Equivalence Transformation for Neutral Differential Equations: Oscillation of Solutions

Abstract

We introduce an equivalence transformation to study the oscillation behavior of solutions for linear neutral differential equations of canonical and noncanonical types. The new approach leads to several novel oscillation criteria. Moreover, we show that the same arguments can be applied to nonlinear neutral equations under suitable monotonicity conditions. The importance of the results is also supported by examples.

1. Introduction

The qualitative behavior of solutions has been a significant area of research in differential equations. Neutral-type delay differential equations (NDEs) that are characterized by the presence of derivatives of the dependent variable with both delayed and non-delayed arguments have also attracted a lot of attention in recent years. For many valuable contributions and background material in this area, the reader is referred to [1,2,3,4,5,6,7,8,9] and the references therein, as well as the seminal monographs [10,11,12]. Subsequent investigations have significantly expanded the understanding of such equations; see, for instance, numerical developments for such equations [13,14].

In particular, second-order neutral differential equations arise in engineering and physical sciences, including vibration systems with delay, transmission lines, and control systems with memory, where delayed derivatives affect stability and dynamics. A comprehensive treatment of such models is given in [15,16].

Recently, Agarwal et al. [1] studied the second-order NDE of the form

$$\begin{array}{c}\hfill {\left(a\left(x\right)u^{\prime }\right)}^{\prime }+l\left(x\right)y_{\sigma }=0,x\ge x_0,\end{array}$$

(1)

where $u\left(x\right)=y\left(x\right)+b\left(x\right)y_{\tau }^{\alpha }\left(x\right)$, $y_{\sigma }$ denotes the composition function $y\circ \sigma$, $\alpha \in (0,1]$, a is positive, b and l are non-negative, and $\tau ,\sigma$ are delayed arguments.

As in ordinary differential equations, (1) is said to be of canonical type if

$${\int }_{x_0}^{\infty }{\displaystyle \frac{ds}{a\left(s\right)}}=\infty ,$$

(2)

and of noncanonical type if

$${\int }_{x_0}^{\infty }{\displaystyle \frac{ds}{a\left(s\right)}}<\infty .$$

(3)

The following oscillation theorem was obtained for (1) when it is of canonical type. For our purpose in this work, we state it for the case $\alpha =1$. In what follows, for simplicity, we put

$${\int }_a^{\infty }f\left(x\right)dx=\underset{T\to \infty }{lim\; sup}{\int }_a^{T}f\left(x\right)dx.$$

Theorem 1

(Theorem 1, [1]). Assume that (2) holds. If there exists a positive function $\eta \in C^1([t_0,\infty ),\mathbb{R})$ such that

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(1-b_{\sigma }\left(s\right)\right)l\left(s\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds=\infty ,\end{array}$$

(4)

then (1) is oscillatory.

In the same paper, the authors also proved an oscillation theorem for the noncanonical case. Namely, if, in addition to (4),

$$1-b\left(x\right)A_{\tau }\left(x\right)/A\left(x\right)>0$$

and

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(1-{\displaystyle \frac{A_{\tau }\left(\sigma \left(s\right)\right)}{a_{\sigma }\left(s\right)}}{b}_{\sigma }\left(s\right)\right)l\left(s\right)A\left(s\right)-{\displaystyle \frac{1}{4a\left(s\right)A\left(s\right)}}\right)ds=\infty ,\end{array}$$

where

$$\begin{array}{c}\hfill A\left(x\right)={\int }_x^{\infty }{\displaystyle \frac{ds}{a\left(s\right)}},\end{array}$$

then (1) is oscillatory.

The aim in this work is to find oscillation theorems that are alternative to the above ones and further improve them even in their present settings. It is shown that the additional conditions mentioned for the noncanonical case can be significantly simplified. Our motivation comes from the work in [17], where the present authors obtained several improved conditions for the oscillation of solutions of delay differential equations by employing an oscillation-preserving transformation introduced in [18] to improve the famous Leighton-type oscillation criteria. In the present study, we will show that a similar equivalence transformation can be derived to investigate the NDE of the form

$$\begin{array}{c}\hfill {\left(a\left(x\right)u^{\prime }\right)}^{\prime }+k\left(x\right)u+l\left(x\right)y_{\sigma }=0,x\ge x_0,\end{array}$$

(5)

where $u\left(x\right)=y\left(x\right)+b\left(x\right)y_{\tau }\left(x\right).$ The following assumptions will hold throughout:

• $a,k,l\in C([t_0,\infty ),(0,\infty ))$; $b\in C([t_0,\infty ),[0,\infty ))$ with $b\left(x\right)<1$.
• $\tau$ and $\sigma$ are delay arguments; $\tau \in C([t_0,\infty ),\mathbb{R})$, $\sigma \in C^1([t_0,\infty ),\mathbb{R})$; ${\sigma }^{\prime }\left(x\right)>0,$ and ${\displaystyle \underset{x\to \infty }{lim}\tau \left(x\right)=\underset{x\to \infty }{lim}\sigma \left(x\right)=\infty }$.

A nontrivial solution $y\left(x\right)$ of (5) is called oscillatory if it is eventually positive or negative.

2. Main Results

We first extend Theorem 1 to include (5).

Theorem 2.

Suppose condition (2) is satisfied. If for some positive differentiable function η,

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(k\left(s\right)+(1-b_{\sigma }\left(s\right))l\left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds=\infty ,\end{array}$$

(6)

then every nontrivial solution of (5) oscillates.

Proof. 

Suppose that $y\left(x\right)$ is an eventually positive solution of (5). We may suppose that $y\left(x\right)>0$, $y_{\tau }\left(x\right)>0$ and $y_{\sigma }\left(x\right)>0$ for all $x\ge x_1$ for some $x_1\ge x_0$. From (5), we see that

$$\begin{array}{c}\hfill {\left(a\left(x\right)u^{\prime }\left(x\right)\right)}^{\prime }=-k\left(x\right)u\left(x\right)-l\left(x\right)y_{\sigma }\left(x\right)\le 0,x\ge x_1.\end{array}$$

(7)

By integrating (7) and using (2), we see that $u^{\prime }\left(x\right)$ is eventually positive, for instance for $x\ge x_1$, increasing the size of $x_1$ if necessary, $u\left(x\right)$ is increasing for $x\ge x_1$. Then, we may write that

$$\begin{array}{c}\hfill y\left(x\right)=u\left(x\right)-b\left(x\right)y_{\tau }\left(x\right)\ge u\left(x\right)-b\left(x\right)u_{\tau }\left(x\right)\ge (1-b\left(x\right))u\left(x\right),\end{array}$$

and so (5) results in

$$\begin{array}{c}\hfill {\left(a\left(x\right)u^{\prime }\left(x\right)\right)}^{\prime }+\left(k\left(x\right)+(1-b_{\sigma }\left(x\right))l\left(x\right)\right)u_{\sigma }\left(x\right)\le 0,x\ge x_1.\end{array}$$

(8)

Furthermore, since $\sigma \left(x\right)\le x$ and $a\left(x\right)u^{\prime }\left(x\right)$ is nonincreasing, we have

$$\begin{array}{c}\hfill a_{\sigma }\left(x\right)u_{\sigma }^{\prime }\left(x\right)\ge a\left(x\right)u^{\prime }\left(x\right).\end{array}$$

(9)

Define

$$\begin{array}{c}\hfill w\left(x\right)=\eta \left(x\right){\displaystyle \frac{a\left(x\right)u^{\prime }\left(x\right)}{u_{\sigma }\left(x\right)}},x\ge x_1.\end{array}$$

(10)

It follows that $w\left(x\right)>0$ and

$$\begin{array}{cc}\hfill w^{\prime }=& {\eta }^{\prime }\left(x\right){\displaystyle \frac{a\left(x\right)u^{\prime }}{u_{\sigma }}}+\eta \left(x\right){\displaystyle \frac{{\left(a\left(x\right)u^{\prime }\right)}^{\prime }}{u_{\sigma }}}-\eta \left(x\right){\displaystyle \frac{{\sigma }^{\prime }\left(x\right)a\left(x\right)u^{\prime }{u}_{\sigma }^{\prime }}{u_{\sigma }^2}}.\hfill \end{array}$$

(11)

Thus, using (8)–(10) in (11) yields

$$\begin{array}{cc}\hfill w^{\prime }\le & -\left(k\left(x\right)+(1-b_{\sigma }\left(x\right))l\left(x\right)\right)\eta \left(x\right)+{\displaystyle \frac{{\eta }^{\prime }\left(x\right)}{\eta \left(x\right)}}w-{\displaystyle \frac{{\sigma }^{\prime }\left(x\right)}{a_{\sigma }\left(x\right)\eta \left(x\right)}}{w}^2\hfill \\ \hfill \le & -\left(k\left(x\right)+(1-b_{\sigma }\left(x\right))l\left(x\right)\right)\eta \left(x\right)+{\displaystyle \frac{a_{\sigma }\left(x\right){\left({\eta }^{\prime }\left(x\right)\right)}^2}{4\eta \left(x\right){\sigma }^{\prime }\left(x\right)}}.\hfill \end{array}$$

Integration of the latter inequality from $x_1$ to x gives

$$\begin{array}{c}\hfill {\int }_{x_1}^x\left(\left(k\left(s\right)+(1-b_{\sigma }\left(s\right))l\left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds\le w\left(x_1\right),\end{array}$$

which contradicts (6) as $x\to \infty$, thereby completing the proof. □

Example 1.

Consider

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{1}{x}}{u}^{\prime }\left(x\right)\right)}^{\prime }+{\displaystyle \frac{1}{x}}u\left(x\right)+{\displaystyle \frac{2}{3x^2}}y(x-3\pi /2)=0,x>2\pi ,\end{array}$$

(12)

where $u\left(x\right)=y\left(x\right)+1/3y(x-\pi ).$ It is clear that (2) is true. We will show that (6) is satisfied with $\eta =1$. Indeed,

$$\begin{array}{cc}\hfill & {\int }_{x_0}^{\infty }\left(k\left(s\right)+(1-b_{\sigma }\left(s\right))l\left(s\right)\right)\eta \left(s\right)ds={\int }_{2\pi }^{\infty }\left({\displaystyle \frac{1}{s}}+{\displaystyle \frac{4}{9s^2}}\right)ds=\infty .\hfill \end{array}$$

Thus, we can conclude by Theorem 2 that every nontrivial solution of (12) oscillates. Note that $y\left(x\right)=cosx$ is an oscillatory solution of (12). To the best of our knowledge, none of the available oscillation criteria are applicable for this example.

Next, we will introduce an oscillation-preserving transformation for the NDE (5). Let $v\left(x\right)$ be a positive differentiable function such that $a\left(x\right)v^{\prime }\left(x\right)$ is also differentiable. For a solution y and the corresponding u, we make the following transformations:

$$\tilde{y}={\displaystyle \frac{1}{v\left(x\right)}}y,\tilde{u}={\displaystyle \frac{1}{v\left(x\right)}}u.$$

(13)

Note that we may write

$$\tilde{u}=\tilde{y}+\tilde{b}\left(x\right){\tilde{y}}_{\tau },$$

where

$$\tilde{b}\left(x\right)=b\left(x\right){\displaystyle \frac{v_{\tau }\left(x\right)}{v\left(x\right)}}.$$

Suppressing x for clarity, it is not difficult to see that

$$a{v}^2{\left({\displaystyle \frac{u}{v}}\right)}^{\prime }=a(u^{\prime }v-u{v}^{\prime })$$

and so

$$\begin{array}{cc}\hfill {\left(a{v}^2{\tilde{u}}^{\prime }\right)}^{\prime }& ={\left(a{v}^2{\left({\displaystyle \frac{u}{v}}\right)}^{\prime }\right)}^{\prime }\hfill \\ & ={\left(a{u}^{\prime }\right)}^{\prime }v+a{u}^{\prime }{v}^{\prime }-u^{\prime }a{v}^{\prime }-u{\left(a{v}^{\prime }\right)}^{\prime }\hfill \\ \hfill & ={\left(a{u}^{\prime }\right)}^{\prime }v-u{\left(a{v}^{\prime }\right)}^{\prime }\hfill \\ \hfill & =-k\left(x\right)v^2\tilde{u}-l\left(x\right)v{v}_{\sigma }{\tilde{y}}_{\sigma }-v{\left(a{v}^{\prime }\right)}^{\prime }\tilde{u}.\hfill \end{array}$$

Thus, we obtain a transformed NDE:

$$\begin{array}{c}\hfill {\left(\tilde{a}\left(x\right){\tilde{u}}^{\prime }\right)}^{\prime }+\tilde{k}\left(x\right)\tilde{u}+\tilde{l}\left(x\right){\tilde{y}}_{\sigma }=0,x\ge x_0,\end{array}$$

(14)

where

$$\begin{array}{cc}\hfill \tilde{a}\left(x\right)& =a\left(x\right)v^2\left(x\right),\hfill \\ \hfill \tilde{k}\left(x\right)& ={\left(a\left(x\right)v^{\prime }\left(x\right)\right)}^{\prime }v\left(x\right)+k\left(x\right)v^2\left(x\right),\hfill \\ \hfill \tilde{l}\left(x\right)& =l\left(x\right)v\left(x\right)v_{\sigma }\left(x\right).\hfill \end{array}$$

Lemma 1.

The oscillation of nontrivial solutions of neutral delay Equations (5) and (14) is equivalent. Moreover, the zeros of $y\left(x\right)$ and $\tilde{y}\left(x\right)$ coincide.

In view of Lemma 1, applying Theorem 2 to the transformed neutral delay Equation (14), we can directly obtain the following theorem.

Theorem 3.

Suppose that $\tilde{k}\left(x\right)\ge 0$ and

$${\int }_{x_0}^{\infty }{\displaystyle \frac{ds}{\tilde{a}\left(s\right)}}=\infty .$$

(15)

If, for some positive differentiable function η,

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(\tilde{k}\left(s\right)+(1-{\tilde{b}}_{\sigma }\left(s\right))\tilde{l}\left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{{\tilde{a}}_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds=\infty ,\end{array}$$

(16)

then every nontrivial solution of (5) oscillates.

We now give an example where Theorem 2 may fail but Theorem 3 gives a conclusion.

Example 2.

Consider

$$\begin{array}{c}\hfill ({e^{-x}{u}^{\prime }\left(x\right))}^{\prime }+e^{-x}u\left(x\right)+2e^{-2x}y(x/2)=0,x>0,\end{array}$$

(17)

where $u\left(x\right)=y\left(x\right)+(1/2)y(x/3).$ Clearly, (2) holds. Since

$$\begin{array}{cc}\hfill & {\int }_{x_0}^{\infty }\left(\left(k\left(s\right)+(1-b_{\sigma }\left(s\right))l\left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds\hfill \\ \hfill & ={\int }_0^{\infty }\left(\left(e^{-s}+e^{-2s}\right)\eta \left(s\right)-{\displaystyle \frac{e^{-s/2}{\left({\eta }^{\prime }\left(s\right)\right)}^2}{2\eta \left(s\right)}}\right)ds,\hfill \end{array}$$

it seems quite challenging to find a positive function $\eta \left(x\right)$ satisfying the hypotheses of Theorem 2. However, using the transformation $v\left(x\right)=e^{x/2}$, we will apply Theorem 3. Note that

$$\begin{array}{cc}\hfill \tilde{b}\left(x\right)& =b\left(x\right){\displaystyle \frac{v_{\tau }\left(x\right)}{v\left(x\right)}}={\displaystyle \frac{e^{-x/3}}{2}},\hfill \\ \hfill \tilde{a}\left(x\right)& =a\left(x\right)v^2\left(x\right)=1,\hfill \\ \hfill \tilde{k}\left(x\right)& ={\left(a\left(x\right)v^{\prime }\left(x\right)\right)}^{\prime }v\left(x\right)+k\left(x\right)v^2\left(x\right)=3/4,\hfill \\ \hfill \tilde{l}\left(x\right)& =l\left(x\right)v\left(x\right)v_{\sigma }\left(x\right)=2e^{-5x/4}.\hfill \end{array}$$

We can now verify that Equation (16) holds by taking $\eta \left(x\right)=1$. In fact,

$$\begin{array}{cc}\hfill & {\int }_{x_0}^{\infty }\left(\left(\tilde{k}\left(s\right)+(1-{\tilde{b}}_{\sigma }\left(s\right))\tilde{l}\left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{{\tilde{a}}_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds\hfill \\ \hfill \\ \hfill & ={\int }_0^{\infty }\left({\displaystyle \frac{3}{4}}+2e^{-5s/4}-e^{-17s/12}\right)ds=\infty .\hfill \end{array}$$

Thus, we may deduce that every nontrivial solution of (17) oscillates. This example underscores the critical role of the equivalence transformation in our analysis.

In the next subsections, we specify certain v functions that are suitable for the canonical and noncanonical cases. In each case, a specific admissible function v is chosen to suit the particular case.

2.1. Refinements for the Canonical Case

We present an oscillation theorem analogous to Theorem 3 that addresses case (2). We choose $v={\varphi }^{n/2}\left(x\right)$, $n<1$, where

$$\begin{array}{c}\hfill \varphi \left(x\right)=1+{\int }_{x_0}^x{\displaystyle \frac{ds}{a\left(s\right)}}.\end{array}$$

Note that since

$$\begin{array}{c}\hfill {\int }_{x_0}^x{\displaystyle \frac{ds}{\tilde{a}\left(s\right)}}={\int }_{x_0}^x{\displaystyle \frac{{\varphi }^{\prime }\left(s\right)}{{\varphi }^n\left(s\right)}}ds,\end{array}$$

condition (15) is automatically satisfied if (2) holds; see [17]. Define

$$\begin{array}{cc}\hfill & P_j(x,z)=k\left(x\right)z^j\left(x\right)-{\displaystyle \frac{j(2-j)}{4a\left(x\right)}}{z}^{j-2}\left(x\right),\hfill \\ & Q_j(x,z)=l\left(x\right)z^{j/2}\left(x\right)z_{\sigma }^{j/2}\left(x\right).\hfill \end{array}$$

Applying Theorem 3 with the above special function v, we have the following result.

Theorem 4.

Assume that (2) and $P_n(x,\varphi \left(x\right))>0$ hold. If, for some positive differentiable function η, one has

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(P_n(s,\varphi \left(s\right))+(1-{\tilde{b}}_{\sigma }\left(s\right))Q_n(s,\varphi \left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\varphi }_{\sigma }^n\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds=\infty ,\end{array}$$

(18)

then every nontrivial solution of (5) oscillates.

Example 3.

Consider

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{e^{-2x}}{2}}{u}^{\prime }\left(x\right)\right)}^{\prime }+{\displaystyle \frac{3e^{-x}}{4}}u\left(x\right)+e^{-x}y(x/4)=0,x>0,\end{array}$$

(19)

where $u\left(x\right)=y\left(x\right)+(3/4)y(x/2).$ Clearly, (2) holds. Since

$$\begin{array}{cc}\hfill & {\int }_{x_0}^{\infty }\left(\left(k\left(s\right)+(1-b_{\sigma }\left(s\right))l\left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds\hfill \\ \hfill & ={\int }_0^{\infty }\left(e^{-s}\eta \left(s\right)-{\displaystyle \frac{e^{-s/2}{\left({\eta }^{\prime }\left(s\right)\right)}^2}{2\eta \left(s\right)}}\right)ds,\hfill \end{array}$$

it seems quite challenging to find a positive function $\eta \left(x\right)$ satisfying the hypotheses of Theorem 2. However, using the transformation $\varphi \left(x\right)=1+{\int }_{x_0}^x{\displaystyle \frac{ds}{a\left(s\right)}}=e^{2x},$ we will apply Theorem 4. Note that

$$\begin{array}{cc}\hfill & \tilde{b}\left(x\right)={\displaystyle \frac{b\left(x\right){\varphi }_{\tau }\left(x\right)}{\varphi \left(x\right)}}={\displaystyle \frac{3}{4}}{e}^{-x},\hfill \\ \hfill & P_{1/2}(x,\varphi \left(x\right))=k\left(x\right){\varphi }^{1/2}\left(x\right)-{\displaystyle \frac{1/2(2-1/2)}{4a\left(x\right)}}{\varphi }^{1/2-2}\left(x\right)={\displaystyle \frac{3}{4}}-{\displaystyle \frac{3}{8}}{e}^{-x},\hfill \\ & {\tilde{Q}}_{1/2}(x,\varphi \left(x\right))=l\left(x\right){\varphi }^{1/4}\left(x\right){\varphi }_{\sigma }^{1/4}\left(x\right)=e^{-3x/8}.\hfill \end{array}$$

We can now verify that Equation (18) holds by taking $\eta \left(x\right)=1$. In fact,

$$\begin{array}{cc}\hfill & {\int }_{x_0}^{\infty }\left(\left(P_{1/2}(s,\varphi \left(s\right))+(1-{\tilde{b}}_{\sigma }\left(s\right))Q_{1/2}(s,\varphi \left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\varphi }_{\sigma }^{1/2}\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds\hfill \\ \hfill & ={\int }_0^{\infty }\left({\displaystyle \frac{3}{4}}+e^{-3s/8}-{\displaystyle \frac{3e^{-s}}{8}}-{\displaystyle \frac{3e^{-5s/8}}{4}}\right)ds=\infty .\hfill \end{array}$$

We conclude that the conditions of Theorem 4 are satisfied, where we put $\eta \left(x\right)=1$. Thus, every nontrivial solution of (19) oscillates.

2.2. Refinements for the Noncanonical Case

In this case, we choose $v={\psi }^{m/2}\left(x\right)$, $m>1$, where

$$\begin{array}{c}\hfill \psi \left(x\right)={\int }_x^{\infty }{\displaystyle \frac{ds}{a\left(s\right)}}.\end{array}$$

(20)

Again, condition (15) is automatically satisfied.

We will distinguish three possibilities:

$$1<m<2,m=2,m>2.$$

Theorem 5.

Suppose condition (3) holds and $P_m(x,\psi \left(x\right))>0$ is satisfied for some $1<m<2$. If, for some positive differentiable function η, one has

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(P_m(s,\psi \left(s\right))+(1-{\tilde{b}}_{\sigma }\left(s\right))Q_m(s,\psi \left(s\right))\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\psi }^m\left(\sigma \left(s\right)\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds=\infty ,\end{array}$$

then every nontrivial solution of (5) oscillates.

Theorem 6.

Suppose condition (3) holds. If, for some positive differentiable function η, one has

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(P_2(s,\psi \left(s\right))+(1-{\tilde{b}}_{\sigma }\left(s\right))Q_2(s,\psi \left(s\right))\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\psi }^2\left(\sigma \left(s\right)\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds=\infty ,\end{array}$$

(21)

then every nontrivial solution of (5) oscillates.

Theorem 7.

Suppose condition (3) holds and $m>2$. If, for some positive differentiable function η, one has

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(P_m(s,\psi \left(s\right))+(1-{\tilde{b}}_{\sigma }\left(s\right))Q_m(s,\psi \left(s\right))\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\psi }^m\left(\sigma \left(s\right)\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds=\infty ,\end{array}$$

then every nontrivial solution of (5) oscillates.

Example 4.

Consider

$$\begin{array}{c}\hfill {\left(x^2{u}^{\prime }\left(x\right)\right)}^{\prime }+\left(x^2-{\displaystyle \frac{x}{2}}\right)u\left(x\right)+{\displaystyle \frac{17x}{8}}y(x-5\pi /2)=0,x>4\pi ,\end{array}$$

(22)

where $u\left(x\right)=y\left(x\right)+(1/4)y(x-\pi /2).$ It is clear that (3) is valid and $\psi \left(x\right)=1/x$.

We will apply Theorem 6. Note that

$$\begin{array}{cc}\hfill & \tilde{b}\left(x\right)={\displaystyle \frac{b\left(x\right){\psi }_{\tau }\left(x\right)}{\psi \left(x\right)}}={\displaystyle \frac{x}{2(2x-\pi )}},\hfill \\ \hfill & P_2(x,\psi \left(x\right))=k\left(x\right){\psi }^2\left(x\right)=1-{\displaystyle \frac{1}{2x}},\hfill \\ & Q_2(x,\psi \left(x\right))=l\left(x\right)\psi \left(x\right){\psi }_{\sigma }\left(x\right)={\displaystyle \frac{17}{4(2x-5\pi )}}.\hfill \end{array}$$

Taking $\eta \left(x\right)=1$, condition (21) is also satisfied:

$$\begin{array}{cc}\hfill & {\int }_{x_0}^{\infty }\left(P_2(s,\psi \left(s\right))+(1-{\tilde{b}}_{\sigma }\left(s\right))Q_2(s,\psi \left(s\right))\right)\eta \left(s\right)ds\hfill \\ \hfill & ={\int }_{4\pi }^{\infty }\left(\left(1-{\displaystyle \frac{1}{2s}}\right)+\left(1-{\displaystyle \frac{2s-5\pi }{8(s-3\pi }}\right){\displaystyle \frac{17}{4(2s-5\pi )}}\right)ds\hfill \\ \hfill & \ge {\int }_{4\pi }^{\infty }\left(1-{\displaystyle \frac{1}{2s}}\right)ds=\infty ,\hfill \end{array}$$

and by Theorem 6, every nontrivial solution of (22) oscillates. Indeed, $y\left(x\right)=sinx$ is an oscillatory solution of (22).

3. Nonlinear Extensions

In this last section, we will demonstrate that the new method can be used to encompass nonlinear NDEs. We will consider the following nonlinear version of (5):

$$\begin{array}{c}\hfill {\left(a\left(x\right)u^{\prime }\right)}^{\prime }+k\left(x\right)f\left(u\right)+l\left(x\right)g\left(y_{\sigma }\right)=0,x\ge x_0,\end{array}$$

(23)

where $u\left(x\right)=y\left(x\right)+b\left(x\right)y_{\tau }\left(x\right)$, and f and g are continuous functions subject to the following constraints:

• $yf\left(y\right)>0$ and $yg\left(y\right)>0$ for $y\ne 0$.
• f is differentiable and $f^{\prime }\left(y\right)\ge 1$ for $y\ne 0$; g is differentiable and non decreasing for $y\ne 0$.
• g satisfies $\left|g\left(rt\right)\right|\ge g_1\left(r\right)\left|g_2\left(t\right)\right|$ for all $r>0$ and $t\ne 0$ with $|g_2\left(t\right)|\ge L|f\left(t\right)|$ for some $L>0$.

We first state and prove a theorem analogous to Theorem 2.

Theorem 8.

Assume that (2) holds. If, for some positive differentiable function η, one has

$$\begin{array}{c}\hfill {\int }_{x_0}^{\infty }\left(\left(k\left(s\right)+L{g}_1\left(1-b_{\sigma }\left(s\right)\right)l\left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds=\infty ,\end{array}$$

(24)

then every nontrivial solution of (23) oscillates.

Proof. 

Suppose that there is a solution $y\left(x\right)>0$, $y_{\tau }\left(x\right)>0$ and $y_{\sigma }\left(x\right)>0$ for all $x\ge x_1$ for some $x_1\ge x_0$. From (23), we see, as in the linear case, that for $x\ge x_1$,

$$\begin{array}{c}\hfill y\left(x\right)\ge (1-b(x\left)\right)u\left(x\right),\end{array}$$

and

$$\begin{array}{c}\hfill a_{\sigma }\left(x\right)u^{\prime }\left(\sigma \left(x\right)\right)\ge a\left(x\right)u^{\prime }\left(x\right).\end{array}$$

Using the assumptions on the nonlinear functions, we also have

$$\begin{array}{c}\hfill {\left(a\left(x\right)u^{\prime }\right)}^{\prime }+\left(k\left(x\right)+L{g}_1\left(1-b_{\sigma }\left(x\right)\right)l\left(x\right)\right)f\left(u_{\sigma }\left(x\right)\right)\le 0.\end{array}$$

We define

$$\begin{array}{c}\hfill w\left(x\right)=\eta \left(x\right){\displaystyle \frac{a\left(x\right)u^{\prime }\left(x\right)}{f\left(u_{\sigma }\left(x\right)\right)}},x\ge x_1.\end{array}$$

Clearly, the function $w\left(x\right)$ is positive and satisfies

$$\begin{array}{cc}\hfill w^{\prime }\le & -\left(k\left(x\right)+L{g}_1\left(1-b_{\sigma }\left(x\right)\right)l\left(x\right)\right)\eta \left(x\right)+{\displaystyle \frac{{\eta }^{\prime }\left(x\right)}{\eta \left(x\right)}}w-{\displaystyle \frac{{\sigma }^{\prime }\left(x\right)}{a_{\sigma }\left(x\right)\eta \left(x\right)}}{w}^2\hfill \\ \hfill \le & -\left(k\left(x\right)+L{g}_1\left(1-b_{\sigma }\left(x\right)\right)l\left(x\right)\right)\eta \left(x\right)+{\displaystyle \frac{a_{\sigma }\left(x\right){\left({\eta }^{\prime }\left(x\right)\right)}^2}{4\eta \left(x\right){\sigma }^{\prime }\left(x\right)}}.\hfill \end{array}$$

Integration to the latter inequality from $x_1$ to x leads to

$$\begin{array}{c}\hfill {\int }_{x_1}^x\left(\left(k\left(s\right)+L{g}_1\left(1-b_{\sigma }\left(s\right)\right)l\left(s\right)\right)\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}}\right)ds\le w\left(x_1\right),\end{array}$$

which contradicts (24) as $x\to \infty$, thereby completing the proof. □

Example 5.

Consider

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{1}{\sqrt{x}}}{u}^{\prime }\left(x\right)\right)}^{\prime }+{\displaystyle \frac{1}{x}}u\left(x\right)+{\displaystyle \frac{1}{x^2}}y\left(\sqrt{x}\right)(2+y^2\left(\sqrt{x}\right))=0,x>1,\end{array}$$

(25)

where $u\left(x\right)=y\left(x\right)+(1/x^2)y(x/2).$

We see that $f\left(y\right)=y$, $g_1\left(r\right)=r$ and $g_2\left(t\right)=2t.$ It is clear that (2) holds. We will show that (24) is satisfied. We take $\eta \left(x\right)=1$, and observe that

$$\begin{array}{cc}\hfill & {\int }_{x_0}^{\infty }\left(k\left(s\right)+L{g}_1\left(1-b\left(\sigma \right(s)\right)l\left(s\right)\right)\eta \left(s\right)ds={\int }_1^{\infty }\left({\displaystyle \frac{1}{s}}+2\left({\displaystyle \frac{s-1}{s}}\right){\displaystyle \frac{1}{s^2}}\right)ds=\infty ,\hfill \end{array}$$

Thus, by Theorem 8, we conclude that every nontrivial solution of (25) oscillates.

Next, we consider the noncanonical case (3). Note that using (13) and (23), we may write

$$\begin{array}{cc}\hfill {\left(a{v}^2{\tilde{u}}^{\prime }\right)}^{\prime }& =-kvf\left(v\tilde{u}\right)-lvg\left(v_{\sigma }{\tilde{y}}_{\sigma }\right)-v{\left(a{v}^{\prime }\right)}^{\prime }\tilde{u}.\hfill \end{array}$$

(26)

Setting $v=\psi \left(x\right)$, where $\psi$ is given by (20), the last term in (26) becomes zero. Then, employing the transformation (13), (23) results in the following nonlinear NDE:

$$\begin{array}{c}\hfill {\left(a\left(x\right){\psi }^2\left(x\right){\tilde{u}}^{\prime }\right)}^{\prime }+k\left(x\right)\psi \left(x\right)f\left(\psi \left(x\right)\tilde{u}\right)+l\left(x\right)\psi \left(x\right)g\left(\psi \left(\sigma \left(x\right)\right){\tilde{y}}_{\sigma }\right)=0,x\ge x_0.\end{array}$$

(27)

We assume that there exist functions $f_1,f_2,g_1$, and $g_2$ such that

$$\left|f\left(rt\right)\right|\ge f_1\left(r\right)|f_2\left(t\right)|,|g\left(rt\right)|\ge g_1\left(r\right)\left|g_2\left(t\right)\right|,r>0,t\ne 0.$$

(28)

In addition, we assume $t{f}_2\left(t\right)>0$ and $t{g}_2\left(t\right)>0$ for $t\ne 0$; $f_2$ is differentiable and $f_2^{\prime }\left(t\right)\ge 1$ for all $t\ne 0$; $|g_2\left(t\right)|\ge L|f_2\left(t\right)|$ for some $L>0$ and for all $t\ne 0$; and $f_1\left(r\right)>0$ and $g_1\left(r\right)>0$ for all $r>0$.

We now apply Theorem 8 to obtain an oscillation theorem for the noncanonical case.

Theorem 9.

Assume that (3) holds. If, for some positive differentiable function η, one has

$$\begin{array}{cc}\hfill {\int }_{x_0}^{\infty }\left(\right(\tilde{k}\left(s\right)& +\tilde{l}\left(s\right))\eta \left(s\right)-{\displaystyle \frac{a_{\sigma }\left(s\right){\psi }^2\left(\sigma \left(s\right)\right){\left({\eta }^{\prime }\left(s\right)\right)}^2}{4\eta \left(s\right){\sigma }^{\prime }\left(s\right)}})ds=\infty ,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \tilde{k}\left(x\right)& =k\left(x\right)\psi \left(x\right)f_1\left(\psi \left(x\right)\right),\tilde{l}\left(x\right)=l\left(x\right)\psi \left(x\right)L{g}_1\left((1-{\tilde{b}}_{\sigma }\left(x\right))\psi \left(\sigma \left(x\right)\right)\right),\hfill \end{array}$$

then every nontrivial solution of (23) oscillates.

Proof. 

Let $y\left(x\right)$ be an eventually positive solution of (23). Following the steps in the proof of Theorem 8, and using (28) in (27), we find that $\tilde{y}$ satisfies the neutral delay differential inequality

$$\begin{array}{c}\hfill {\left(\tilde{a}\left(x\right){\tilde{u}}^{\prime }\left(x\right)\right)}^{\prime }+\tilde{k}\left(x\right)f_2\left(\tilde{u}\left(x\right)\right)+\tilde{l}\left(x\right)g_2\left({\tilde{u}}_{\sigma }\left(x\right)\right)\le 0,\end{array}$$

(29)

where

$$\begin{array}{c}\hfill \tilde{a}\left(x\right)=a\left(x\right){\psi }^2\left(x\right).\end{array}$$

Notice that, because of (3), we have

$${\int }_{x_0}^{\infty }{\displaystyle \frac{ds}{\tilde{a}\left(s\right)}}={\int }_{x_0}^{\infty }{\displaystyle \frac{ds}{a\left(s\right){\psi }^2\left(s\right)}}=-{\int }_{x_0}^{\infty }{\displaystyle \frac{{\psi }^{\prime }\left(s\right)}{{\psi }^2\left(s\right)}}ds=\infty .$$

This establishes the fulfillment of condition (2) in Theorem 8. The remainder of the proof follows arguments analogous to Theorem 8, employing inequality (29). □

4. Conclusions

In this work, we investigated a class of second-order neutral differential equations (NDEs) and established novel sufficient conditions under which all solutions exhibit oscillatory behavior. Our analysis was based on a carefully constructed transformation technique that preserves the inherent oscillatory nature of the original equations. This methodological approach led to the derivation of new oscillation criteria that extend and refine existing results in the literature. The primary advantage of our method lies in its generality and simplicity. Unlike many classical approaches, which often rely on restrictive assumptions or specific forms of the neutral term, our technique remains effective even in more complex and generalized settings. This significantly broadens the range of applicability of the oscillation criteria, making them relevant for a wider class of NDEs that arise in various scientific and engineering contexts. The theoretical results were also supported by illustrative examples that demonstrated the practical utility of the proposed conditions. These examples showed that the new criteria are capable of identifying oscillatory behavior in cases where previously known results fail to do so.

Future research may include extension to higher-order NDEs as well as to other types of functional differential equations, and weakening the restriction imposed on the coefficient functions and the nonlinearities. Solving such equations numerically is also an interesting area of research. Indeed, there are many recent studies on the numerical treatment of such equations; see [19]. In addition, it is an open problem whether or not one can expand the space of v functions in the nonlinear case.