Nonlinear differential equations with neutral term: Asymptotic behavior of solutions

1.   Introduction

Differential equations (DEs) are mathematical models used to study phenomena that occur in nature, where each dependent variable represents a quantity in the modeled phenomenon. Differential equations made it possible to understand many complex phenomena in our daily lives and play a pivotal role in many applications in engineering ^[1,2,3]. They have become important tools in applied sciences and technology, used for studying telephone signals, media, conversations, and the statistics of online purchasing. More traditionally, they were used in astronomy to describe the orbits of planets and the motion of stars ^[4,5,6]. They also have many applications in biology and the medical sciences. By describing those phenomena with variables that symbolize time and place, differential equations can provide insights about the phenomena on future.

Due to the huge advantage of neutral differential equations in describing several neutral phenomena, there is great scientific and academic values theoretically and practically for studying neutral differential equations ^[7,8,9]. Hence, a large amount of research attention has been focused on the oscillation problem of third-order linear and nonlinear neutral differential equations in recent years; see, for example ^[10,11,12].

Recent years have seen a surge in research on the oscillation and non-oscillation of solutions to third/fourth-order differential equations ^[13,14,15]. For further exploration, readers can refer to the references provided ^[16,17,18].

The authors in ^[19,20,21] discussed several oscillatory properties of higher-order equations in canonical form, and used different methods to find those properties, such as Riccati transformations. Moreover, they applied the comparison method to inequalities of different orders that are oscillatory ^[22,23].

The purpose of this work is to investigate the oscillatory and asymptotic behavior of the third-order neutral delay differential equations

where $w(t) = x(t)+b(t)x(g(t))$. We also assume that the following conditions are satisfied:

$(H_{1}) \; \varsigma _{i}, g\in C^{\prime }\left(\left[ t _{0}, \infty \right), \mathbb{R}\right), \varsigma _{i}(t) < t, g(t) < t, g^{\prime }(t)\geq 0$ and $\lim_{t \rightarrow \infty }g(t) = \lim_{t \rightarrow \infty }\varsigma _{i}(t) = \infty, i = 1, 2, .., j$;

$(H_{2}) \; b, a_{i}\in C\left(\left[ t _{0}, \infty \right), \mathbb{R}^{+}\right), 0\leq b(t)\leq b_{0} < \infty$ and $a_{i}$ does not vanish identically;

$(H_{3}) \; r_{1}, r_{2}\in C\left(\left[ t _{0}, \infty \right), (0, \infty)\right)$ satisfy

that is, (E) is in noncanonical form;

$(H_{4}) \; g$ and $\varsigma _{i}$ commute.

By a solution of (E), we mean a function $x\in C\left(\left[ T_{x}, \infty \right), \mathbb{R}\right)$ with $T_{x} > t _{0},$ which has the property $L_{i} \; w\in C^{1}\left(\left[ T_{x}, \infty \right), \mathbb{R} \right), i = 0, 1, 2$, and satisfies $(E)on\left[ T_{x}, \infty \right).$ We only consider those solutions of (E) which exist on some half-line $[T_{x}, \infty)$ and satisfy the condition $\sup \{|x(t)|:T\leq t < \infty \} > 0$ for any $T\geq T_{x}$. We assume that (E) possesses such a solution. A solution of (E) is said to be $oscillatory$ if it is neither eventually negative nor eventually positive, and it is called $nonoscillatory$ otherwise. The equation itself is referred to as oscillatory if all of the solutions are oscillatory.

The main motivation for studying this paper is to contribute to the development of the oscillation theory for third-order equations by finding sufficient conditions that guarantee that the solutions of this type of equations are oscillatory.

Chatzarakis et al. ^[24] established new oscillation criteria for the differential equation

in the canonical form. Recently, techniques have been developed to study the oscillatory behavior of solutions to third-order equations.

Candan ^[25] established some sufficient conditions for oscillation of the following class of third-order neutral differential equations

under conditions

Li et al. ^[26] also studied the cases

also under conditions

So during these years, it was found that sufficient criteria were found to ensure that the solutions of (E) were oscillatory. The first of these findings for (E) was reported in ^[3], in canonical type under the conditions $0\leqslant b(t)\leqslant b_{0} < \infty$ and $\varsigma _{i}og = go\varsigma _{i}.$ Very recently in ^[4,5], the authors provided enough parameters for (E) to oscillate.in the noncanonical or semi-canonical case with an unbounded neutral coefficient, that is, $b(t)\geq b_{0} > 1$ since in this case one can easily find the relation between $x(t)$ and $w(t).$ This is generally essential to obtain oscillation criteria for neutral-type differential equations.

Our literature review indicates a scarcity of research on the oscillatory behavior of solutions to Eq (E) when it takes the semi-canonical form. This paper tackles Eq (E) in its less-studied semi-canonical form. We begin by transforming it into the more common canonical form. This transformation allows us to then establish new criteria for determining when solutions to Eq (E) oscillate.

In the sequel, we use the following notations for a compact presentation of our results:

We remark that in the study of the asymptotic behaviour of the positive solutions of (E), there are four cases:

2.   Main results

In view of $(H_{3}),$ one can use the following notations:

for all $t > u\geq t _{1}\geq t _{0}$.

From the form of (E), it is enough to consider positive solutions for nonoscillatory solutions of (E). The following is a standard one and can be found in ^[1].

Hence, if we want to derive oscillation conditions for (E), we have to eliminate the above mentioned four cases. However, if we transform (E) into semi-canonical type, then the number of cases is reduced to three without making any additional assumptions. Thus, this greatly streamlines the analysis of (E) oscillation.

Theorem 1. The noncanonical operator ${L_{3}}w$ has the semi-canonical representation

Proof. Direct calculation shows that

that is,

Further note that

and

Hence $L_{3}w$ transformed into semi-canonical form. This ends the proof.

Now it follows from Theorem 2.2 that (E) can be written in the equivalent semi-canonical form

By letting $\gamma (t) = \frac{w(t)}{\varpi _{1}(t)},$ the following result is at once.

Theorem 2. Noncanonical equation (E) has a solution $x(t)$ if and only if the semi-canonical Eq $(E_{s})$

has the solution $x(t)$.

Corollary 1. The function $x$ is identified as the ultimate positive solution to $(E)$ if and only if the semi-canonical Eq $(E_s)$ has the same solution $x$.

Now set

Corollary 2.4 clearly simplifies the investigation of (E) since for $(E_{s})$ we deal with only three cases of positive solutions; see, for example [3, Theorem 2.2], namely

eventually.

Lemma 1. Let $x$ be an eventually positive solution of (E) then the corresponding function satisfies the inequality

for all $t \geq t _{1}\geq t _{0}.$

Proof. The function $\theta$ is identified as the ultimate positive solution to $(E)$ Let $\theta$ be an eventually positive solution of $(E)$. Then we have that $\theta(\zeta) > 0, \theta(\tau(\zeta)) > 0$ and $\theta(\delta(\zeta)) > 0$ for all $t \geq \zeta_1$. From Corollary 2.4, the function $\alpha(\zeta)$ is a positive solution of $(E_s)$ for all $\zeta \geq \zeta_1.$ Now, from $(E_s), \; (H_1)$ and $(H_4),$ we see that

Combining $(E_s)$ along with the last inequality, we obtain

Using $(H_2)$ in the definition of $\phi(\zeta),$ we obtain

In view of the latter, inequality (2.3) becomes

or

which proves(2.2). □

Before we state and prove our main results, let us define

for all $t \geq t _{0}.$

Theorem 3. Given that $\gamma$ constitutes the final positive solution of $(E_{s}).$ If

then class $\mathrm{O}_{2}$ is empty.

Proof. Assume to the contrary that class $\mathrm{O}_2$ is not empty. Then there exists a $\zeta_1 \geq \zeta_0$ such that $\alpha(\zeta) > 0$, $\alpha(\tau(\zeta)) > 0$, $\alpha(\delta(\zeta)) > 0$ for all $\zeta \geq \zeta_1$, such that the function $\alpha(\zeta)$ in class $\mathrm{O}_2$ for all $\zeta \geq \zeta_1.$ Since $\beta_1(\zeta) \alpha^{\prime}(\zeta) > 0$ is increasing, we have

Dividing this inequality by $\beta_1(\zeta)$, then integrating the resulting inequality, we obtain

Integrating $(E_s)$ from $\zeta_2$ to $\zeta$ and using (2.6) in the resulting inequality, we obtain

which tends $\zeta_0 \rightarrow \infty$ as $\zeta \rightarrow \infty.$ This contradiction ends the proof. □

Lemma 2. Let $\gamma$ be an eventually positive increasing solution of $(E_{s}).$ If

then $\gamma$ satisfies the class $\mathrm{O}_{3}$ for $t \geq t _{1}$ for some $t _{1}\geq t _{0}$ and further

Proof. Since $\alpha$ is a positive increasing solution, so class $\mathrm{O}_1$ is empty, and hence, by Theorem 2.6, $\alpha \in \mathrm{O}_2 \cup \mathrm{O}_3$ for $\zeta \geq \zeta_1$, where $\zeta_1 \geq \zeta_0$ is such that $\alpha(\delta(\zeta)) > 0$ and $\alpha(\tau(\zeta)) > 0$ for $\zeta\geq \zeta_1.$ In view of $(H_3),$ we see that (2.7) implies (2.5), and hence $\alpha$ satisfies class $\mathrm{O}_3$ for $\zeta \geq \zeta_1.$ Since $D_1 \alpha$ is positive and decreasing, we see that

This ends the proof. □

Theorem 4. Let $\gamma$ be an eventually positive solution of $(E_{s}).$ If

then the classes $\mathrm{O}_{2}$ and $\mathrm{O}_{3}$ are empty.

Proof. Assume that (2.9) holds but $\alpha$ belongs to classes $\mathrm{O}_2$ and $\mathrm{O}_3.$ Pick $\zeta_1 \geq \zeta_o$ such that $\alpha(\tau(\zeta)) > 0$ and $\alpha(\delta(\zeta)) > 0$ for $\zeta \geq \zeta_1.$ Clearly, it is necessary for the validity of (2.9) that (2.7) holds. Hence, by Theorem 2.6 and Lemma 2.7, one can see that $\alpha$ satisfies class $\mathrm{O}_3$. Proceeding as in the proof of Lemma 2.7, we see that (2.8) holds, and so we obtain

for $\zeta \geq \zeta_2$ for some $\zeta_2 \geq \zeta_1.$ From the latter inequality and Eq $(E_s),$ we observe that

Integrating from $\zeta_2$ to $\zeta$, we obtain

Since $D_2 \alpha(\zeta)$ is decreasing and $\tau(\zeta) < \zeta$, we have $D_2 \alpha(\zeta) \leq D_2 \alpha(\tau (\zeta)$), and using this in $(2.10),$ we obtain

Let $\omega(\zeta) = \beta_1(\zeta) \alpha^{\prime}(\zeta) > 0$ is a positive solution of the first-order delay differential inequality

However, by [13, Theorem 2.11], the inequality $(2.12)$ does not have a positive solution. This contradicts our initial assumption, and the proof is complete. □

Theorem 5. Assume that (2.5) holds. If

then the classes $\mathrm{O}_2$ and $\mathrm{O}_3$ are empty.

Proof. Assume to the contrary that $\alpha$ satisfies class $\mathrm{O}_2$ or $\mathrm{O}_3$ for $\zeta \geq \zeta_1.$ First note that $\lim _{\zeta \rightarrow \infty} B(\zeta) = 0$ holds, which together with (2.13) implies (2.5). So by Lemma 2.7, we conclude that $\alpha$ satisfies $\mathrm{O}_3$ and the asymptotic property (2.8) for all $\zeta \geq \zeta_1 \geq \zeta_0.$ Proceeding as in the proof of Theorem 2.8, we arrive at (2.11). Now from the monotonicity of $D_2 \alpha(\zeta),$ we obtain

and using this in (2.11), we obtain

where we have used $\beta_1\left(\delta(\zeta) \alpha^{\prime}(\delta(\zeta)) \geq \beta_1(\zeta) \alpha^{\prime}(\zeta) \right..$ From (2.14) we obtain

But the last inequality contradicts (2.13), and the proof is complete. □

Theorem 6. Let $\gamma$ be an eventually positive solution of $(E_{s}).$ If $\varsigma _{i}(t) < g(g(t))$ and

then the class $\mathrm{O}_{1}$ is empty.

Proof. Assume the contrary that (2.15) holds, but $\alpha$ belongs to class $\mathrm{O}_1$. Choose $\zeta_1 \geq \zeta_0$ such that $\delta(\zeta) \geq \zeta_1$ for $\zeta \geq \zeta_1.$ From the monotonicity of $D_2 \alpha(\zeta)$ that for $v \geq u$

Dividing by $\beta_1(u)$ and then integrating from $u$ to $v \geq u$ in $u$ for the resulting inequality, we find

Integrating (2.2) from $\tau(\zeta)$ to $\zeta$ and using $(2.16)$ with $u = \delta(s)$ and $v = \delta(\zeta),$ we obtain

From $\delta(\zeta) < \tau(\tau(\zeta))$ and $\tau(\tau(\zeta)) < \tau(\zeta),$ we find

and using these in (2.17), we obtain

which contradicts (2.15) and the proof is complete. □

Theorem 7. Given that $\gamma$ constitutes the final positive solution of $(E_{s}).$ If the function $\sigma (t)\in C\left(\left[ t _{0}, \infty \right), (0, \infty)\right)$ satisfying $\varsigma _{i}(t) < \sigma (t) < g(t)$ such that

then the class $\mathrm{O}_1$ is empty.

Proof. Let (2.18) holds, but $\alpha$ belongs to class $\mathrm{O}_1$. Proceeding as in the prof of Theorem 2.10 we arrive at (2.10). Setting $u = \delta(\zeta)$ and $v = \xi(\zeta), \zeta \geq x \geq \zeta_1$, in (2.10), we obtain

On the other hand, using (2.19) in (2.4) yields

Now, let

Using the fact that $\tau(\zeta) < \zeta$ and $D_2 \alpha(\zeta)$ is nonincreasing, we have

or equivalently

From (2.21) and (2.20), we see that $w(\zeta)$ is a positive solution of the first-order delay differential inequality

If we apply [13, Theorem 2.11], we obtain that $w(t)$ is not a positive solution to (2.22), and thus the proof is complete. □

The primary outcome of the study is as follows: oscillation condition for (E).

Theorem 8. Assume that $(H_1)$–$(H_4)$ hold. If (2.9) (or (2.13)) and (2.15) (or (2.18)) satisfied, then Eq (E) is oscillatory.

Proof. Let $\theta$ be a nonoscillatory solution of (E), and without loss of generality, assume that there exists a $\zeta_1 \geq \zeta_0$ such that $\theta(\zeta) > 0, \theta(\tau(\zeta)) > 0$ and $\theta(\delta(\zeta)) > 0$ for all $\zeta \geq \zeta_1.$ Then, by Corollary 2.4, the function $\theta(\zeta)$ is also a positive solution of $(E_s)$ as well as the related function $\alpha(\zeta)$, which satisfies one of the three classes $\mathrm{O}_1$ or $\mathrm{O}_2$ or $\mathrm{O}_3$ for $\zeta \geq \zeta_1.$

In view of Theorem 2.8 (or Theorem 2.9), the classes $\mathrm{O}_2$ and $\mathrm{O}_3$ are empty. On the other hand from Theorem 2.10 (or Theorem 2.11), the class $\mathrm{O}_1$ is empty. This contradiction implies that the Eq (E) is oscillatory. This concludes the proof. □

We provide an example at the end of this section to highlight the significance of our primary findings.

Example 1. Examine the third-order Euler type neutral differential equation

where $a_{0} > 0, b_{0} > 0, \beta _{1}\in (0, 1)$ and $\beta _{2}\in (0, 1).$ A simple calculation shows that

We apply this data to obtain the transformed equation in semi-canonical form

so, we find

The condition (2.9) becomes.

that is, condition (2.9) satisfied if

Choose $\beta _{3}$ such that $\beta _{2} < \beta _{3} < \beta _{1}$ then the condition (2.18) becomes

that is, condition (2.18) is satisfied if

Therefore Eq (2.23) is oscillatory if

and

In particular if we assume $\beta _{1} = 1/2\quad \beta _{2} = 1/4\quad \beta _{3} = 1/3\quad b_{0} = 1/2$ then we get $a_{0} > 29.033674.$ So in this case, the Eq (2.23) is oscillatory if $a_{0} > 29.033674$.

Take note that none of the outcomes listed in ^[2,3,4] can yield this conclusion since $b_{0} < 1$ and the equation is noncanonical.

3.   Conclusions

The aim of this paper is to investigate the oscillatory characteristics inherent in third-order differential equations featuring a noncanonical term. This investigation is conducted through the application of integral averaging and comparison techniques, ultimately leading to the derivation of oscillation criteria. The study culminates in the establishment of a central theorem pertaining to the oscillation behavior of equations. Additionally, three examples of the effectiveness of these criteria were discussed. In future work, we will study fractional order delay differential equations in their non-canonical form to find oscillatory properties that will contribute to enriching oscillation theory.

Acknowledgments

This research received no external funding.

Competing interests

There are no competing interests.