Nonlinear third-order differential equations with distributed delay: Some new oscillatory solutions

Saeed Althubiti

doi:10.1016/j.jksus.2023.102730

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Original article

08 2023

:35;

102730

doi:

10.1016/j.jksus.2023.102730

Nonlinear third-order differential equations with distributed delay: Some new oscillatory solutions

Saeed Althubiti

Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 2022-8-20, Accepted: 2023-5-25,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

We consider a certain class of third order nonlinear delay differential equations in this work. The results that we obtained are an improvement and extension of some results mentioned in previous literature, as the criteria we obtained are less restrictive compared to the previous results reported in literature. An example is provided to illustrate new results.

Keywords

Third-order nonlinear differential equations

Delay

Oscillation

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1 Introduction

In this paper, we consider the third-order neutral nonlinear differential equation of the form

(1)

{(ι (ξ) {(ψ^{''} (ξ))}^{α})}^{'} + \int_{a}^{b} ϑ (ξ, ω) χ^{α} (τ (ξ, ω)) d ω = 0, for ξ ⩾ ξ_{0},

where

α

is a ratio of positive odd integers and

ψ (ξ) = χ (ξ) + p_{0} χ (ξ - δ_{0}) .

Throughout this work, we will assume the following:

(I₁) $δ_{0}, p_{0}$ are constants such that $p_{0}, δ_{0} ⩾ 0$ ;
(I₂) $ι \in C^{1} ([ξ_{0}, \infty), (0, \infty)), ϑ, τ \in C ([ξ_{0}, \infty) \times [a, b], ℝ), τ (ξ, s) < ξ, \lim_{ξ \to \infty} τ (ξ, ω) = \infty, ϑ (ξ, ω)$ don’t not vanish identically, and $\int_{ξ_{0}}^{\infty} ι^{- 1 / α} (ξ) d ξ = \infty .$

When studying the behavior of positive solutions of (1), we note that there are only two cases for $ξ > ξ_{1}$ is sufficiently large:

$Case (i) :$ $ψ (ξ) > 0, ψ^{'} (ξ) > 0, ψ^{''} (ξ) > 0$ ;

$Case (ii) :$ $ψ (ξ) > 0, ψ^{'} (ξ) < 0, ψ^{''} (ξ) > 0$ .

Definition 1

A solution of (1) means $χ \in C ([ξ_{a}, \infty), R)$ where $ξ_{a} ≔ \min \{ξ_{0} - δ_{0}, \min_{ξ \in I} τ (ξ)\}$ which satisfies (1) and the property $ι {(ψ^{''})}^{α} \in C^{1} (I, R)$ on I. We consider the nontrivial solutions of (1) that satisfy the condition $\sup {|χ (ξ)| : ξ ⩾ ξ_{1}} > 0$ for all $ξ_{1} ⩾ ξ_{a}$ .

Definition 2

A solution $χ$ of (1) is said to be nonoscillatory if it isneither positive nor negative eventually. Otherwise, it is oscillatory.

Definition 3

If (1) has property D, then we say that solution $χ$ of (1) is either oscillatory or satisfies $\lim_{ξ \to \infty} χ (ξ) = 0$ .

A long time ago, third order differential equations have been involved in many mathematical models in various field of applied sciences where the famous isoperimetric problem was formulated. Later, a solution was found based on a third-order differential equation. Thus, third-order differential equations have become the target of researchers and those interested in their effectiveness in modeling many phenomena of economic and scientific life, especially physical, engineering and biological ones, we refer to Agarwal et al. (2000), Agarwal et al. (2001), Baculikova and Dzurina (2010), Baculikova and Dzurina (2011), Baculikova and Dzurina (2012), Baculikova and Dzurina (2014), Stability and Gorain (1998), Candan (2015), Dzurina et al. (2012), Erbe et al. (1995), Grace (1984), Győi et al. (1991), Hale (1977), Karpuz et al. (2010), Kitamura and Kusano (1980), Ladde et al. (1987), Li (1996), Li and Rogovchenko (2014), Li and Rogovchenko (2020), Li and Thandapani (2011), Li et al. (2010), Li et al. (2012), Liu and Triggiani (2013), Marchand et al. (2012), Philos (1981), Rath et al. (2004), Shang (2012), Tang (2002), Tekin (2021), Wu et al. (2016), Wu et al. (2018), Xing et al. (2011), Xu and Xia (2004), Yang and Xu (2014), Zhang and Wang (2010).

When considering partial differential equations and their applications, Adeye- mo and Khalique studied an extended Kadomtsev–Petviashvili-like equation (Adeyemo and Khalique, 2022a), and a higher-dimensional soliton equation (Adeyemo and Khalique, 2022b) achieving analytic solutions. They emphasized the importance of these solutions in scientific fields. The authors of Adeyemo et al. (2022) obtained travelling wave solutions for a (3 + 1)-dimensional generalized Zakharov-Kuznetsov equation. In Adeyemo (2022), a generalized extended (2 + 1)-D quantum Zakharov-Kuznetsov equation was analytically studied, where the author outlined the applications of cnoidal and snoidal waves of the model in ocean engineering and oceanography. A (2 + 1)-D generalized Bogoyavlensky-Konopelchenko equation was investigated in Adeyemo et al. (2022).

In a bounded domain in $R^{n}$ with a smooth boundary, the boundary stabilization of the problem satisfying the differential equation $θ^{″} + λ θ^{‴} = k^{2} (Δ θ + μ Δ θ^{'})$ is investigated in Stability and Gorain (1998). As modeled by the Standard linear model of viscoelasticity, these equations appear in the vibrations of flexible structures with internal material damping. Under mixed boundary conditions, the authors examined an exponential energy decay for their stated problem. An inverse problem for the linearized Jordan-Moore-Gibson-Thompson equation, a third-order in time partial differential equation that appears in nonlinear acoustic waves modeling high-intensity ultrasound, is introduced in Liu and Triggiani (2013). By using only one boundary measurement, the two canonical recovery issues of uniqueness and stability are examined. The Jordan-Moore-Gibson-Thompson equation’s dynamical decomposition is the foundation of the suggested method. The authors of Marchand et al. (2012) investigated the Moore-Gibson-Thompson equation, which arises in high-intensity ultrasound. They presented an abstract third-order equation in a Hilbert space. The authors provided that this third-order abstract equation with unbounded free dynamical operator is not well-posed in its simplest form, with a particular set of parameter values. In Tekin (2021), the authors examined the inverse problem of recovering a time-dependent coefficient of a nonlinear third order in time partial differential equation, also known as the Moore-Gibson-Thompson equation, from a single boundary measurement.

Despite the importance of third-order differential equations, the literature that has appeared so far is few compared to second-order differential equations, we recommend recent monographs to the reader (Baculikova and Dzurina, 2011; Wu et al., 2018; Wu et al., 2016 and Zhang and Wang, 2010 ).

It is worth noting that we find that third-order delay differential equations may have only oscillatory solutions, while they may have oscillatory and non-oscillatory solutions; for instant, the solutions for the eqaution $ψ^{'''} (ξ) + ψ (ξ - σ) = 0,$ are oscillatory if and only if, $σ > 3 / e$ . On the other hand, the equation of the form $ψ^{'''} (ξ) + 2 z^{'} (ξ) - ψ (ξ - \frac{1}{2} 3 π) = 0$ has an oscillatory and a nonoscillatory solution ( $\sin ξ, e^{λ ξ}$ ) such that $λ^{3} + 2 λ = e^{- (3 π l 2) λ}, λ > 0$ .

Recently, Baculikova and Dzurina (2010) and Yang and Xu (2014) established some different sufficient criteria which ensure that all nonoscillatory solutions to the equation

(2)

{(ι (ξ) {({(χ (ξ) + p_{0} χ (σ (ξ)))}^{''})}^{α})}^{'} + ϑ (ξ) χ^{α} (τ (ξ)) d s = 0

tend to zero. Also, Li et al. (2012) and Baculikova and Dzurina (2010) investigated Eq. (2) for

α = 1

under the condition

(3)

0 ⩽ p_{0} < 1 .

Assuming

τ (ξ) = ξ - δ_{0}

, Li and Rogovchenko (2014) studied asymptotics of Eq. (2) with the condition

0 ⩽ p_{0} < \infty .

In the present paper, by using different techniques (comparison with first order delay equations and the technique of Riccati transformation), we obtain the conditions that ensure the oscillation of the solutions of this equation. Moreover, we extend and improve previous results.

Furthermore, we present new criteria that ensure the oscillation of all the solutions of Eq. (1), these criteria are an improvement of previous results, as the conditions mentioned are less restrictive and easier to apply. By taking advantage of the results obtained recently and the current results, the conditions achieved ensure the oscillation of all solutions of Eq. (1).

We state the following lemma, which we will need to prove our results later.

Lemma 1

Let $h_{1}, h_{2} \in [0, \infty)$ and $m > 0$ . Then

(4)

{(h_{1} + h_{2})}^{m} ⩽ μ (h_{1}^{m} + h_{2}^{m}) .

where

\begin{matrix} μ = & 1 when m ⩽ 1; \\ μ = & 2^{m - 1} when m > 1 . \end{matrix}

Lemma 2

Let $H, F \in C (I, R), F_{*} \in R$ and $H (ξ) = F (ξ) + aF (ξ - k), ξ ⩾ ξ_{0} + \max {0, k}$ , where a and k are constants and $a \neq 1$ . Assume that $\exists$ $l \in R$ such that $\lim_{ξ \to \infty} H (ξ) = l$ , moreover, $l = \{\begin{matrix} (1 + a) F_{*} & if & {lim inf}_{ξ \to \infty} F (ξ) = F_{*} \\ (1 + a) F^{*} & if & {lim sup}_{ξ \to \infty} F (ξ) = F^{*} . \end{matrix}$ where $F_{*}, F_{*} \in R$ .

2 Main results

Lemma 3

Let $χ (ξ) > 0$ be a solution of Eq. (1). Assume that $ψ (ξ)$ satisfies case $(ii)$ . If

(5)

\int_{ξ_{2}}^{\infty} \int_{υ}^{\infty} {(\frac{1}{ι (u)} \int_{u}^{\infty} \int_{a}^{b} ϑ (κ, ω) d ω d κ)}^{1 / α} d ud υ = \infty,

then,

(6)

\lim_{ξ \to \infty} χ (ξ) = 0 .

Proof

$ψ (ξ)$ is a nonincreasing positive function, then there exists a $ψ_{0} ⩾ 0$ such that $\lim_{ξ \to \infty} ψ (ξ) = ψ_{0} ⩾ 0 .$ We claim that $ψ_{0} = 0$ . Otherwise, by the above Lemma, we get $\lim_{ξ \to \infty} χ (ξ) = ψ_{0} / (1 + p_{0}) > 0 .$ Therefore, there exists a $ξ_{2} ⩾ ξ_{0}$ such that, for all $ξ ⩾ ξ_{2}$

(7)

χ (τ (ξ, s)) > \frac{ψ_{0}}{2 (1 + p_{0})} > 0 .

From (1) and (7), it follows that

{(ι (ξ) {(ψ^{''} (ξ))}^{α})}^{'} ⩽ - \int_{a}^{b} ϑ (ξ, ω) {(\frac{ψ_{0}}{2 (1 + p_{0})})}^{α} d ω .

Integrating the above inequality from

ξ

\infty

, we get

ι (ξ) {(ψ^{''} (ξ))}^{α} ⩾ {(\frac{ψ_{0}}{2 (1 + P_{0})})}^{α} \int_{ξ}^{\infty} \int_{a}^{b} ϑ (κ, ω) d ω d κ .

It follows that

(8)

ψ^{''} (ξ) ⩾ \frac{ψ_{0}}{2 (1 + P_{0})} {(\frac{1}{ι (ξ)} \int_{ξ}^{\infty} \int_{a}^{b} ϑ (κ, ω) d ω d κ)}^{\frac{1}{α}} .

Integrating (8) from

ξ

\infty

, yields

- ψ^{'} (ξ) ⩾ \frac{ψ_{0}}{2 (1 + P_{0})} \int_{ξ}^{\infty} {(\frac{1}{ι (u)} \int_{u}^{\infty} \int_{a}^{b} ϑ (κ, ω) d ω d κ)}^{1 / α} d u .

Integrating again from

ξ_{2}

\infty

, gives

ψ (ξ_{2}) ⩾ \frac{ψ_{0}}{2 (1 + P_{0})} \int_{ξ_{2}}^{\infty} \int_{υ}^{\infty} {(\frac{1}{ι (u)} \int_{u}^{\infty} \int_{a}^{b} ϑ (κ, ω) d ω d κ)}^{1 / α} d ud υ,

this contradicts (5). Therefore,

\lim_{ξ \to \infty} ψ (ξ) = 0

, and from

0 < χ (ξ) ⩽ ψ (ξ)

, we have (6).

Theorem 1

Let (5) be satisfied and assume that there exists a function $ϱ \in C (I, R)$ where $ϱ (ξ) ⩽ τ (ξ, ω), ϱ (ξ) < ξ$ and $\lim_{ξ \to \infty} ϱ (ξ) = \infty$ . If the first-order delay differential equation

(9)

y^{'} (ξ) + \frac{1}{{(1 + p_{0})}^{α}} \int_{a}^{b} R (ϱ) ϑ (ξ, ω) d ω y (ϱ (ξ)) = 0,

where

R (ξ) ≔ {(\int_{ξ_{2}}^{ϱ (ξ)} \int_{ξ_{1}}^{ρ} ι^{- 1 / α} (ω) d ω d ρ)}^{α},

is oscillatory; eventually, then (1) has property D.

Proof

Suppose that $χ (ξ)$ is a positive solution of (1); there exists a $ξ_{1} ⩾ ξ_{0}$ such that either $(i)$ or $(ii)$ holds for all $ξ ⩾ ξ_{1}$ . Let $ψ$ satisfies case $(ii)$ , by Lemma 3, we see that (6) holds. Assume that $ψ$ satisfies case $(i)$ , Since $ι (ξ) {(ψ^{''} (ξ))}^{α}$ is nonincreasing, we have

(10)

\begin{matrix} ψ^{'} (ξ) ⩾ & \int_{ξ_{1}}^{ξ} \frac{1}{ι^{1 / α} (ω)} ι^{1 / α} (ω) (ψ^{''} (ω)) d ω \\ ⩾ & ι^{1 / α} (ξ) (ψ^{''} (ξ)) \int_{ξ_{1}}^{ξ} \frac{1}{ι^{1 / α} (ω)} d ω, \end{matrix}

Integrating (10) from

ξ_{2}

ξ

, where

ξ_{2} > ξ_{1}

, we get

(11)

ψ (ξ) ⩾ ι {(ξ)}^{1 / α} (ψ^{''} (ξ)) \int_{ξ_{2}}^{ξ} (\int_{ξ_{1}}^{ρ} \frac{1}{ι^{1 / α} (ω)} d ω) d ρ .

Now, Since

ψ^{'} (ξ)

is a nondecreasing positive function. There exists a constant

c_{0}

such that

\lim_{ξ \to \infty} ψ^{'} (ξ) = c_{0} > 0

(or

c_{0} = \infty

). By Lemma 2, we have

\lim_{ξ \to \infty} χ^{'} (ξ) = c_{0} / (1 + p_{0}) > 0,

this implies that

χ (ξ)

is a nondecreasing function,we get

ψ (ξ) = χ (ξ) + p_{0} χ (ξ - δ_{0}) ⩽ (1 + p_{0}) χ (ξ) .

Therefore,

χ (ξ) ⩾ \frac{1}{1 + p_{0}} ψ (ξ) .

Since

τ (ξ, s) ⩾ ϱ (ξ)

, we obtain

χ (τ (ξ, s)) ⩾ \frac{1}{1 + p_{0}} ψ (ϱ (ξ)) .

From (1), we have

(12)

{(ι (ξ) {(ψ^{''} (ξ))}^{α})}^{'} + \frac{ψ^{α} (ϱ (ξ))}{{(1 + p_{0})}^{α}} \int_{a}^{b} ϑ (ξ, ω) d ω ⩽ 0 .

Using (12) and (11), we arrive at

{(ι (ξ) {(ψ^{''} (ξ))}^{α})}^{'} + \frac{\int_{a}^{b} ϑ (ξ, ω) d ω}{{(1 + p_{0})}^{α}} {(\int_{ξ_{2}}^{ϱ (ξ)} \int_{ξ_{1}}^{ρ} ι^{- 1 / α} (ω) d ω d ρ)}^{α} (ι (ϱ (ξ)) {(ψ^{''} (ϱ (ξ)))}^{α}) ⩽ 0 .

Therefore, we have

y (ξ) = ι (ξ) {(ψ^{''} (ξ))}^{α}

is a positive solution of (9).

Corollary 1

Let (5) holds, and suppose that there exists a function $ϱ \in C (I, R)$ such that $ϱ (ξ) ⩽ τ (ξ, ω), ϱ (ξ) < ξ$ and $\lim_{ξ \to \infty} ϱ (ξ) = \infty$ . If

(13)

\lim \inf_{ξ \to \infty} \int_{ϱ (ξ)}^{ξ} R (u) (\int_{a}^{b} ϑ (ξ, ω) d ω) d u > \frac{{(1 + p_{0})}^{α}}{e},

then Eq. (1) has property D.

Proof

In view of Győi et al. (1991); Erbe et al. (1995) condition (13) implies the oscillation of the delay differential Eq. (1).

Theorem 2

If a function $θ \in C ([ξ_{0}, \infty), (0, \infty))$ exists, where $θ (ξ) ⩽ ξ, τ (ξ, ω) - δ_{0} = τ (ξ - δ_{0}, ω), τ (ξ, ω) ⩽ (θ (ξ) - δ_{0})$ and

(14)

\underset{ξ \to \infty}{\lim \sup} (\int_{ξ_{2}}^{ξ} (\int_{ξ_{1}}^{ρ} \frac{1}{ι^{1 / α} (ω)} d ω) d ρ) \int_{θ (ξ)}^{ξ} \int_{a}^{b} \tilde{ϑ} (u, ω) d ω d u . > μ (1 + p_{0}^{α}),

where,

(15)

\tilde{ϑ} (ξ, ω) ≔ \min {ϑ (ξ, ω), ϑ (ξ - δ_{0}, ω)}

then case

(ii)

is impossible to satisfy.

Proof

Let $χ > 0$ be a solution of (1). Then, $χ (ξ), χ (τ (ξ))$ and $χ (ξ - δ_{0})$ are positive functions for $ξ ⩾ ξ_{1}$ is sufficiently large. By using Lemma 1, we obtain $ψ^{α} (ξ) ⩽ μ (χ^{α} (ξ) + p_{0}^{α} χ^{α} (ξ - δ_{0})),$ and

(16)

ψ^{α} (τ (ξ, ω)) ⩽ μ (χ^{α} (τ (ξ, ω)) + p_{0}^{α} χ^{α} (τ (ξ, ω) - δ_{0})) .

Now, from (1) we have

(17)

{(ι (ξ - δ_{0}) {(ψ^{''} (ξ - δ_{0}))}^{α})}^{'} + \int_{a}^{b} ϑ (ξ - δ_{0}, ω) χ^{α} (τ (ξ - δ_{0}, ω)) d ω = 0 .

Using (1), (16) and (17), we have

(18)

\begin{matrix} 0 ⩾ & {(ι (ξ) {(ψ^{''} (ξ))}^{α})}^{'} + \int_{a}^{b} ϑ (ξ, s) χ^{α} (τ (ξ, s)) d s + p_{0}^{α} {(ι (ξ - δ_{0}) {(ψ^{''} (ξ - δ_{0}))}^{α})}^{'} \\ + p_{0}^{α} \int_{a}^{b} ϑ (ξ - δ_{0}, ω) χ^{α} (τ (ξ, ω) - δ_{0}) d ω \\ ⩾ & {(ι (ξ) {(ψ^{''} (ξ))}^{α})}^{'} + p_{0}^{α} {(ι (ξ - δ_{0}) {(ψ^{''} (ξ - δ_{0}))}^{α})}^{'} \\ + \int_{a}^{b} \tilde{ϑ} (ξ, ω) (χ^{α} (τ (ξ, ω)) + p_{0}^{α} χ^{α} (τ (ξ) - δ_{0}, ω)) d ω . \end{matrix}

Thus

(19)

{(ι (ξ) {(ψ^{''} (ξ))}^{α} + p_{0}^{α} (ι (ξ - δ_{0}) {(ψ^{''} (ξ - δ_{0}))}^{α}))}^{'} + \frac{1}{μ} \int_{a}^{b} \tilde{ϑ} (ξ, ω) ψ^{α} (τ (ξ, ω)) d ω ⩽ 0 .

Integrating (19) from

θ (ξ)

ξ

, we see that

\begin{matrix} ι (θ (ξ)) {(ψ^{''} (θ (ξ)))}^{α} + p_{0}^{α} ι (θ (ξ) - δ_{0}) {(ψ^{''} (θ (ξ) - δ_{0}))}^{α} \\ ⩽ & ι (ξ) {(ψ^{''} (ξ))}^{α} + p_{0}^{α} ι (ξ - δ_{0}) {(ψ^{''} (ξ - δ_{0}))}^{α} \\ - \frac{1}{μ} \int_{θ (ξ)}^{ξ} \int_{a}^{b} \tilde{ϑ} (u, ω) ψ^{α} (τ (u, ω)) d ω d u, \end{matrix}

Since

ι (ξ) {(ψ^{''} (ξ))}^{α}

is nonincreasing, we have

(1 + p_{0}^{α}) ι (θ (ξ) - δ_{0}) {(ψ^{''} (θ (ξ) - δ_{0}))}^{α} ⩾ \frac{1}{μ} ψ^{α} (τ (ξ, s)) \int_{θ (ξ)}^{ξ} \int_{a}^{b} \tilde{ϑ} (u, ω) d ω d u .

From (11), we obtain

\begin{matrix} (1 + p_{0}^{α}) ι (θ (ξ) - δ_{0}) {(ψ^{''} (θ (ξ) - δ_{0}))}^{α} \\ ⩾ & \frac{1}{μ} ι (τ (ξ, ω)) {(ψ^{''} (τ ((ξ, ω)))}^{α} \\ \times (\int_{ξ_{2}}^{ξ} (\int_{ξ_{1}}^{ρ} \frac{1}{ι^{1 / α} (ω)} d ω) d ρ) \int_{θ (ξ)}^{ξ} \int_{a}^{b} \tilde{ϑ} (u, ω) d ω d u . \end{matrix}

This gives

(1 + p_{0}^{α}) ⩾ \frac{1}{μ} (\int_{ξ_{2}}^{ξ} (\int_{ξ_{1}}^{ρ} \frac{1}{ι^{1 / α} (ω)} d ω) d ρ) \int_{θ (ξ)}^{ξ} \int_{a}^{b} \tilde{ϑ} (u, ω) d ω d u .

Applying the

\lim \sup

on both sides of the previous inequality, we gain a contradiction to (14).

Combining Corollary 1 with Theorem 2, we get the Theorem of oscillation for (1) as follows.

Theorem 3

Assume that there exists a functions $ϱ \in C (I, R)$ and $θ \in C ([ξ_{0}, \infty), (0, \infty))$ such that $ϱ (ξ) ⩽ τ (ξ, ω), ϱ (ξ) < ξ, \lim_{ξ \to \infty} ϱ (ξ) = \infty, θ (ξ) ⩽ ξ, τ (ξ, ω) - δ_{0} = τ (ξ - δ_{0}, ω), τ (ξ, ω) ⩽ (θ (ξ) - δ_{0})$ . If (13) and (14) hold, then Eq. (1) is oscillatory.

Example 1

Consider the following third-order neutral differential equation

(20)

{(χ (ξ) + p_{0} χ (ξ - δ_{0}))}^{'''} + (1 + p_{0} e^{δ_{0}}) χ (ξ) = 0 .

Choose

ϱ (ξ) = ξ - 1

, by Corollary 1, we see that Eq. (20) has property D. Note that the solution

χ (ξ) = e^{- ξ}

is satisfying (6).

3 Conclusions

In this paper, we consider the oscillatory behavior of third-order neutral differential equation with distributed deviating arguments which is commonly used in the engineering and natural sciences for modeling various problems. Through this investigation, we were able to improve and extend upon previous results in the literature. In contrast to previous results, we obtained less restrictive conditions as where we do not need the conditions $0 ⩽ p_{0} < 1$ and $either ι^{'} (ξ) ⩾ 0 or ι^{'} (ξ) ⩽ 0,$ which is an improvement compared to Baculikova and Dzurina (2010); Baculikova and Dzurina (2012); Candan (2015); Dzurina et al. (2012), and can be applied more widely in this field of study.

Acknowledgment

The researchers would like to acknowledge the Deanship of Scientific Research, Taif University for funding this work.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References

Adeyemo O.D., . Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+ 1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering. J. Ocean Eng. Sci. 2022
[Google Scholar]
Adeyemo O.D., Khalique C.M., . Dynamics of soliton waves of group-invariant solutions through optimal system of an extended KP-like equation in higher dimensions with applications in medical sciences and mathematical physics. J. Geometry Phys.. 2022;177:104502.
[Google Scholar]
Adeyemo O.D., Khalique C.M., . Dynamical soliton wave structures of one-dimensional lie subalgebras via group-invariant solutions of a higher-dimensional soliton equation with various applications in ocean physics and mechatronics engineering. Commun. Appl. Mathe. Comput.. 2022;4(4):1531-1582.
[Google Scholar]
Adeyemo O.D., Zhang L., Khalique C.M., . Optimal solutions of Lie subalgebra, dynamical system, travelling wave solutions and conserved currents of (3+ 1)-dimensional generalized Zakharov-Kuznetsov equation type I. Eur. Phys. J. Plus. 2022;137(8):954.
[Google Scholar]
Adeyemo O.D., Zhang L., Khalique C.M., . Bifurcation theory, lie group-invariant solutions of subalgebras and conservation laws of a generalized (2+ 1)-dimensional BK equation Type II in plasma physics and fluid mechanics. Mathematics. 2022;10(14):2391.
[Google Scholar]
Agarwal R.P., Grace S.R., Regan D., . Oscillation Theory for Difference and Functional Differential equations. Kluwer Academic, Dordrecht: Marcel Dekker; 2000.
Agarwal R.P., Grace S.R., O’Regan D., . Oscillation criteria for certain nth order differential equations with deviating arguments. J. Math. Anal. Appl.. 2001;262:601-622.
[Google Scholar]
Baculikova B., Dzurina J., . Oscillation of third-order neutral differential equations. Math. Comput. Modell.. 2010;52:215-226.
[Google Scholar]
Baculikova B., Dzurina J., . Oscillation of third-order nonlinear differential equations. J. Appl. Math. Lett.. 2011;24:466-470.
[Google Scholar]
Baculikova B., Dzurina J., . Oscillation Theorems for third order neutral differential equations. Carpathian J. Math.. 2012;28:199-206.
[Google Scholar]
Baculikova B., Dzurina J., . On the oscillation of odd order advanced differential equations. Boundary Value Probl. 2014
[Google Scholar]
Bose S.K., Gorain. G.C., . Stability of the boundary stabilised internally damped wave equation $y^{″} + λ y^{‴} = c^{2} (Δ y + μ Δ y^{'})$ in a bounded domain in $R^{n}$ . Indian J. Math.. 1998;40:1-15.
[Google Scholar]
Candan T., . Oscillation criteria and asymptotic properties of solutions of third-order nonlinear neutral differential equations. Math. Methods Appl.. 2015;38:1379-1392.
[Google Scholar]
Dzurina J., Thandapani E., Tamilvanan S., . Oscillation of solutions to third-order half-linear neutral differential equations. Electron. J. Differential Eqs.. 2012;2012:1-9.
[Google Scholar]
Erbe L., Kong Q., Zhang B.G., . Oscillation Theory for Functional Differential Eqautions. New York: Marcel Dekker; 1995.
Grace S.R., . Oscillation Theorems for nth-order differential equations with deviating arguments. J. Math Appl. Anal.. 1984;101:268-296.
[Google Scholar]
Győi I., Gyori I., Ladas G.E., . Oscillation Theory of Delay Differential Equations: With Applications. Clarendon Press; 1991.
Hale J.K., . Theory of Functional Differential equations. New York: Spring-Verlag; 1977.
Karpuz B., Ocalan O., Ozturk S., . Comparison Theorems on the oscillation and asymptotic behavior of higher-order neutral differential equations. Glasgow Math J.. 2010;52:107-114.
[Google Scholar]
Kitamura Y., Kusano T., . Oscillation of first-order nonlinear differential equations with deviating arguments. Proc. Amer. Math.. 1980;78(1):64-68.
[Google Scholar]
Ladde G.S., Lakshmikantham V., Zhang B., . G: Oscillation theory of Differential Equations with Deviating Arguments. New York: Marcel Dekker; 1987.
Li B., . Oscillation of first order delay differential equations. Am. Mathe. Soc.. 1996;124(12):3729-3737.
[Google Scholar]
Li T., Rogovchenko Yu., . V: Asymptotic behavior of higher-order quasilinear neutral differential equations. Abstr. Appl. Anal.. 2014;2014:1-11.
[Google Scholar]
Li T., Rogovchenko Y.V., . On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appl. Mathe. Lett.. 2020;1(105):106293.
[Google Scholar]
Li T., Thandapani E., . Oscillation of solutions to odd-order nonlinear neutral functional differential equations. Electron. J. Differ. Equ.. 2011;2011:23.
[Google Scholar]
Li T., Han Z., Zhao P., Sun S., . Oscillation of even-order neutral delay differential equations. Adv Differ Equ.. 2010;2010:1-9.
[Google Scholar]
Li T., Zhang C., Xing G., . Oscillation of third-order neutral delay differential equations. Abstr. Appl. Anal.. 2012;2012:1-11.
[Google Scholar]
Liu S., Triggiani R., . An inverse problem for a third order PDE arising in high-intensity ultrasound: Global uniqueness and stability by one boundary measurement. J. Inverse Ill-Posed Probl.. 2013;21:825-869.
[Google Scholar]
Marchand R., McDevitt T., Triggiani R., . An abstract semigroup approach to the third-order Moore–Gibson–Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability. Math. Meth. Appl. Sci.. 2012;35:1896-1929.
[Google Scholar]
Philos C., . On the existence of nonoscillatory solutions tending to zero at $\infty$ for differential equations with positive delay. Arch. Math.. 1981;36:168-178.
[Google Scholar]
Rath R.N., Padhy L.N., Misra N., . Oscillation of solutions of non-linear neutral delay differential equations of higher order for $p (ξ) = \pm 1$ . Arch Math.. 2004;40:359-366.
[Google Scholar]
Shang Y., . Functions of $α$ -slow increase. Bull. Mathe. Anal. Appl.. 2012;4(1):226-230.
[Google Scholar]
Tang X.H., . Oscillation for first order superlinear delay differential equations. J. London Mathe. Soc.. 2002;65(1):115-122.
[Google Scholar]
Tekin I., . Inverse problem for a nonlinear third order in time partial differential equation. Math. Meth. Appl. Sci.. 2021;44:9571-9581.
[Google Scholar]
Wu Y., Yu Y., Zhang J., Xiao J., . Oscillation criteria for second order Emden-Fowler functional differential equations of neutral type. J. Inequal. Appl. 2016
[Google Scholar]
Wu Y., Yu Y., Xiao J., . Oscillation of second-order Emden-Fowler neutral delay differential equations. Electronic J. Diff. Eqs.. 2018;18:1-15.
[Google Scholar]
Xing G., Li T., Zhang C., . Oscillation of higher-order quas-linear neutral differential equations. Adv. Diff. Eqs.. 2011;2011:2011.
[Google Scholar]
Xu Z., Xia Y., . Integral averaging technique and oscillation of certain even order delay differential equations. J. Math Appl. Anal.. 2004;292:238-246.
[Google Scholar]
Yang L., Xu Z., . Oscillation of certain third-order quasilinear neutral differential equations. Math. Slovaca. 2014;64:85-100.
[Google Scholar]
Zhang S.Y., Wang Q.R., . Oscillation of second-order nonlinear neutral dynamic equations on time scales. Appl. Math. Comput.. 2010;216(10):2837-2848.
[Google Scholar]

Introduction

Main results

Conclusions

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[1] Adeyemo O.D., . Applications of cnoidal and snoidal wave solutions via optimal system of subalgebras for a generalized extended (2+ 1)-D quantum Zakharov-Kuznetsov equation with power-law nonlinearity in oceanography and ocean engineering. J. Ocean Eng. Sci. 2022
[Google Scholar]

[2] Adeyemo O.D., Khalique C.M., . Dynamics of soliton waves of group-invariant solutions through optimal system of an extended KP-like equation in higher dimensions with applications in medical sciences and mathematical physics. J. Geometry Phys.. 2022;177:104502.
[Google Scholar]

[3] Adeyemo O.D., Khalique C.M., . Dynamical soliton wave structures of one-dimensional lie subalgebras via group-invariant solutions of a higher-dimensional soliton equation with various applications in ocean physics and mechatronics engineering. Commun. Appl. Mathe. Comput.. 2022;4(4):1531-1582.
[Google Scholar]

[4] Adeyemo O.D., Zhang L., Khalique C.M., . Optimal solutions of Lie subalgebra, dynamical system, travelling wave solutions and conserved currents of (3+ 1)-dimensional generalized Zakharov-Kuznetsov equation type I. Eur. Phys. J. Plus. 2022;137(8):954.
[Google Scholar]

[5] Adeyemo O.D., Zhang L., Khalique C.M., . Bifurcation theory, lie group-invariant solutions of subalgebras and conservation laws of a generalized (2+ 1)-dimensional BK equation Type II in plasma physics and fluid mechanics. Mathematics. 2022;10(14):2391.
[Google Scholar]

[6] Agarwal R.P., Grace S.R., Regan D., . Oscillation Theory for Difference and Functional Differential equations. Kluwer Academic, Dordrecht: Marcel Dekker; 2000.

[7] Agarwal R.P., Grace S.R., O’Regan D., . Oscillation criteria for certain nth order differential equations with deviating arguments. J. Math. Anal. Appl.. 2001;262:601-622.
[Google Scholar]

[8] Baculikova B., Dzurina J., . Oscillation of third-order neutral differential equations. Math. Comput. Modell.. 2010;52:215-226.
[Google Scholar]

[9] Baculikova B., Dzurina J., . Oscillation of third-order nonlinear differential equations. J. Appl. Math. Lett.. 2011;24:466-470.
[Google Scholar]

[10] Baculikova B., Dzurina J., . Oscillation Theorems for third order neutral differential equations. Carpathian J. Math.. 2012;28:199-206.
[Google Scholar]

[11] Baculikova B., Dzurina J., . On the oscillation of odd order advanced differential equations. Boundary Value Probl. 2014
[Google Scholar]

[12] Bose S.K., Gorain. G.C., . Stability of the boundary stabilised internally damped wave equation $y^{″} + λ y^{‴} = c^{2} (Δ y + μ Δ y^{'})$ in a bounded domain in $R^{n}$ . Indian J. Math.. 1998;40:1-15.
[Google Scholar]

[13] Candan T., . Oscillation criteria and asymptotic properties of solutions of third-order nonlinear neutral differential equations. Math. Methods Appl.. 2015;38:1379-1392.
[Google Scholar]

[14] Dzurina J., Thandapani E., Tamilvanan S., . Oscillation of solutions to third-order half-linear neutral differential equations. Electron. J. Differential Eqs.. 2012;2012:1-9.
[Google Scholar]

[15] Erbe L., Kong Q., Zhang B.G., . Oscillation Theory for Functional Differential Eqautions. New York: Marcel Dekker; 1995.

[16] Grace S.R., . Oscillation Theorems for nth-order differential equations with deviating arguments. J. Math Appl. Anal.. 1984;101:268-296.
[Google Scholar]

[17] Győi I., Gyori I., Ladas G.E., . Oscillation Theory of Delay Differential Equations: With Applications. Clarendon Press; 1991.

[18] Hale J.K., . Theory of Functional Differential equations. New York: Spring-Verlag; 1977.

[19] Karpuz B., Ocalan O., Ozturk S., . Comparison Theorems on the oscillation and asymptotic behavior of higher-order neutral differential equations. Glasgow Math J.. 2010;52:107-114.
[Google Scholar]

[20] Kitamura Y., Kusano T., . Oscillation of first-order nonlinear differential equations with deviating arguments. Proc. Amer. Math.. 1980;78(1):64-68.
[Google Scholar]

[21] Ladde G.S., Lakshmikantham V., Zhang B., . G: Oscillation theory of Differential Equations with Deviating Arguments. New York: Marcel Dekker; 1987.

[22] Li B., . Oscillation of first order delay differential equations. Am. Mathe. Soc.. 1996;124(12):3729-3737.
[Google Scholar]

[23] Li T., Rogovchenko Yu., . V: Asymptotic behavior of higher-order quasilinear neutral differential equations. Abstr. Appl. Anal.. 2014;2014:1-11.
[Google Scholar]

[24] Li T., Rogovchenko Y.V., . On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appl. Mathe. Lett.. 2020;1(105):106293.
[Google Scholar]

[25] Li T., Thandapani E., . Oscillation of solutions to odd-order nonlinear neutral functional differential equations. Electron. J. Differ. Equ.. 2011;2011:23.
[Google Scholar]

[26] Li T., Han Z., Zhao P., Sun S., . Oscillation of even-order neutral delay differential equations. Adv Differ Equ.. 2010;2010:1-9.
[Google Scholar]

[27] Li T., Zhang C., Xing G., . Oscillation of third-order neutral delay differential equations. Abstr. Appl. Anal.. 2012;2012:1-11.
[Google Scholar]

[28] Liu S., Triggiani R., . An inverse problem for a third order PDE arising in high-intensity ultrasound: Global uniqueness and stability by one boundary measurement. J. Inverse Ill-Posed Probl.. 2013;21:825-869.
[Google Scholar]

[29] Marchand R., McDevitt T., Triggiani R., . An abstract semigroup approach to the third-order Moore–Gibson–Thompson partial differential equation arising in high-intensity ultrasound: structural decomposition, spectral analysis, exponential stability. Math. Meth. Appl. Sci.. 2012;35:1896-1929.
[Google Scholar]

[30] Philos C., . On the existence of nonoscillatory solutions tending to zero at $\infty$ for differential equations with positive delay. Arch. Math.. 1981;36:168-178.
[Google Scholar]

[31] Rath R.N., Padhy L.N., Misra N., . Oscillation of solutions of non-linear neutral delay differential equations of higher order for $p (ξ) = \pm 1$ . Arch Math.. 2004;40:359-366.
[Google Scholar]

[32] Shang Y., . Functions of $α$ -slow increase. Bull. Mathe. Anal. Appl.. 2012;4(1):226-230.
[Google Scholar]

[33] Tang X.H., . Oscillation for first order superlinear delay differential equations. J. London Mathe. Soc.. 2002;65(1):115-122.
[Google Scholar]

[34] Tekin I., . Inverse problem for a nonlinear third order in time partial differential equation. Math. Meth. Appl. Sci.. 2021;44:9571-9581.
[Google Scholar]

[35] Wu Y., Yu Y., Zhang J., Xiao J., . Oscillation criteria for second order Emden-Fowler functional differential equations of neutral type. J. Inequal. Appl. 2016
[Google Scholar]

[36] Wu Y., Yu Y., Xiao J., . Oscillation of second-order Emden-Fowler neutral delay differential equations. Electronic J. Diff. Eqs.. 2018;18:1-15.
[Google Scholar]

[37] Xing G., Li T., Zhang C., . Oscillation of higher-order quas-linear neutral differential equations. Adv. Diff. Eqs.. 2011;2011:2011.
[Google Scholar]

[38] Xu Z., Xia Y., . Integral averaging technique and oscillation of certain even order delay differential equations. J. Math Appl. Anal.. 2004;292:238-246.
[Google Scholar]

[39] Yang L., Xu Z., . Oscillation of certain third-order quasilinear neutral differential equations. Math. Slovaca. 2014;64:85-100.
[Google Scholar]

[40] Zhang S.Y., Wang Q.R., . Oscillation of second-order nonlinear neutral dynamic equations on time scales. Appl. Math. Comput.. 2010;216(10):2837-2848.
[Google Scholar]

Nonlinear third-order differential equations with distributed delay: Some new oscillatory solutions

Abstract

Keywords

Third-order nonlinear differential equations

Delay

Oscillation

1 Introduction

2 Main results

3 Conclusions

Acknowledgment

Declaration of Competing Interest

References

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