Oscillatory Analysis of Third-Order Hybrid Trinomial Delay Differential Equations via Binomial Transform

Abstract

The oscillatory behavior of a class of third-order hybrid-type delay differential equations—used to model various real-world phenomena in fluid dynamics, control systems, biology, and beam deflection—is investigated in this study. A novel method is proposed, whereby these complex trinomial equations are reduced to a simpler binomial form by employing solutions of the corresponding linear differential equations. Through the use of comparison techniques and integral averaging methods, new oscillation criteria are derived to ensure that all solutions exhibit oscillatory behavior. These results are shown to extend and enhance existing theories in the oscillation analysis of functional differential equations. The effectiveness and originality of the proposed approach are illustrated by means of two representative examples.

1. Introduction

The study of third-order differential equations, particularly those involving delay or functional components, has garnered significant interest due to their wide-ranging applications in the natural sciences, engineering, and mathematical biology. These equations are often employed to model complex dynamic systems in which past states influence future behavior, as observed in population dynamics, control systems, and signal processing. Among these, hybrid trinomial delay differential equations are distinguished by their intricate structure, which combines polynomial and delay features with both positive and negative terms, thereby making their qualitative analysis especially challenging.

Accordingly, in this paper, attention is focused on a third-order functional differential equation involving both positive and negative terms, expressed in the following form:

$${\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\xi }_1\left(t\right)\eta \left(t\right)+{\xi }_2\left(t\right){\eta }^{\alpha }\left(\tau \left(t\right)\right)=0,t⩾t_0$$

(1)

subject to the following conditions:

(C1)

${\mu }_2\left(t\right),{\mu }_1\left(t\right),{\xi }_1\left(t\right),{\xi }_2\left(t\right)\in C\left([t_0,\infty )\right),{\mu }_2\left(t\right)>0,{\mu }_1\left(t\right)>0,{\xi }_1\left(t\right)>0$, and ${\xi }_2\left(t\right)>0$;

(C2)

${\displaystyle \tau \left(t\right)\in C^{\prime }\left([t_0,\infty )\right),{\tau }^{\prime }\left(t\right)>0,\tau \left(t\right)\le t,\mathrm{and}\underset{t\to \infty }{\lim }\tau \left(t\right)=\infty }$;

(C3)

$\alpha$ is a ratio of odd positive integers;

(C4)

Equation (1) is in canonical form, that is,

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\mu }_2\left(t\right)}}dt={\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\mu }_1\left(t\right)}}dt=\infty .$$

(2)

Given the initial point $t_0>0$, define $t_{-1}={\inf }_{t⩾t_0}\tau \left(t\right)$. By a solution of (1), we mean a function $\eta \left(t\right)\in C\left([t_{-1},\infty ),\mathbb{R}\right)$ such that ${\mu }_1\left(t\right){\eta }^{\prime }\left(t\right),{\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\in C^{\prime }\left([t_0,\infty ),\mathbb{R}\right)$ and satisfies (1) for all $t⩾t_0$. Our focus is restricted to the solutions $\eta \left(t\right)$ of (1) that exist on a half-line $[t_0,\infty )$ and satisfy $\sup \left|\eta \right(t\left)\right|:t⩾T>0$ for all $T⩾t_0$. We tacitly assume that Equation (1) possesses such solutions.

The oscillatory nature of a solution is understood in the usual sense; that is, a nontrivial solution is said to be oscillatory if it has infinitely many zeros, and it is nonoscillatory otherwise. Equation (1) is said to be oscillatory if all of its solutions are oscillatory.

Oscillatory properties are crucial for understanding and modeling real-world phenomena, as many systems naturally exhibit periodic or repeating behaviors—ranging from the simple back-and-forth motion of a pendulum to complex biological and economic cycles. Analyzing such oscillations enables the prediction of system behavior, the control of unwanted vibrations, and the optimization of performance across a wide range of applications. By studying oscillatory behavior, engineers and scientists can better anticipate system responses under varying conditions and design systems that either minimize or exploit oscillations effectively.

By letting either ${\xi }_1\left(t\right)=0$ or ${\xi }_2\left(t\right)=0$, Equation (1) reduces to simpler binomial differential equations, with or without delay, in the following forms:

$${\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+{\xi }_2\left(t\right){\eta }^{\alpha }\left(\tau \left(t\right)\right)=0$$

(3)

and

$${\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\xi }_1\left(t\right)\eta \left(t\right)=0.$$

(4)

Therefore, Equation (1) is referred to as a hybrid-type third-order delay differential equation.

This type of equation can be used to model the behavior of fluids, including phenomena such as turbulence and wave propagation. The positive and negative terms in the equations represent different forces or interactions within the fluid. In control theory, these equations model the behavior of control systems, where feedback loops with both positive and negative gains are employed to stabilize or regulate system behavior. In biological systems, third-order equations can describe population dynamics, where growth (positive term) is influenced by factors such as resource limitations (negative term). A key area of research involves studying the oscillation and stability of solutions to such equations, including the determination of conditions under which solutions oscillate, converge to a steady state, or diverge.

The oscillatory and asymptotic properties of Equations (3) and (4) have been studied by many authors; see, for example, papers [1,2,3,4,5,6,7,8,9,10], the monograph [11], and the references therein. This interest is driven by their numerous applications in the natural sciences and engineering; see, for instance, [12,13,14], which present models from mathematical biology where oscillation and/or delay effects may be formulated through cross-diffusion terms.

By a well-known result of Kiguradze [15] (Lemma 1), the possible nonoscillatory solutions of Equations (3) and (4) can be easily classified, and these solutions differ significantly. If S denotes the set of all nonoscillatory solutions of the studied equations, then for Equation (3), the set S admits the following decomposition:

$$S=S_0\cup S_2,$$

where a positive solution satisfies

$$\eta \left(t\right)\in S_0\iff {\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)<0,{\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }>0,{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }<0,$$

$$\eta \left(t\right)\in S_2\iff {\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)>0,{\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }>0,{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }<0.$$

On the other hand, for Equation (4), the set S has the following decomposition:

$$S=S_1\cup S_3,$$

with a positive solution satisfying

$$\eta \left(t\right)\in S_1\iff {\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)>0,{\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }<0,{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0,$$

$$\eta \left(t\right)\in S_3\iff {\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)>0,{\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }>0,{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0.$$

Hence, from the above discussion, the structure of the nonoscillatory solution space of Equation (1), which contains both positive and negative terms, remains unclear.

Recently, in [16,17,18,19], the authors considered an equation related to (1) of the following form:

$${\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\xi }_1\left(t\right)f\left(\eta \left(\sigma \left(t\right)\right)\right)+{\xi }_2\left(t\right)h\left(\eta \left(\tau \left(t\right)\right)\right)=0$$

(5)

and studied the oscillatory and asymptotic behavior of solutions to (5), assuming either that f is bounded or h is bounded with the following:

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\mu }_1\left(t\right)}}{\int }_t^{\infty }{\displaystyle \frac{1}{{\mu }_2\left(s\right)}}{\int }_s^{\infty }{\xi }_1\left(s_1\right)d{s}_{1}dsdt<\infty ,$$

or

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\mu }_1\left(t\right)}}{\int }_t^{\infty }{\displaystyle \frac{1}{{\mu }_2\left(s\right)}}{\int }_s^{\infty }{\xi }_2\left(s_1\right)d{s}_{1}dsdt<\infty .$$

Another method frequently used in the oscillation theory of trinomial differential equations is to omit one term (see [20,21,22,23,24,25]). This approach yields the following differential inequalities for Equation (1):

$$\left\{{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+{\xi }_2\left(t\right){\eta }^{\alpha }\left(\tau \left(t\right)\right)\right\}\mathrm{sgn}\eta \left(t\right)\ge 0,$$

$$\left\{{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\xi }_1\left(t\right)\eta \left(t\right)\right\}\mathrm{sgn}\eta \left(t\right)\le 0,$$

which are opposite in sign to those required for our analysis. Hence, using the above techniques, there exist only a limited number of studies dealing with Equation (1) involving both positive and negative terms.

It should be noted that the equations considered in [16,17,18,19,22,23] involve deviating arguments in both the positive and negative terms. However, Equation (1) contains a positive term with delay and a negative term without delay. Therefore, Equation (1) studied in this paper is fundamentally different from those previously examined in the literature.

In light of these distinctions, a different and novel method is employed in this paper to reduce the trinomial-type equation to a binomial-type equation, enabling a clearer understanding of the structure of nonoscillatory solutions. This reduction is essential for establishing criteria that guarantee the oscillation of all solutions to the studied Equation (1). By applying comparison methods and integral averaging techniques, several new oscillation criteria are derived for the transformed binomial-type equation, which, in turn, imply the oscillation of all solutions of Equation (1). Two examples are provided to illustrate the importance and significance of the main results.

2. Auxiliary Results

The main results are established via a series of lemmas, which relate properties of solutions of (1) to those of the auxiliary differential equations as follows:

$${\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\xi }_1\left(t\right)y\left(t\right)=0,$$

(6)

and

$${\left({\displaystyle \frac{{\mu }_2\left(t\right)}{y\left(t\right)}}{V}^{\prime }\left(t\right)\right)}^{\prime }+{\displaystyle \frac{{\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }}{{\mu }_1\left(t\right)y^2\left(t\right)}}V\left(t\right)=0.$$

(7)

We begin with our first result based on an equivalent representation for the linear differential operator

$${\mathcal{L}}_{\eta }\left(t\right)={\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\xi }_1\left(t\right)\eta \left(t\right)$$

(8)

in terms of positive solutions $y\left(t\right)$ and $V\left(t\right)$ of (6) and (7), respectively.

Lemma 1.

Let $y\left(t\right)$ be a positive solution of (6). Then operator (8) can be written as

$${\mathcal{L}}_{\eta }\left(t\right)={\left({\displaystyle \frac{{\mu }_2\left(t\right)}{y\left(t\right)}}{\left({\mu }_1\left(t\right)y^2\left(t\right){\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }+{\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }{\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right)}^{\prime }.$$

(9)

Proof. 

Direct computation shows that the right-hand side of (9) equals

$$\begin{array}{cc}\hfill & {\left({\displaystyle \frac{{\mu }_2\left(t\right)}{y\left(t\right)}}{\left({\mu }_1\left(t\right)y\left(t\right){\eta }^{\prime }\left(t\right)-{\mu }_1\left(t\right)y^{\prime }\left(t\right)\eta \left(t\right)\right)}^{\prime }\right)}^{\prime }+{\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }{\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right)}^{\prime }\hfill \\ \hfill & ={\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }{\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\hfill \\ \hfill & ={\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right){\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\xi }_1\left(t\right)\eta \left(t\right),\hfill \end{array}$$

which completes the proof. □

Lemma 2.

Let $y\left(t\right)$ be a positive solution of (6) and suppose equation

$${\left({\displaystyle \frac{{\mu }_2\left(t\right)}{y\left(t\right)}}{V}^{\prime }\left(t\right)\right)}^{\prime }+{\displaystyle \frac{{\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }}{{\mu }_1\left(t\right)y^2\left(t\right)}}V\left(t\right)=0$$

(10)

possesses a positive solution. Then the operator (8) can be written as

$${\mathcal{L}}_{\eta }\left(t\right)={\displaystyle \frac{1}{V\left(t\right)}}{\left[{\displaystyle \frac{{\mu }_2\left(t\right)V^2\left(t\right)}{y\left(t\right)}}{\left({\displaystyle \frac{{\mu }_1\left(t\right)y^2\left(t\right)}{V\left(t\right)}}{\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right]}^{\prime }.$$

(11)

Proof. 

By direct calculation, the right-hand side of (11) becomes

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{V\left(t\right)}}{\left[{\displaystyle \frac{{\mu }_2\left(t\right)}{y\left(t\right)}}V\left(t\right){\left({\mu }_1\left(t\right)y^2\left(t\right){\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right)}^{\prime }\right)}^{\prime }-{\displaystyle \frac{{\mu }_2\left(t\right)}{y\left(t\right)}}{V}^{\prime }\left(t\right)\left({\mu }_1\left(t\right)y^2\left(t\right){\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right)}^{\prime }\right)\right]}^{\prime }\hfill \\ \hfill & ={\left({\displaystyle \frac{{\mu }_2\left(t\right)}{y\left(t\right)}}{\left({\mu }_1\left(t\right)y^2\left(t\right){\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right)}^{\prime }\right)}^{\prime }\right)}^{\prime }-{\displaystyle \frac{{\mu }_1\left(t\right)y^2\left(t\right)}{V\left(t\right)}}\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right){\left({\displaystyle \frac{{\mu }_2\left(t\right)V^{\prime }\left(t\right)}{y\left(t\right)}}\right)}^{\prime }.\hfill \end{array}$$

(12)

Using (9) in (12), we get the following:

$${\mathcal{L}}_{\eta }\left(t\right)-{\displaystyle \frac{{\mu }_1\left(t\right)y^2\left(t\right)}{V\left(t\right)}}\left({\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}\right)\left[{\left({\displaystyle \frac{{\mu }_2\left(t\right)V^{\prime }\left(t\right)}{y\left(t\right)}}\right)}^{\prime }+{\displaystyle \frac{{\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }}{{\mu }_1\left(t\right)y^2\left(t\right)}}V\left(t\right)\right]={\mathcal{L}}_{\eta }\left(t\right),$$

since $V\left(t\right)$ satisfies (10). This completes the proof. □

From Lemmas 1 and 2, Equation (1) can be rewritten in the following binomial form:

$${\left({\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }+\Omega \left(t\right){\omega }^{\alpha }\left(\tau \left(t\right)\right)=0,$$

(13)

where

$$\begin{array}{cc}\hfill {\beta }_2\left(t\right)& ={\displaystyle \frac{{\mu }_2\left(t\right)V^2\left(t\right)}{y\left(t\right)}},{\beta }_1\left(t\right)={\displaystyle \frac{{\mu }_1\left(t\right)y^2\left(t\right)}{V\left(t\right)}},\hfill \\ \hfill \Omega \left(t\right)& =V\left(t\right){\xi }_2\left(t\right)y^{\alpha }\left(\tau \left(t\right)\right),\omega \left(t\right)={\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}.\hfill \end{array}$$

Remark 1.

Note that a trinomial equation of the form (1) can be transformed to a binomial form if either the positive or negative term is free from deviating arguments.

Following Trench [26], we say that (13) is in canonical form if

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\beta }_2\left(t\right)}}dt={\int }_{t_0}^{\infty }{\displaystyle \frac{y\left(t\right)}{{\mu }_2\left(t\right)V^2\left(t\right)}}dt=\infty ,$$

(14)

and

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\beta }_1\left(t\right)}}dt={\int }_{t_0}^{\infty }{\displaystyle \frac{V\left(t\right)}{{\mu }_1\left(t\right)y^2\left(t\right)}}dt=\infty .$$

(15)

It is important to find conditions that guarantee the existence of positive solutions of (6) and (7) such that conditions (14) and (15) hold, ensuring that (13) is in canonical form.

According to the well-known Kiguradze lemma [15], the set S of all nonoscillatory positive solutions of (6) can be decomposed as follows:

$$S=S_1\cup S_3,$$

where

$$\begin{array}{c}\hfill y\left(t\right)\in S_1\iff y\left(t\right)>0,y^{\prime }\left(t\right)>0,{\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }<0,{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0,\\ \hfill y\left(t\right)\in S_3\iff y\left(t\right)>0,y^{\prime }\left(t\right)>0,{\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }>0,{\left({\mu }_2\left(t\right){\left({\mu }_1\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }>0.\end{array}$$

Lemma 3.

Assume that

$$\underset{t\to \infty }{\lim }\sup {\pi }_1\left(t\right){\pi }_{21}\left(t\right){\mu }_1\left(t\right){\xi }_1\left(t\right)<{\displaystyle \frac{2}{3\sqrt{3}}},$$

(16)

where

$${\pi }_1\left(t\right)={\int }_{t_0}^t{\displaystyle \frac{1}{{\mu }_1\left(s\right)}}ds,{\pi }_{21}\left(t\right)={\int }_{t_0}^t{\displaystyle \frac{{\pi }_1\left(s\right)}{{\mu }_2\left(s\right)}}ds.$$

Then all solutions of (6) are nonoscillatory, and the equation admits two linearly independent solutions belonging to $S_1$ and $S_3$, respectively.

Proof. 

The result follows from Lemma 2 of [27]; details are omitted. □

To obtain our main results, it is convenient to work with $y\left(t\right)\in S_1$. Hence, we assume that (16) holds throughout. It is known (see [15,27]) that if (6) admits a solution $y\left(t\right)\in S_1$, then the corresponding Equation (7) possesses two positive solutions as follows:

$$\begin{array}{c}\hfill V\left(t\right)\in S_0\iff V\left(t\right)>0,V^{\prime }\left(t\right)<0,{\left({\displaystyle \frac{{\mu }_2\left(t\right)V^{\prime }\left(t\right)}{y\left(t\right)}}\right)}^{\prime }>0,\\ \hfill V\left(t\right)\in S_2\iff V\left(t\right)>0,V^{\prime }\left(t\right)>0,{\left({\displaystyle \frac{{\mu }_2\left(t\right)V^{\prime }\left(t\right)}{y\left(t\right)}}\right)}^{\prime }>0.\end{array}$$

For our purposes, we work with $V\left(t\right)\in S_0$, which we say is *associated* to $y\left(t\right)$.

Lemma 4.

Let $y\left(t\right)\in S_1$ be a positive solution of (6), and let $V\left(t\right)$ be associated to $y\left(t\right)$. Then conditions (14) and (15) are satisfied.

Proof. 

Condition (14) follows from the monotonicity of $y\left(t\right)$ and $V\left(t\right)$ and from condition (2). Condition (15) follows from Lemma 4 in [27]. This completes the proof. □

Note that if $y_1\left(t\right)$ and $y_2\left(t\right)$ are two solutions in $S_1$, to achieve the canonical form of (13), we take the solution $y_1\left(t\right)$ such that ${\lim }_{t\to \infty }{y}_1\left(t\right)/y_2\left(t\right)=0$. We call such a solution the *principal solution* of (6).

Combining Lemmas 3 and 4, we obtain the following result:

Corollary 1.

Let (16) hold. Let $y\left(t\right)\in S_1$ be a principal solution of (6), and let $V\left(t\right)$ be its associated solution of (7). Then Equation (1) admits an equivalent binomial representation (13) such that (13) is in canonical form.

3. Oscillation Results

In this section, we study the oscillation properties of (1) with the aid of its equivalent form (13). Hereafter, without loss of generality, we consider only positive solutions of (1). In view of the familiar Kiguradze’s lemma [15], we have the following structure for the nonoscillatory solutions of (13).

Lemma 5.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). If $\eta \left(t\right)$ is an eventually positive solution of (1), then the corresponding function $\omega \left(t\right)={\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}$ satisfies either

$$\omega \left(t\right)\in {\overline{S}}_0\iff {\omega }^{\prime }\left(t\right)<0,{\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }>0,{\left({\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }<0,$$

or

$$\omega \left(t\right)\in {\overline{S}}_2\iff {\omega }^{\prime }\left(t\right)>0,{\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }>0,{\left({\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }<0.$$

Consequently, the set $\overline{S}$ of all positive solutions of (13) (as well as of (1)) has the following decomposition:

$$\overline{S}={\overline{S}}_0\cup {\overline{S}}_2.$$

We now present a criterion under which class ${\overline{S}}_2$ is empty.

Define the following auxiliary functions:

$$\begin{array}{cc}\hfill B_1\left(t\right)& ={\int }_{t_1}^t{\displaystyle \frac{1}{{\beta }_1\left(s\right)}}ds,B_2\left(t\right)={\int }_{t_1}^t{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}ds,\hfill \\ \hfill B_{12}\left(t\right)& ={\int }_{t_1}^t{\displaystyle \frac{B_2\left(s\right)}{{\beta }_1\left(s\right)}}ds,{\Omega }_1\left(t\right)=\Omega \left(t\right)B_{12}^{\alpha }\left(\tau \left(t\right)\right),\hfill \end{array}$$

where $t_1\ge t_0$ is sufficiently large.

Theorem 1.

Let (16) hold, $y\left(t\right)\in S_1$ be a principal solution of (6), and $V\left(t\right)$ its associated solution of (7). If the first-order nonlinear delay differential equation

$$z^{\prime }\left(t\right)+{\Omega }_1\left(t\right)z^{\alpha }\left(\tau \left(t\right)\right)=0$$

(17)

is oscillatory, then class ${\overline{S}}_2$ is empty.

Proof. 

Assume, on the contrary, that $\omega \left(t\right)$ is a positive solution of Equation (13) belonging to class ${\overline{S}}_2$ for all $t\ge t_1\ge t_0$. Define

$$z\left(t\right):={\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }>0.$$

Since $z\left(t\right)$ is decreasing, we have

$${\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)={\int }_{t_1}^t{\left({\beta }_1\left(s\right){\omega }^{\prime }\left(s\right)\right)}^{\prime }ds\ge {\int }_{t_1}^t{\displaystyle \frac{{\beta }_2\left(s\right){\left({\beta }_1\left(s\right){\omega }^{\prime }\left(s\right)\right)}^{\prime }}{{\beta }_2\left(s\right)}}ds\ge B_2\left(t\right)z\left(t\right).$$

Integrating again, we get

$$\omega \left(t\right)\ge {\int }_{t_1}^t{\displaystyle \frac{B_2\left(s\right)}{B_1\left(s\right)}}z\left(s\right)ds\ge B_{12}\left(t\right)z\left(t\right).$$

Hence,

$${\omega }^{\alpha }\left(\tau \left(t\right)\right)\ge B_{12}^{\alpha }\left(\tau \left(t\right)\right)z^{\alpha }\left(\tau \left(t\right)\right).$$

Substituting into (13), we obtain

$$-z^{\prime }\left(t\right)\ge {\Omega }_1\left(t\right)z^{\alpha }\left(\tau \left(t\right)\right).$$

Thus, $z\left(t\right)$ satisfies the differential inequality

$$z^{\prime }\left(t\right)+{\Omega }_1\left(t\right)z^{\alpha }\left(\tau \left(t\right)\right)\le 0.$$

However, by Theorem 1 in [28], the corresponding differential Equation (17) must also have a positive solution, which contradicts its assumed oscillatory nature. This completes the proof. □

We now provide explicit conditions under which class ${\overline{S}}_2$ is empty.

Corollary 2.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). If $\alpha =1$ and

$$\underset{t\to \infty }{\lim }\inf {\int }_{\tau \left(t\right)}^t{\Omega }_1\left(s\right)ds>{\displaystyle \frac{1}{e}},$$

(18)

then class ${\overline{S}}_2$ is empty.

Corollary 3.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). If $0<\alpha <1$ and

$${\int }_{t_0}^{\infty }{\Omega }_1\left(t\right)dt=\infty ,$$

(19)

then class ${\overline{S}}_2$ is empty.

Corollary 4.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). Suppose $\alpha >1$ and $\tau \left(t\right)=\theta t$, with $\theta \in (0,1)$. If there exists $\lambda >{\displaystyle \frac{\ln \left(\alpha \right)}{\ln \left(\theta \right)}}$ such that

$$\underset{t\to \infty }{\lim }\inf \left[{\Omega }_1\left(t\right)\exp (-t^{\lambda })\right]>0,$$

(20)

then class ${\overline{S}}_2$ is empty.

Corollary 5.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). Suppose $\alpha >1$ and $\tau \left(t\right)=t^{\theta }$, with $\theta \in (0,1)$. If there exists $\lambda >{\displaystyle \frac{\ln \left(\alpha \right)}{\ln \left(\theta \right)}}$ such that

$$\underset{t\to \infty }{\lim }\inf \left[{\Omega }_1\left(t\right)\exp \left(-{\left(\ln t\right)}^{\lambda }\right)\right]>0,$$

(21)

then class ${\overline{S}}_2$ is empty.

The proofs of Corollaries 2–5 follow from known oscillation criteria for Equation (17), corresponding to different ranges of $\alpha$: for $\alpha =1$, see [29]; for $0<\alpha <1$, see [30]; and for $\alpha >1$, see [31].

In the following, we state and prove a result ensuring that any solution $\omega \left(t\right)={\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}$ of (13) belonging to class ${\overline{S}}_0$ converges to zero asymptotically.

Theorem 2.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). Assume that either

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\beta }_2\left(t\right)}}{\int }_t^{\infty }\Omega \left(s\right)dsdt=\infty ,$$

(22)

or

$${\int }_{t_0}^{\infty }{\displaystyle \frac{1}{{\beta }_1\left(t\right)}}{\int }_t^{\infty }{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}{\int }_s^{\infty }\Omega \left(u\right)dudsdt=\infty .$$

(23)

If $\omega \left(t\right)\in {\overline{S}}_0$ is a solution of Equation (13), then

$$\underset{t\to \infty }{\lim }{\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}=\underset{t\to \infty }{\lim }\omega \left(t\right)=0.$$

Proof. 

Let $\omega \left(t\right)$ be a positive solution of (13) belonging to class ${\overline{S}}_0$. Choose $t_1\ge t_0$ such that $\omega \left(\tau \right(t\left)\right)>0$ for all $t\ge t_1$. Clearly, there exists a finite limit ${\omega }_0$ such that

$$\underset{t\to \infty }{\lim }\omega \left(t\right)={\omega }_0\ge 0.$$

Assume, for contradiction, that ${\omega }_0>0$. Then there exists $t_2\ge t_1$ such that $\omega \left(\tau \left(t\right)\right)\ge {\omega }_0$ for all $t\ge t_2$.

First, suppose that condition (22) holds. Integrating Equation (13) from t to ∞, we obtain

$${\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }\ge {\int }_t^{\infty }\Omega \left(s\right){\omega }^{\alpha }\left(\tau \left(s\right)\right)ds\ge {\omega }_0^{\alpha }{\int }_t^{\infty }\Omega \left(s\right)ds,$$

which implies

$${\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }\ge {\displaystyle \frac{{\omega }_0^{\alpha }}{{\beta }_2\left(t\right)}}{\int }_t^{\infty }\Omega \left(s\right)ds.$$

(24)

Integrating (24) from $t_2$ to t, we get

$$-{\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\le -{\beta }_1\left(t_2\right){\omega }^{\prime }\left(t_2\right)-{\int }_{t_2}^t{\displaystyle \frac{{\omega }_0^{\alpha }}{{\beta }_2\left(s\right)}}{\int }_s^{\infty }\Omega \left(u\right)duds\to -\infty \mathrm{as}t\to \infty ,$$

which contradicts the positivity of $-{\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)$. Now suppose that condition (23) holds. Integrating (24) from t to ∞, we obtain

$$-{\omega }^{\prime }\left(t\right)\ge {\displaystyle \frac{{\omega }_0^{\alpha }}{{\beta }_1\left(t\right)}}{\int }_t^{\infty }{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}{\int }_s^{\infty }\Omega \left(u\right)duds,$$

and consequently,

$$\omega \left(t\right)\le \omega \left(t_2\right)-{\int }_{t_2}^t{\displaystyle \frac{{\omega }_0^{\alpha }}{{\beta }_1\left(s\right)}}{\int }_s^{\infty }{\displaystyle \frac{1}{{\beta }_2\left(u\right)}}{\int }_u^{\infty }\Omega \left(s_1\right)d{s}_{1}duds\to -\infty \mathrm{as}t\to \infty ,$$

which contradicts the positivity of $\omega \left(t\right)$. Hence, our assumption that ${\omega }_0>0$ must be false. We conclude that

$$\underset{t\to \infty }{\lim }\omega \left(t\right)=0.$$

This completes the proof. □

Remark 2.

The above theorem is independent of the value of α, and it thus applies to linear, sublinear, and superlinear equations, as well as to both delay and advanced-type equations.

Combining Corollaries 2–5 with Theorem 2, we immediately obtain the following result.

Theorem 3.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). If either (22) or (23) holds along with Corollary 2, 3, 4, or 5, then every nonoscillatory solution $\eta \left(t\right)$ of Equation (1) satisfies

$$\underset{t\to \infty }{\lim }{\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}=0.$$

Remark 3.

The conclusion of the above theorem can be reformulated as every nonoscillatory solution $\eta \left(t\right)$ of (1) satisfies the inequality $\left|\eta \right(t\left)\right|\le My\left(t\right)$ for some positive constant M.

Next, we obtain conditions under which class ${\overline{S}}_0$ is empty. Define

$${\Omega }_2\left(t\right)={\displaystyle \frac{1}{{\beta }_1\left(t\right)}}{\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}{\int }_s^{\sigma \left(s\right)}\Omega \left(s_1\right)d{s}_{1}ds.$$

Theorem 4.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). Assume there exists a function $\sigma \left(t\right)\in C^{\prime }\left([t_0,\infty ),\mathbb{R}\right)$ such that

$${\sigma }^{\prime }\left(t\right)\ge 0,\sigma \left(t\right)>t,\delta \left(t\right)=\tau \left(\sigma \left(\sigma \left(t\right)\right)\right)<t.$$

(25)

If the first-order delay differential equation

$${\chi }^{\prime }\left(t\right)+{\Omega }_2\left(t\right){\chi }^{\alpha }\left(\delta \left(t\right)\right)=0$$

(26)

is oscillatory, then class ${\overline{S}}_0$ is empty.

Proof. 

Assume, on the contrary, that $\omega \left(t\right)$ is an eventually positive solution of (13), belonging to class ${\overline{S}}_0$ for all $t\ge t_1$. Integrating (13) from t to $\sigma \left(t\right)$, yields

$$\begin{array}{cc}\hfill {\beta }_2\left(t\right){\left({\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\right)}^{\prime }& \ge {\int }_t^{\sigma \left(t\right)}\Omega \left(s\right){\omega }^{\alpha }\left(\tau \left(s\right)\right)ds\hfill \\ & \ge {\omega }^{\alpha }\left(\tau \left(\sigma \left(t\right)\right)\right){\int }_t^{\sigma \left(t\right)}\Omega \left(s\right)ds.\hfill \end{array}$$

Dividing both sides by ${\beta }_2\left(t\right)$ and integrating from t to $\sigma \left(t\right)$, we obtain

$$-{\beta }_1\left(t\right){\omega }^{\prime }\left(t\right)\ge {\omega }^{\alpha }\left(\delta \left(t\right)\right){\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}{\int }_s^{\sigma \left(s\right)}\Omega \left(s_1\right)d{s}_{1}ds.$$

Integrating again from t to ∞, we find

$$\omega \left(t\right)\ge {\int }_t^{\infty }{\displaystyle \frac{{\omega }^{\alpha }\left(\delta \left(s\right)\right)}{{\beta }_1\left(s\right)}}{\int }_s^{\sigma \left(s\right)}{\displaystyle \frac{1}{{\beta }_2\left(s_1\right)}}{\int }_{s_1}^{\sigma \left(s_1\right)}\Omega \left(s_2\right)d{s}_{2}d{s}_{1}ds.$$

Let the right-hand side be denoted by $\chi \left(t\right)$. Then $\omega \left(t\right)\ge \chi \left(t\right)>0$, and we find that

$$\begin{array}{cc}\hfill 0& ={\chi }^{\prime }\left(t\right)+\left({\displaystyle \frac{1}{{\beta }_1\left(t\right)}}{\int }_t^{\sigma \left(t\right)}{\displaystyle \frac{1}{{\beta }_2\left(s\right)}}{\int }_s^{\sigma \left(s\right)}\Omega \left(s_1\right)d{s}_{1}ds\right){\omega }^{\alpha }\left(\delta \left(t\right)\right)\hfill \\ & \ge {\chi }^{\prime }\left(t\right)+{\Omega }_2\left(t\right){\chi }^{\alpha }\left(\delta \left(t\right)\right).\hfill \end{array}$$

Hence, Theorem 1 from [28] implies that Equation (26) has a positive solution $\chi \left(t\right)$, contradicting its assumed oscillation. Therefore, class ${\overline{S}}_0$ must be empty. This completes the proof. □

Corollary 6.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). Assume that there exists a function $\sigma \left(t\right)\in C^{\prime }\left([t_0,\infty ),\mathbb{R}\right)$ such that (23) holds. If $\alpha =1$ and

$$\underset{t\to \infty }{\lim }\inf {\int }_{\delta \left(t\right)}^t{\Omega }_2\left(s\right)ds>{\displaystyle \frac{1}{e}},$$

(27)

then class ${\overline{S}}_0$ is empty.

Corollary 7.

Under the same assumptions as Corollary 6, if $0<\alpha <1$ and

$${\int }_{t_0}^{\infty }{\Omega }_2\left(t\right)dt=\infty ,$$

(28)

then class ${\overline{S}}_0$ is empty.

Corollary 8.

Under the same assumptions as Corollary 6, if $\alpha >1$, $\delta \left(t\right)=\theta t$ with $\theta \in (0,1)$, and there exists $\lambda >{\displaystyle \frac{\ln \left(\alpha \right)}{\ln \left(\theta \right)}}$ such that

$$\underset{t\to \infty }{\lim }\inf \left[{\Omega }_2\left(t\right)\exp (-t^{\lambda })\right]>0,$$

(29)

then class ${\overline{S}}_0$ is empty.

Corollary 9.

Under the same assumptions as Corollary 6, if $\alpha >1$, $\delta \left(t\right)=t^{\theta }$ with $\theta \in (0,1)$, and there exists $\lambda >{\displaystyle \frac{\ln \left(\alpha \right)}{\ln \left(\theta \right)}}$ such that

$$\underset{t\to \infty }{\lim }\inf \left[{\Omega }_2\left(t\right)\exp (-{\left(\ln t\right)}^{\lambda })\right]>0,$$

(30)

then class ${\overline{S}}_0$ is empty.

The sufficient conditions for the oscillation of (26) for $\alpha =1$, $0<\alpha <1$, and $\alpha >1$ in the above corollaries can be found in [29], [30], and [31], respectively.

Combining the criteria for ${\overline{S}}_0$ and ${\overline{S}}_2$ to be empty, we now present the following oscillation criterion for Equation (1).

Theorem 5.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). Assume that there exists a function $\sigma \left(t\right)\in C^{\prime }\left([t_0,\infty ),\mathbb{R}\right)$ such that (23) holds. Let $\alpha =1$ (or $\alpha <1$). If (19) (respectively, (20)) and (25) (respectively, (26)) hold, then Equation (1) is oscillatory.

Proof. 

Let $\eta \left(t\right)$ be an eventually positive solution of (1) such that $\eta \left(\tau \right(t\left)\right)>0$ for all $t\ge t_1$. Then, by Corollary 1, the function $\omega \left(t\right)={\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}$ is also a positive solution of (13), and by Lemma 1, we have $\omega \left(t\right)\in {\overline{S}}_0$ or $\omega \left(t\right)\in {\overline{S}}_2$ for all $t\ge t_1$. Corollary 2 (or 3) ensures that ${\overline{S}}_2$ is empty, and Corollary 6 (or 7) implies that ${\overline{S}}_0$ is empty. Therefore, by the oscillation-preserving transformation $\eta \left(t\right)=\omega \left(t\right)y\left(t\right)$, it follows that Equation (1) is oscillatory. □

Theorem 6.

Let (16) hold, and let $y\left(t\right)\in S_1$ be a principal solution of (6), with $V\left(t\right)$ being its associated solution of (7). Assume that there exists a function $\sigma \left(t\right)\in C^{\prime }\left([t_0,\infty ),\mathbb{R}\right)$ such that (23) holds. Suppose $\alpha >1$, $\tau \left(t\right)={\theta }_{1}t$ (or $t^{{\theta }_1}$), $\delta \left(t\right)={\theta }_{2}t$ (or $t^{{\theta }_2}$) for some ${\theta }_1,{\theta }_2\in (0,1)$. If there exists $\lambda >{\displaystyle \frac{\ln \left(\alpha \right)}{\ln \left({\theta }_1\right)}}$ (or $\lambda >{\displaystyle \frac{\ln \left(\alpha \right)}{\ln \left({\theta }_2\right)}}$) such that (21) (or (22)) and (27) (or (28)) hold, then Equation (1) is oscillatory.

Proof. 

The proof is similar to that of Theorem 3 and is omitted for brevity. □

4. Examples

In this section, we present two examples whose oscillatory character cannot be determined by any known results in [16,17,18,19,20,21,22,23,24].

Example 1.

Consider the third-order hybrid delay differential equation

$${\eta }^{‴}\left(t\right)-{\displaystyle \frac{21}{64t^3}}\eta \left(t\right)+{\displaystyle \frac{405}{128t^3}}\eta (t/2)=0,t\ge 1.$$

(31)

The corresponding auxiliary equation is

$$y^{‴}\left(t\right)-{\displaystyle \frac{21}{64t^3}}y\left(t\right)=0.$$

Condition (16) is satisfied, and the required principal solution is $y\left(t\right)=t^{1/4}$. Using this, the associated solution is found to be

$$V\left(t\right)=t^{(5-\sqrt{37})/8}.$$

Further calculations yield the following:

$$\begin{array}{cc}& {\beta }_1\left(t\right)=t^{(\sqrt{37}-1)/8},{\beta }_2\left(t\right)=t^{(3-\sqrt{37})/4},\hfill \\ & \Omega \left(t\right)={\displaystyle \frac{405}{2^{1/4}·128}}{t}^{-(17+\sqrt{37})/8},\hfill \\ & B_{12}\left(t\right)\approx {\displaystyle \frac{32}{(1+\sqrt{37})(11+\sqrt{37})}}{t}^{(11+\sqrt{37})/8},\hfill \\ & {\Omega }_1\left(t\right)={\displaystyle \frac{\mathrm{12,960}}{2^{(13+\sqrt{37})/8}·128(1+\sqrt{37})(11+\sqrt{37})}}{t}^{-3/4}.\hfill \end{array}$$

It is easy to verify that conditions (14), (15), (18), and (23) are satisfied. Therefore, by Theorem 3, we conclude that every nonoscillatory solution of (31) satisfies

$$\underset{t\to \infty }{\lim }{\displaystyle \frac{\eta \left(t\right)}{y\left(t\right)}}=0.$$

In fact, $\eta \left(t\right)={\displaystyle \frac{1}{t}}$ is one such solution of (31).

Example 2.

Consider the third-order hybrid delay differential equation

$${\left({\displaystyle \frac{1}{t}}{\left({\displaystyle \frac{1}{t}}{\eta }^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\displaystyle \frac{3}{t^5}}\eta \left(t\right)+\left({\displaystyle \frac{t^2+3}{t^3}}\right)\eta \left(t-{\displaystyle \frac{3\pi }{2}}\right)=0,t\ge 1.$$

(32)

The corresponding auxiliary equation is

$${\left({\displaystyle \frac{1}{t}}{\left({\displaystyle \frac{1}{t}}{y}^{\prime }\left(t\right)\right)}^{\prime }\right)}^{\prime }-{\displaystyle \frac{3}{t^5}}y\left(t\right)=0,$$

which clearly satisfies condition (16). The principal solution is $y\left(t\right)=t^{\delta }$, where $\delta ={\displaystyle \frac{5-\sqrt{13}}{2}}$. Using this, the associated solution is given by

$$V\left(t\right)=t^{-{\delta }_1},\mathit{where}{\delta }_1={\displaystyle \frac{\sqrt{13}+\sqrt{22+6\sqrt{13}}-9}{4}}\approx 0.305.$$

Further computations give the following:

$$\begin{array}{cc}& {\beta }_1\left(t\right)=t^{2{\delta }_1+\delta -1}\approx t^{0.7},{\beta }_2\left(t\right)=t^{-2{\delta }_1-\delta -1}\approx t^{-2.717},\hfill \\ & \Omega \left(t\right)={\displaystyle \frac{t^2+3}{t^3}}{t}^{-{\delta }_1}{\left(t-{\displaystyle \frac{3\pi }{2}}\right)}^{\delta },\hfill \\ & B_{12}\left(t\right)={\displaystyle \frac{t^{4-\delta +{\delta }_1}}{(4-\delta +{\delta }_1)(2+\delta +2{\delta }_1)}},\hfill \end{array}$$

and

$${\Omega }_1\left(t\right)={\displaystyle \frac{t^2+3}{t^{3+{\delta }_1}}}·{\displaystyle \frac{{\left(t-\frac{3\pi }{2}\right)}^{4+{\delta }_1}}{(4-\delta +{\delta }_1)(2+\delta +2{\delta }_1)}}.$$

Conditions (14) and (15) are clearly satisfied. For condition (18), we observe the following:

$$\underset{t\to \infty }{\lim }\inf {\int }_{t-{\displaystyle \frac{3\pi }{2}}}^t{\Omega }_1\left(s\right)ds>\underset{t\to \infty }{\lim }{\displaystyle \frac{3\pi }{2(4-\delta +{\delta }_1)(2+\delta +2{\delta }_1)}}·{\displaystyle \frac{{(t+3\pi )}^{4+{\delta }_1}}{t^{1+{\delta }_1}}}\to \infty .$$

Thus, condition (18) is satisfied.

Now choose $\sigma \left(t\right)=t+{\displaystyle \frac{\pi }{2}}$. Then, $\delta \left(t\right)=t-{\displaystyle \frac{\pi }{2}}<t$. For condition (24), we estimate the following:

$$\begin{array}{c}\hfill \underset{t\to \infty }{\lim }\inf {\int }_{t-\pi /2}^t{s}^{1+{\delta }_1-2\delta }{\int }_s^{s+\pi /2}{s}_1^{1+\delta +2{\delta }_1}{\int }_{s_1}^{s_1+\pi /2}{\displaystyle \frac{s_2^2+3}{s_2^3}}{s}_2^{-{\delta }_1}{\left(s_2-{\displaystyle \frac{3\pi }{2}}\right)}^{\delta }d{s}_{2}d{s}_{1}ds\\ \hfill >\underset{t\to \infty }{\lim }{\displaystyle \frac{{\pi }^2}{4}}{t}^{1+\delta }\to \infty .\end{array}$$

Hence, condition (24) is satisfied. Therefore, by Theorem 3, Equation (32) is oscillatory. In fact, $\eta \left(t\right)=t\sin t$ is one such oscillatory solution of (32).

5. Conclusions

In this study, we investigated the oscillatory behavior of a class of third-order hybrid-type trinomial delay differential equations. By employing a novel transformation approach—converting these complex equations into a simpler binomial form via solutions of associated linear auxiliary equations—we established new oscillation criteria. These criteria were derived using comparison techniques and integral averaging methods, enabling the effective handling of variable coefficients and delays beyond the reach of traditional approaches.

Our main results demonstrate that the oscillation of solutions can be guaranteed under broad and practically verifiable conditions involving integral bounds on the coefficient functions. This advances existing theories by generalizing prior criteria, which often relied on restrictive assumptions or specific structural forms. Moreover, the new criteria are applicable to a wide range of models, including those with variable delays and mixed-sign coefficients. Two examples are provided as follows: one illustrates that every nonoscillatory solution tends to zero asymptotically, while the other shows that every solution is oscillatory.

Despite these contributions, the study opens several avenues for future research. A promising direction involves developing oscillation criteria that do not require explicit solutions of the auxiliary linear equations, potentially broadening applicability to more general classes of equations. Furthermore, extending the framework to encompass neutral-type, higher-order, or stochastic delay equations remains an open and intriguing challenge.

In conclusion, the transformation-based methodology introduced herein offers a powerful tool for oscillation analysis, enriching the existing literature and paving the way for further theoretical developments and practical applications in the study of delay differential equations.