(E)

where $z\left(t\right)=x\left(t\right)+b\left(t\right)x\left(\tau \right(t\left)\right)$, subject to the following conditions:

(H1)

$p\left(t\right),q\left(t\right),b\left(t\right)\in C\left([t_0,\infty ),\mathbb{R}\right)$, with $0\le b\left(t\right)<1$, $p\left(t\right)>0$, and $q\left(t\right)\ge 0$;

(H2)

$a\left(t\right)\in C^{\prime }\left([t_0,\infty ),(0,\infty )\right)$ and $\alpha$ is a ratio of odd positive integers;

(H3)

$\sigma \left(t\right),\tau \left(t\right)\in C\left([t_0,\infty ),\mathbb{R}\right)$, with $\sigma \left(t\right)\le t$, $\tau \left(t\right)\le t$,

$$\underset{t\to \infty }{lim}\sigma \left(t\right)=\underset{t\to \infty }{lim}\tau \left(t\right)=\infty ,$$

and let $\delta \left(t\right)=min\left\{\tau \right(t),\sigma (t\left)\right\}$.

As usual, by a proper solution of equation $\left(E\right)$ we mean a function that satisfies $\left(E\right)$ for all sufficiently large t and for which $sup\left\{\right|x\left(t\right)|:t\ge T\}>0$ for all $T\ge t_0$. A proper solution is called oscillatory if it has infinitely many zeros; otherwise, it is called nonoscillatory.

Note that if $p\left(t\right)=0$, then $\left(E\right)$ reduces to the second-order nonlinear neutral delay differential equation

$${\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }+q\left(t\right)x^{\alpha }\left(\sigma \left(t\right)\right)=0,t\ge t_0$$

(E₁)

and if $q\left(t\right)=0$, then we obtain the linear neutral differential equation

$${\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }-p\left(t\right)x\left(t\right)=0,t\ge t_0.$$

(E₂)

Therefore, we may refer to equation $\left(E\right)$ as a hybrid-type nonlinear neutral differential equation.

These equations model systems in which the state at a given time depends not only on its current behavior but also on its past states. The “neutral” term, where the derivative also depends on delayed versions of the unknown function, captures these memory effects. The presence of both positive and negative coefficients allows for the modeling of feedback mechanisms within the system. This can lead to a richer array of behaviors, including stable, unstable, or oscillatory dynamics observed in many natural and engineered systems (see [3,4] and the references therein).

A significant focus in the study of these equations is the analysis of oscillation of their solutions. The presence of positive and negative terms can lead to solutions that oscillate around an equilibrium point, a critical characteristic in many physical and biological phenomena. Oscillatory solutions are particularly important as they model repeating behaviors found in natural and engineered systems, such as waves, vibrations, and population dynamics. Studying these solutions provides insight into system stability and behavior, allowing scientists and engineers to predict and understand phenomena across physics, biology, and engineering. By analyzing oscillatory solutions, one can determine the conditions under which systems oscillate, thereby aiding in the design and control of such systems.

Because of their practical importance, many researchers have studied oscillation criteria for second-order neutral differential equations of different types. Numerous results have appeared in the literature concerning oscillation and asymptotic behavior of second-order differential equations with linear, sub-linear, and super-linear neutral terms (see the monographs [5,6] and the papers [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]).

Oscillation criteria for second-order differential equations involving both positive and negative terms remain a topic of ongoing research. The presence of both positive and negative terms complicates the analysis, as the structure of the set of nonoscillatory solutions of the hybrid neutral equation $\left(E\right)$ is not well understood. Hence, only a limited number of criteria have been developed to study the oscillatory behavior of such equations.

In [3,25,26], the authors examined the oscillation and nonoscillation of the first-order neutral differential equation

$${(x\left(t\right)-R\left(t\right)x(t-r))}^{\prime }+P\left(t\right)x(t-\tau )-Q\left(t\right)x(t-\sigma )=0,$$

(E₃)

using the integral averaging method and fixed point theorems. Furthermore, second-order neutral differential equations of the form

$${\left(r\left(t\right){(x\left(t\right)\pm C\left(t\right)h\left(x(t-\tau )\right))}^{\prime }\right)}^{\prime }+P\left(t\right)f\left(x(t-\delta )\right)-Q\left(t\right)g\left(x(t-\sigma )\right)=0$$

(E₄)

and their generalizations were studied in [3,27,28,29,30,31,32] via the comparison method and fixed point theorems.

More recently, in [33], the authors investigated oscillation criteria for the second-order linear neutral differential equation

$${\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }+p\left(t\right)z\left(t\right)+q\left(t\right)x\left(\sigma \left(t\right)\right)=0,t\ge t_0,$$

(E₅)

where $z\left(t\right)=x\left(t\right)+b\left(t\right)x\left(\tau \right(t\left)\right)$, using the Riccati transformation technique and the integral averaging method.

Observe that the equations considered in [3,27,28,29,30,31,32] involve deviating arguments in both positive and negative terms, while the equation considered in [33] is linear and contains only positive terms. However, our equation $\left(E\right)$ is nonlinear and contains a positive term with delay and a negative term without delay. Therefore, equation $\left(E\right)$ is fundamentally and structurally different from those previously examined in the literature.

Moreover, we employ a different method to transform the trinomial hybrid equation into a binomial form, which allows a clear identification of the structure of nonoscillatory solutions of the transformed equation. This symmetry transformation is crucial for establishing oscillation criteria for equation $\left(E\right)$ since it preserves the oscillation of original equation and the transformed equation.

By employing the comparison method, integral averaging technique and arithmetic-geometric mean inequality, we derive new sufficient conditions ensuring the oscillation of all solutions of the transformed binomial-type equation, which in turn guarantees the oscillation of all solutions of equation $\left(E\right)$. Three illustrative examples are provided to demonstrate the applicability and novelty of the main results. Notably, for one illustrative example, an explicit solution is obtained and shown to be oscillatory.

2. Preliminary Results

In this section, we present some preliminary results that will be needed to establish our main theorems. First, we transform the hybrid equation $\left(E\right)$ into a binomial-type equation using a positive solution of the related auxiliary second-order linear ordinary differential equation

$${\left(a\left(t\right)u^{\prime }\left(t\right)\right)}^{\prime }=p\left(t\right)u\left(t\right),t\ge t_0.$$

(1)

Lemma 1. 

Let $\left(H_1\right)$ and $\left(H_2\right)$ hold. Then, equation (1) has a positive decreasing solution on $[t_0,\infty )$.

Proof. 

The proof is similar to Theorem 2.46 of [34], and hence the details are omitted. □

Next, note that equation $\left(E\right)$ can be rewritten in the equivalent form

$${\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }-p\left(t\right)z\left(t\right)+p\left(t\right)b\left(t\right)x\left(\tau \left(t\right)\right)+q\left(t\right)x^{\alpha }\left(\sigma \left(t\right)\right)=0.$$

(2)

Our next result is based on an equivalent representation of the differential operator

$$L\left(z\left(t\right)\right)={\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }-p\left(t\right)z\left(t\right)$$

(3)

in terms of a positive solution $u\left(t\right)$ of (1).

Lemma 2. 

Let $u\left(t\right)$ be a positive decreasing solution of (1) on $[t_0,\infty )$. Then the operator (3) can be represented as

$$L\left(z\left(t\right)\right)=\frac{1}{u\left(t\right)}{\left(a\left(t\right)u^2\left(t\right){\left(\frac{z\left(t\right)}{u\left(t\right)}\right)}^{\prime }\right)}^{\prime }.$$

(4)

Proof. 

By direct computation, we obtain

$$\begin{array}{cc}\hfill L\left(z\right(t\left)\right)& =\frac{1}{u\left(t\right)}{\left(a\left(t\right)u\left(t\right)z^{\prime }\left(t\right)-a\left(t\right)z\left(t\right)u^{\prime }\left(t\right)\right)}^{\prime }\hfill \\ & ={\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }-{\left(a\left(t\right)u^{\prime }\left(t\right)\right)}^{\prime }\frac{z\left(t\right)}{u\left(t\right)}\hfill \\ & ={\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }-p\left(t\right)z\left(t\right),\hfill \end{array}$$

where we have used the fact that $u\left(t\right)$ is a positive solution of (1). The proof of the lemma is complete. □

Lemma 3. 

Let $u\left(t\right)$ be a positive decreasing solution of (1) on $[t_0,\infty )$. Then equation $\left(E\right)$ can be written in the form

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+u\left(t\right)p\left(t\right)b\left(t\right)x\left(\tau \left(t\right)\right)+u\left(t\right)q\left(t\right)x^{\alpha }\left(\sigma \left(t\right)\right)=0,$$

(5)

where

$$\eta \left(t\right)=a\left(t\right)u^2\left(t\right),\mu \left(t\right)=\frac{z\left(t\right)}{u\left(t\right)}.$$

Proof. 

The result follows immediately by combining (2) with (4). □

For our investigation, it is convenient to assume that (5) is in canonical form, i.e.,

$${\int }_{t_0}^{\infty }\frac{1}{\eta \left(t\right)}dt={\int }_{t_0}^{\infty }\frac{1}{a\left(t\right)u^2\left(t\right)}dt=\infty .$$

(6)

Next, we establish the structure of nonoscillatory solutions of (5) when condition (6) holds.

Lemma 4. 

Let $u\left(t\right)$ be a positive decreasing solution of (1) on $[t_0,\infty )$ satisfying (6). If $x\left(t\right)$ is a positive solution of $\left(E\right)$, then the corresponding function $\mu \left(t\right)$ satisfies

$$\mu \left(t\right)>0,\eta \left(t\right){\mu }^{\prime }\left(t\right)>0,{\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }\le 0$$

(7)

for all $t\ge t_1\ge t_0$.

Proof. 

Let $x\left(t\right)$ be an eventually positive solution of $\left(E\right)$, i.e., $x\left(t\right)>0$, $x\left(\sigma \right(t\left)\right)>0$, and $x\left(\tau \right(t\left)\right)>0$ for all $t\ge t_1$ for some $t_1\ge t_0$. Then, by definition, $z\left(t\right)>0$ for all $t\ge t_1$. Since $u\left(t\right)>0$ for all $t\ge t_0$, it follows that $\mu \left(t\right)=\frac{z\left(t\right)}{u\left(t\right)}>0$ for all $t\ge t_1$. From (5), we have ${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }\le 0$ for all $t\ge t_1$. Moreover, by condition (6), it follows that $\eta \left(t\right){\mu }^{\prime }\left(t\right)>0$ for all $t\ge t_1$. a positive solution of (1). The proof of the lemma is complete. □

Lemma 5. 

Let $u\left(t\right)$ be a positive decreasing solution of (1) on $[t_0,\infty )$ satisfying (6). Then

$$x\left(t\right)\ge u\left(t\right)F\left(t\right)\mu \left(t\right),t\ge t_1\ge t_0,$$

(8)

where

$$F\left(t\right)=1-b\left(t\right)\frac{u\left(\tau \right(t\left)\right)}{u\left(t\right)}>0.$$

Proof. 

From the definition of $\mu \left(t\right)$ and its monotonicity, we have

$$u\left(t\right)\mu \left(t\right)=z\left(t\right)=x\left(t\right)+b\left(t\right)x\left(\tau \right(t\left)\right).$$

Thus,

$$x\left(t\right)\ge u\left(t\right)\mu \left(t\right)-b\left(t\right)u\left(\tau \right(t\left)\right)\mu \left(\tau \right(t\left)\right)$$

and therefore,

$$x\left(t\right)\ge u\left(t\right)\left(1-b\left(t\right)\frac{u\left(\tau \right(t\left)\right)}{u\left(t\right)}\right)\mu \left(t\right),$$

which proves the result. □

Before stating and proving the next result, let us define

$$\begin{array}{cc}& Q_1\left(t\right)=u\left(t\right)b\left(t\right)p\left(t\right)u\left(\tau \left(t\right)\right)F\left(\tau \left(t\right)\right),Q_2\left(t\right)=u\left(t\right)q\left(t\right)u^{\alpha }\left(\sigma \left(t\right)\right)F^{\alpha }\left(\sigma \left(t\right)\right),\hfill \\ & Q_3\left(t\right)=Q_1\left(t\right)+Q_2\left(t\right),\mathsf{\Omega }\left(t\right)={\int }_{t_1}^t\frac{1}{\eta \left(s\right)}ds,F\left(t\right)>0,\hfill \end{array}$$

where $t_1\ge t_0$.

Lemma 6. 

Let $u\left(t\right)$ be a positive decreasing solution of (1) on $[t_0,\infty )$ satisfying (6). Then, equation $\left(E\right)$ is oscillatory provided that

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_1\left(t\right)\mu \left(\tau \left(t\right)\right)+Q_2\left(t\right){\mu }^{\alpha }\left(\sigma \left(t\right)\right)=0,t\ge t_0,$$

(9)

is oscillatory.

Proof. 

For the sake of contradiction, assume that $x\left(t\right)$ is an eventually positive solution of $\left(E\right)$, i.e., $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for all $t\ge t_1\ge t_0$. Then the corresponding function $z\left(t\right)>0$ for all $t\ge t_1$. By Lemma 4, we have $\mu \left(t\right)=\frac{z\left(t\right)}{u\left(t\right)}>0$, and it satisfies condition (7) for all $t\ge t_1$. Using (8) in (5), we obtain that $\mu \left(t\right)$ is a positive and increasing solution of

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_1\left(t\right)\mu \left(\tau \left(t\right)\right)+Q_2\left(t\right){\mu }^{\alpha }\left(\sigma \left(t\right)\right)\le 0.$$

However, by Theorem 2 of [35], the corresponding equation (9) also has a positive solution, which yields a contradiction. This completes the proof. □

3. Oscillation Results

In this section, we derive oscillation criteria for equation $\left(E\right)$ with the help of equation (9). We begin with the following theorem.

Theorem 1. 

Let $u\left(t\right)$ be a positive decreasing solution of (1) on $\left[t_0,\infty \right)$ satisfying (6). If

$${\int }_{t_0}^{\infty }{Q}_1\left(t\right)dt=\infty \mathrm{or}{\int }_{t_0}^{\infty }{Q}_2\left(t\right)dt=\infty ,$$

(10)

then equation $\left(E\right)$ is oscillatory.

Proof. 

Assume, for the sake of contradiction, that $x\left(t\right)$ is an eventually positive solution of $\left(E\right)$, i.e., $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for all $t⩾t_1$ for some $t_1⩾t_0$. Then the corresponding function satisfies $z\left(t\right)>0$, $z\left(\tau \right(t\left)\right)>0$, and $z\left(\sigma \right(t\left)\right)>0$ for all $t⩾t_1$.

From Lemma 4, the function $\mu \left(t\right)=\frac{z\left(t\right)}{u\left(t\right)}>0$ satisfies condition (7). Since $\mu \left(t\right)$ is increasing, there exists a constant $M>0$ such that $\mu \left(t\right)⩾M>0$ for all $t⩾t_2⩾t_1$.

Using this in (9) and integrating from $t_2$ to t, we obtain

$${\int }_{t_2}^t\left(M{Q}_1\left(s\right)+M^{\alpha }{Q}_2\left(s\right)\right)ds\le \eta \left(t_2\right){\mu }^{\prime }\left(t_2\right)-\eta \left(t\right){\mu }^{\prime }\left(t\right).$$

As $t\to \infty$, it follows that

$${\int }_{t_2}^{\infty }\left(M{Q}_1\left(t\right)+M^{\alpha }{Q}_2\left(t\right)\right)dt\le \eta \left(t_2\right){\mu }^{\prime }\left(t_2\right)<\infty ,$$

which contradicts (10).The proof of the theorem is complete. □

Remark 1. 

Theorem 1 is independent of α and the delay argument. Hence, it is applicable to linear, sub-linear, and super-linear equations, as well as to both delay and advanced type differential equations.

In the following, we present several oscillation criteria for equation $\left(E\right)$ when condition (10) fails to hold.

Theorem 2. 

Assume $\alpha =1$ and let $u\left(t\right)$ be a positive decreasing solution of (1) on $\left[t_0,\infty \right)$ satisfying (6). If

$$\underset{t\to \infty }{lim}inf{\int }_{\delta \left(t\right)}^t{Q}_3\left(s\right)\mathsf{\Omega }\left(\delta \left(s\right)\right)ds>\frac{1}{e},$$

(11)

then equation $\left(E\right)$ is oscillatory.

Proof. 

Assume, for the sake of contradiction, that $x\left(t\right)$ is an eventually positive solution of $\left(E\right)$, i.e., $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for all $t⩾t_1⩾t_0$. Then the corresponding function satisfies $z\left(t\right)>0$, $z\left(\sigma \right(t\left)\right)>0$, and $z\left(\tau \right(t\left)\right)>0$ for all $t⩾t_1$.

From Lemma 4, the function $\mu \left(t\right)=\frac{z\left(t\right)}{u\left(t\right)}>0$ satisfies condition (7). From the monotonicity of $\eta \left(t\right){\mu }^{\prime }\left(t\right)$, we obtain

$$\mu \left(t\right)⩾{\int }_{t_1}^t\frac{\eta \left(s\right){\mu }^{\prime }\left(s\right)}{\eta \left(s\right)}ds⩾\mathsf{\Omega }\left(t\right)\eta \left(t\right){\mu }^{\prime }\left(t\right)$$

(12)

and hence

$${\left(\frac{\mu \left(t\right)}{\mathsf{\Omega }\left(t\right)}\right)}^{\prime }=\frac{\mathsf{\Omega }\left(t\right)\eta \left(t\right){\mu }^{\prime }\left(t\right)-\mu \left(t\right)}{\eta \left(t\right){\mathsf{\Omega }}^2\left(t\right)}⩽0,$$

which implies that $\frac{\mu \left(t\right)}{\mathsf{\Omega }\left(t\right)}$ is decreasing.

Now, from (9), we have

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_1\left(t\right)\mu \left(\tau \left(t\right)\right)+Q_2\left(t\right)\mu \left(\sigma \left(t\right)\right)=0.$$

(13)

Since $\delta \left(t\right)=min\left\{\tau \right(t),\sigma (t\left)\right\}$ and $\mu \left(t\right)$ is increasing, it follows from (13) that

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_3\left(t\right)\mu \left(\delta \left(t\right)\right)⩽0.$$

(14)

Combining (12) with (14) and letting $\omega \left(t\right)=\eta \left(t\right){\mu }^{\prime }\left(t\right)$, we see that $\omega \left(t\right)$ is a positive solution of the inequality

$${\omega }^{\prime }\left(t\right)+Q_3\left(t\right)\mathsf{\Omega }\left(\delta \left(t\right)\right)\omega \left(\delta \left(t\right)\right)⩽0.$$

(15)

However, by Theorem 2.2.6(i) in [14], condition (11) implies that inequality (15) has no eventually positive solutions, a contradiction. The proof of the theorem is complete. □

Theorem 3. 

Assume that $\alpha =1$ and let $u\left(t\right)$ be a positive decreasing solution of (1) on $\left[t_0,\infty \right)$ satisfying (6). If

$$\underset{t\to \infty }{lim}inf\mathsf{\Omega }\left(\delta \left(t\right)\right){\int }_t^{\infty }{Q}_3\left(s\right)ds>\frac{1}{4},$$

(16)

then equation $\left(E\right)$ is oscillatory.

Proof. 

Suppose, on the contrary, that $x\left(t\right)$ is an eventually positive solution of $\left(E\right)$, that is, $x\left(t\right)>0$, $x\left(e\right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for all $t⩾t_1⩾t_0$. Proceeding as in the proof of Theorem 2, we see that

$$\mu \left(t\right)=\frac{z\left(t\right)}{u\left(t\right)}>0$$

satisfies condition (1) and the inequality

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_3\left(t\right)\mu \left(\delta \left(t\right)\right)\le 0.$$

(17)

However, by Theorem 3 of [36], condition (16) implies that $\mu \left(t\right)$ is oscillatory, which is a contradiction. The proof of the theorem is complete. □

Theorem 4. 

Assume that $\alpha =1$ and let $u\left(t\right)$ be a positive decreasing solution of (1) on $\left[t_0,\infty \right)$ satisfying (6). If

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}sup\{\frac{1}{\mathsf{\Omega }\left(\delta \right(t\left)\right)}{\int }_{t_1}^{\delta \left(t\right)}{Q}_3\left(s\right)\mathsf{\Omega }\left(s\right)\mathsf{\Omega }\left(\delta \left(s\right)\right)ds& +{\int }_{\delta \left(t\right)}^t{Q}_3\left(s\right)\mathsf{\Omega }\left(\delta \left(s\right)\right)ds\hfill \\ \hfill & +\mathsf{\Omega }\left(\delta \left(t\right)\right){\int }_t^{\infty }{Q}_3\left(s\right)ds\}>1,\hfill \end{array}$$

(18)

for some $t_1⩾t_0$, then equation $\left(E\right)$ is oscillatory.

Proof. 

Suppose, on the contrary, that $x\left(t\right)$ is an eventually positive solution of $\left(E\right)$, that is, $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for all $t⩾t_1$, for some $t_1⩾t_0$. Proceeding as in the proof of Theorem 3, we see that

$$\mu \left(t\right)=\frac{z\left(t\right)}{u\left(t\right)}>0$$

satisfies condition (7) and inequality (14) for all $t⩾t_1$. Integrating (14) gives

$${\mu }^{\prime }\left(t\right)⩾\frac{1}{\eta \left(t\right)}{\int }_t^{\infty }{Q}_3\left(s\right)\mu \left(\delta \left(s\right)\right)ds.$$

Integrating once more yields

$$\begin{array}{cc}\hfill \mu \left(t\right)& ⩾{\int }_{t_1}^t\frac{1}{\eta \left(s\right)}{\int }_s^{\infty }{Q}_3\left(s_1\right)\mu \left(\delta \left(s_1\right)\right)d{s}_{1}ds\hfill \\ & ={\int }_{t_1}^t\frac{1}{\eta \left(s\right)}{\int }_s^t{Q}_3\left(s_1\right)\mu \left(\delta \left(s_1\right)\right)d{s}_{1}ds+{\int }_{t_1}^t\frac{1}{\eta \left(s\right)}{\int }_t^{\infty }{Q}_3\left(s_1\right)\mu \left(\delta \left(s_1\right)\right)d{s}_{1}ds.\hfill \end{array}$$

Applying integration by parts, we obtain

$$\mu \left(t\right)⩾{\int }_{t_1}^t{Q}_3\left(s\right)\mathsf{\Omega }\left(s\right)\mu \left(\delta \left(s\right)\right)ds+\mathsf{\Omega }\left(t\right){\int }_t^{\infty }{Q}_3\left(s\right)\mu \left(\delta \left(s\right)\right)ds.$$

Hence,

$$\begin{array}{cc}\hfill \mu \left(\delta \right(t\left)\right)⩾& {\int }_{t_1}^{\delta \left(t\right)}{Q}_3\left(s\right)\mathsf{\Omega }\left(s\right)\mu \left(\delta \left(s\right)\right)ds+\mathsf{\Omega }\left(\delta \left(t\right)\right){\int }_{\delta \left(t\right)}^t{Q}_3\left(s\right)\mu \left(\delta \left(s\right)\right)ds\hfill \\ & +\mathsf{\Omega }\left(\delta \left(t\right)\right){\int }_t^{\infty }{Q}_3\left(s\right)\mu \left(\delta \left(s\right)\right)ds.\hfill \end{array}$$

Since $\mu \left(t\right)$ is increasing and $\mu \left(t\right)/\mathsf{\Omega }\left(t\right)$ is decreasing, the previous inequality becomes

$$\begin{array}{cc}\hfill \mu \left(\delta \right(t\left)\right)⩾& \frac{\mu \left(\delta \right(t\left)\right)}{\mathsf{\Omega }\left(\delta \right(t\left)\right)}{\int }_{t_1}^{\delta \left(t\right)}{Q}_3\left(s\right)\mathsf{\Omega }\left(s\right)\mathsf{\Omega }\left(\delta \left(s\right)\right)ds\hfill \\ & +\mu \left(\delta \left(t\right)\right){\int }_{\delta \left(t\right)}^t{Q}_3\left(s\right)\mathsf{\Omega }\left(\delta \left(s\right)\right)ds+\mathsf{\Omega }\left(\delta \left(t\right)\right)\mu \left(\delta \left(t\right)\right){\int }_t^{\infty }{Q}_3\left(s\right)ds.\hfill \end{array}$$

Dividing through by $\mu \left(\delta \right(t\left)\right)$, we obtain

$$1⩾\frac{1}{\mathsf{\Omega }\left(\delta \right(t\left)\right)}{\int }_{t_1}^{\delta \left(t\right)}{Q}_3\left(s\right)\mathsf{\Omega }\left(s\right)\mathsf{\Omega }\left(\delta \left(s\right)\right)ds+{\int }_{\delta \left(t\right)}^t{Q}_3\left(s\right)\mathsf{\Omega }\left(\delta \left(s\right)\right)ds+\mathsf{\Omega }\left(\delta \left(t\right)\right){\int }_t^{\infty }{Q}_3\left(s\right)ds,$$

which contradicts (18). The proof of the theorem is complete. □

Next, we derive an oscillation condition for equation $\left(E\right)$ in the case $\alpha >1$.

Theorem 5. 

Assume that β is a ratio of odd positive integers such that $\alpha >\beta >1$, and let $u\left(t\right)$ be a positive decreasing solution of (1) on $\left[t_0,\infty \right)$ satisfying (6). If

$${\int }_{t_0}^{\infty }\frac{1}{\eta \left(t\right)}{\int }_t^{\infty }{Q}_4\left(s\right)\frac{{\mathsf{\Omega }}^{\beta }\left(\delta \left(s\right)\right)}{{\mathsf{\Omega }}^{\beta }\left(s\right)}dsdt=\infty ,$$

(19)

where

$$Q_4\left(t\right)={\left(\frac{Q_1\left(t\right)}{{\eta }_1}\right)}^{{\eta }_1}{\left(\frac{Q_2\left(t\right)}{{\eta }_2}\right)}^{{\eta }_2},{\eta }_1=\frac{\alpha -\beta }{\alpha -1},{\eta }_2=\frac{\beta -1}{\alpha -1},$$

(20)

then equation $\left(E\right)$ is oscillatory.

Proof. 

Assume, on the contrary, that $x\left(t\right)$ is an eventually positive solution of $\left(E\right)$, i.e., $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for all $t⩾t_1$, for some $t_1⩾t_0$. Proceeding as in the proof of Lemma 13, we see that $\mu \left(t\right)=\frac{z\left(t\right)}{w\left(t\right)}>0$ satisfies condition (7) and the inequality

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_1\left(t\right)\mu \left(\tau \left(t\right)\right)+Q_2\left(t\right){\mu }^{\alpha }\left(\sigma \left(t\right)\right)⩽0.$$

Since $\mu$ is increasing and $\delta \left(t\right)=min\left\{\tau \right(t),\sigma (t\left)\right\}$, we obtain

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_1\left(t\right)\mu \left(\delta \left(t\right)\right)+Q_2\left(t\right){\mu }^{\alpha }\left(\delta \left(t\right)\right)⩽0.$$

(21)

Recall the well-known arithmetic–geometric mean inequality

$${\eta }_1{y}_1+{\eta }_2{y}_2⩾y_1^{{\eta }_1}{y}_2^{{\eta }_2},y_1,y_2>0,$$

(22)

where ${\eta }_1$ and ${\eta }_2$ are chosen to satisfy (20). Now, taking $y_1={\eta }_1^{-1}{Q}_1\left(t\right){\mu }^{1-\beta }\left(\delta \left(t\right)\right)$ and $y_2={\eta }_2^{-1}{Q}_2\left(t\right){\mu }^{\alpha -\beta }\left(\delta \left(t\right)\right)$ in (22), we get

$$Q_1\left(t\right){\mu }^{1-\beta }\left(\delta \left(t\right)\right)+Q_2\left(t\right){\mu }^{\alpha -\beta }\left(\delta \left(t\right)\right)⩾{\left(\frac{Q_1\left(t\right)}{{\eta }_1}\right)}^{{\eta }_1}{\left(\frac{Q_2\left(t\right)}{{\eta }_2}\right)}^{{\eta }_2}.$$

(23)

Substituting (23) into (21), we obtain

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_4\left(t\right){\mu }^{\beta }\left(\delta \left(t\right)\right)⩽0.$$

(24)

Since $\mu \left(t\right)/\mathsf{\Omega }\left(t\right)$ is decreasing, using this property in (24), we arrive at

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_4\left(t\right)\frac{{\mathsf{\Omega }}^{\beta }\left(\delta \left(t\right)\right)}{{\mathsf{\Omega }}^{\beta }\left(t\right)}{\mu }^{\beta }\left(t\right)⩽0,t⩾t_1.$$

(25)

Integrating (25) from t to ∞ yields

$${\int }_t^{\infty }{Q}_4\left(s\right)\frac{{\mathsf{\Omega }}^{\beta }\left(\delta \left(s\right)\right)}{{\mathsf{\Omega }}^{\beta }\left(s\right)}{\mu }^{\beta }\left(s\right)ds⩽\eta \left(t\right){\mu }^{\prime }\left(t\right),$$

or equivalently,

$$\frac{1}{\eta \left(t\right)}{\int }_t^{\infty }{Q}_4\left(s\right)\frac{{\mathsf{\Omega }}^{\beta }\left(\delta \left(s\right)\right)}{{\mathsf{\Omega }}^{\beta }\left(s\right)}ds⩽\frac{{\mu }^{\prime }\left(t\right)}{{\mu }^{\beta }\left(t\right)}.$$

Integrating this inequality from $t_1$ to t, we obtain

$$\begin{array}{cc}\hfill {\int }_{t_1}^t\frac{1}{\mu \left(s\right)}{\int }_s^{\infty }{Q}_4\left(s_1\right)\frac{{\mathsf{\Omega }}^{\beta }\left(\delta \left(s_1\right)\right)}{{\mathsf{\Omega }}^{\beta }\left(s_1\right)}d{s}_{1}ds& ⩽{\int }_{t_1}^t\frac{{\mu }^{\prime }\left(s\right)}{{\mu }^{\beta }\left(s\right)}ds\hfill \\ & ⩽\frac{{\mu }^{1-\beta }\left(t_1\right)}{\beta -1}-\frac{{\mu }^{1-\beta }\left(t\right)}{\beta -1}.\hfill \end{array}$$

As $t\to \infty$, we obtain

$${\int }_{t_1}^{\infty }\frac{1}{\mu \left(t\right)}{\int }_t^{\infty }{Q}_4\left(s\right)\frac{{\mathsf{\Omega }}^{\beta }\left(\delta \left(s\right)\right)}{{\mathsf{\Omega }}^{\beta }\left(s\right)}dsdt\le \frac{{\mu }^{1-\beta }\left(t_1\right)}{\beta -1}<\infty ,$$

which contradicts (19). The proof of the theorem is complete. □

We conclude this section with the following theorem for the case $\alpha <1$.

Theorem 6. 

Assume that γ is a ratio of odd positive integers such that $1>\gamma >\alpha$, and let $u\left(t\right)$ be a positive decreasing solution of (1) on $\left[t_0,\infty \right)$ satisfying (6). If

$${\int }_{t_0}^{\infty }{\mathsf{\Omega }}^{\gamma }\left(\delta \left(t\right)\right)Q_5\left(t\right)dt=\infty ,$$

(26)

where

$$Q_5\left(t\right)={\left(\frac{Q_1\left(t\right)}{{\eta }_1}\right)}^{{\eta }_1}{\left(\frac{Q_2\left(t\right)}{{\eta }_2}\right)}^{{\eta }_2},{\eta }_1=\frac{\gamma -\alpha }{1-\alpha },{\eta }_2=\frac{1-\gamma }{1-\alpha },$$

(27)

then equation $\left(E\right)$ is oscillatory.

Proof. 

Assume, on the contrary, that $x\left(t\right)$ is an eventually positive solution of $\left(E\right)$, i.e., $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, and $x\left(\sigma \right(t\left)\right)>0$ for all $t⩾t_1$, for some $t_1⩾t_0$. Proceeding as in the proof of Theorem 5, we arrive at (21) for all $t⩾t_1$. Recall the well-known arithmetic–geometric mean inequality

$${\eta }_1{y}_1+{\eta }_2{y}_2⩾y_1^{{\eta }_1}{y}_2^{{\eta }_2},y_1,y_2>0,$$

(28)

where ${\eta }_1$ and ${\eta }_2$ are chosen to satisfy (27).

Now, setting $y_1={\eta }_1^{-1}{Q}_1\left(t\right){\mu }^{1-\gamma }\left(\delta \left(t\right)\right)$ and $y_2={\eta }_2^{-1}{Q}_2\left(t\right){\mu }^{\alpha -\gamma }\left(\delta \left(t\right)\right)$ in (28), we obtain

$$Q_1\left(t\right){\mu }^{1-\gamma }\left(\delta \left(t\right)\right)+Q_2\left(t\right){\mu }^{\alpha -\gamma }\left(\delta \left(t\right)\right)⩾{\left(\frac{Q_1\left(t\right)}{{\eta }_1}\right)}^{{\eta }_1}{\left(\frac{Q_2\left(t\right)}{{\eta }_2}\right)}^{{\eta }_2}.$$

(29)

Using (29) in (21), we obtain

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_5\left(t\right){\mu }^{\prime }\left(\delta \left(t\right)\right)\le 0.$$

(30)

Now, using (12) in (30), we find

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_5\left(t\right){\mathsf{\Omega }}^{\gamma }\left(\delta \left(t\right)\right){\left(\eta \left(\delta \left(t\right)\right){\mu }^{\prime }\left(\delta \left(t\right)\right)\right)}^{\gamma }\le 0,t⩾t_1.$$

Since $\delta \left(t\right)\le t$ and $\eta \left(t\right){\mu }^{\prime }\left(t\right)$ is decreasing, we further obtain

$${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }+Q_5\left(t\right){\mathsf{\Omega }}^{\gamma }\left(\delta \left(t\right)\right){\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\gamma }\le 0,t⩾t_1.$$

Dividing by ${\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{\prime }$ and integrating from $t_1$ to t, we get

$$\begin{array}{cc}\hfill {\int }_{t_1}^t{Q}_5\left(s\right){\mathsf{\Omega }}^{\gamma }\left(\delta \left(s\right)\right)ds& ⩽-{\int }_{t_1}^t\frac{{\left(\eta \left(s\right){\mu }^{\prime }\left(s\right)\right)}^{\prime }}{{\left(\eta \left(s\right){\mu }^{\prime }\left(s\right)\right)}^{\gamma }}ds\hfill \\ \hfill & ⩽\frac{{\left(\eta \left(t_1\right){\mu }^{\prime }\left(t_1\right)\right)}^{1-\gamma }}{1-\gamma }-\frac{{\left(\eta \left(t\right){\mu }^{\prime }\left(t\right)\right)}^{1-\gamma }}{1-\gamma }.\hfill \end{array}$$

(31)

Letting $t\to \infty$ in (31) gives a contradiction with (26). This completes the proof. □

Remark 2. 

Theorems 1–6 extend and complement several known oscillation criteria for second-order nonlinear neutral delay differential equations. In particular, they unify oscillation results for both linear and nonlinear equations with mixed arguments.

4. Examples

In this section, we present three examples to demonstrate the importance and novelty of the main results. In the first example, we provide an explicit solution for the considered equation.

Example 1. 

Consider the second-order hybrid neutral delay differential equation

$$z^{\prime \prime }\left(t\right)-\frac{2}{t^2}x\left(t\right)+\left(\frac{t^2+4}{2t^2}\right)x(t-2\pi )=0,t⩾3\pi ,$$

(32)

where $z\left(t\right)=x\left(t\right)+\frac{1}{2}x(t-\pi )$.

The corresponding auxiliary equation (1) takes the form

$$u^{\prime \prime }\left(t\right)-\frac{2}{t^2}u\left(t\right)=0,t⩾3\pi .$$

(33)

The function $u\left(t\right)=t^{-1}$ is a positive decreasing solution of (33), and $\eta \left(t\right)=\frac{1}{t^2}$ satisfies condition (6). Further calculations show that

$$F\left(t\right)=\frac{t-2\pi }{2(t-\pi )},\delta \left(t\right)=t-2\pi ,Q_1\left(t\right)\approx \frac{1}{t^4},Q_2\left(t\right)\approx \frac{1}{4t^2}+\frac{1}{t^4},\mathsf{\Omega }\left(t\right)=\frac{t^3}{3}.$$

It is easy to verify that condition (16) becomes

$$\underset{t\to \infty }{lim}inf\frac{{(t-2\pi )}^3}{3}{\int }_t^{\infty }\left(\frac{1}{4s^2}+\frac{2}{s^4}\right)ds=\infty >\frac{1}{4},$$

which implies that condition (16) holds. Therefore, by Theorem 3, equation (32) is oscillatory. In fact, $x\left(t\right)=sint$ is one such oscillatory solution.

Example 2. 

Consider the second-order hybrid neutral delay differential equation

$${\left(t^{3/2}{z}^{\prime }\left(t\right)\right)}^{\prime }-\frac{1}{2\sqrt{t}}x\left(t\right)+q_{0}t{x}^{5/3}(t/3)=0,t⩾1,$$

(34)

where $z\left(t\right)=x\left(t\right)+\frac{1}{3}x(t/2)$ and $q_0>0$ is a constant.

The corresponding auxiliary equation (1) takes the form

$${\left(t^{3/2}{u}^{\prime }\left(t\right)\right)}^{\prime }-\frac{1}{2\sqrt{t}}u\left(t\right)=0,t⩾1.$$

(35)

The function $u\left(t\right)=1/t$ is a positive decreasing solution of (35), and $\eta \left(t\right)=\frac{1}{\sqrt{t}}$ satisfies condition (6). Further calculations show that

$$F\left(t\right)=\frac{1}{3},\delta \left(t\right)=t/3,Q_1\left(t\right)=\frac{1}{6}{t}^{-5/2},Q_2\left(t\right)=q_0{t}^{-5/3}.$$

Choose $\beta =7/5$ so that $5/3>7/5>1$. Then ${\eta }_1=2/5,{\eta }_2=3/5$, and

$$Q_4\left(t\right)=q_0^{3/5}\left(\frac{5}{12}\right){\left(\frac{1}{4}\right)}^{1/5}\frac{1}{t^2},\mathsf{\Omega }\left(t\right)\approx \frac{2}{3}{t}^{3/2}.$$

Condition (19) becomes

$$q_0^{3/5}{\int }_1^{\infty }\sqrt{t}{\int }_t^{\infty }\frac{5}{54\sqrt{3}}{(1/4)}^{1/5}\frac{1}{s^2}dsdt=\frac{5}{54\sqrt{3}}{(1/4)}^{1/5}{q}_0^{3/5}{\int }_1^{\infty }\frac{1}{\sqrt{t}}dt=\infty ,$$

which shows that (19) holds for all $q_0>0$. Hence, by Theorem 5, equation (34) is oscillatory for all $q_0>0$.

Example 3. 

Consider the second-order hybrid neutral delay differential equation

$${\left(t{z}^{\prime }\left(t\right)\right)}^{\prime }-\frac{1}{t}x\left(t\right)+q_0{x}^{1/3}(t/2)=0,t⩾1,$$

(36)

where $z\left(t\right)=x\left(t\right)+\frac{1}{4}x(t/3)$ and $q_0>0$ is a constant.

The corresponding auxiliary equation (1) takes the form

$${\left(t{u}^{\prime }\left(t\right)\right)}^{\prime }-\frac{1}{t}u\left(t\right)=0,t⩾1.$$

(37)

The function $u\left(t\right)=\frac{1}{t}$ is a positive decreasing solution of (37), and $\eta \left(t\right)=\frac{1}{t}$ satisfies condition (6). Further calculations show that

$$F\left(t\right)=\frac{1}{4},\delta \left(t\right)=t/3,Q_1\left(t\right)=\frac{3}{16t^3},Q_2\left(t\right)=\frac{q_0}{2^{1/3}}{t}^{-1/3}.$$

Choose $\gamma =\frac{3}{5}$ so that $1>\frac{3}{5}>\frac{1}{3}$. Then ${\eta }_1=\frac{2}{5},{\eta }_2=\frac{3}{5}$, and

$$Q_5\left(t\right)=q_0^{3/5}{\left(\frac{1}{2}\right)}^{1/5}{\left(\frac{15}{32}\right)}^{2/5}{\left(\frac{5}{3}\right)}^{3/5}{t}^{-7/5},\mathsf{\Omega }\left(t\right)\approx \frac{t^2}{2}.$$

Condition (26) becomes

$${\int }_1^{\infty }{q}_0^{3/5}{\left(\frac{1}{2}\right)}^{4/5}{\left(\frac{15}{32}\right)}^{2/5}{\left(\frac{5}{3}\right)}^{3/5}{t}^{-1/5}dt=\infty ,$$

which shows that condition (26) holds whenever $q_0>0$. Therefore, by Theorem 6, equation (36) is oscillatory.

5. Conclusions

In this paper, we investigated the oscillatory properties of solutions to a hybrid second-order nonlinear neutral delay differential equation (E). The studied equation was first transformed into a binomial form. By applying the integral averaging method, comparison technique and the arithmetic-geometric mean inequality, we derived new sufficient conditions ensuring that all solutions of (E) oscillate. The obtained results are novel and complement the existing oscillation theory of functional differential equations. Furthermore, three illustrative examples were provided to demonstrate the significance and novelty of the main findings, since none of the known results are directly applicable to equations (32), (34), and (36).

It would be interesting to extend the results of this work to the case when condition (6) fails to hold. Another promising direction for future research is to study the oscillatory behavior of the following hybrid nonlinear neutral equation:

$${\left(a\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }+p\left(t\right)x\left(t\right)-q\left(t\right)x^{\alpha }\left(\sigma \left(t\right)\right)=0,$$

where $z\left(t\right)=x\left(t\right)+b\left(t\right)x\left(\tau \right(t\left)\right)$.