Property $\bar{A}$ of third-order noncanonical functional differential equations with positive and negative terms

1.   Introduction

In this paper, we are concerned with the third-order differential equation with a positive and negative term

where $t \ge t_0 > 0$ and we need the following assumptions for our work in the sequel.

$(H1)$ $p_1(t), p_2(t), a(t), b(t) \in \mathbb{C}([t_{0}, \infty))$ are positive and continuous functions for all $t \ge t_o$;

$(H2)$ $g(u), h(u) \in \mathbb{C}(\mathbb{R}), ug(u) > 0, uh(u) > 0\; for\; u\neq0,$ h is bounded, g is nondecreasing;

$(H3)$ $-g(-uv)\ge g(uv)\ge g(u)g(v)\; for\; uv > 0;$

$(H4)$ $\tau(t), \sigma(t) \in \mathbb{C}([t_{0}, \infty))$ are continuous functions such that $\tau(t)\le t$, $\lim_{t\to\infty}\tau(t) = \infty$ and $\lim_{t\to\infty}\sigma(t) = \infty$;

$(H5)$ the noncanonical case of (E), that is

By a solution of (E), we mean a function $x(t)$ with derivatives $p_1 (t)x'(t), (p_1 (t) x'(t))'$ continuous on $[t_x, \infty), T_x \ge t_0$, which satisfies (E) on $[t_x, \infty)$. We consider only those solutions $x(t)$ of (E) which satisfy $\sup\{|x(t)|:t \ge T\} > 0$ for any $T\ge T_x$. A solution x of (E) is said to be oscillatory if it is neither eventually positive nor eventually negative, and it is said to be nonoscillatory otherwise. The equation itself is termed oscillatory if all its solutions oscillate.

In the generalization of the Kiguradze lemma ^[21], the fixed sign of the highest derivative is used to derive the structure of probable nonoscillatory solutions. Since (E) contains both positive and negative terms, we cannot fix the sign of the third-order quasi-derivative for an eventually positive solution. Therefore, the authors studied the oscillatory properties of (E) when either $a(t)\equiv0$ or $b(t)\equiv0$; see, for example ^{[6,7,8,9,10,11,12,13,14,15,16,18,19,20,25,26,31,32,33]} and the references cited therein.

Recently, there have been several highly interesting results relating to the oscillatory properties of differential equations ^{[3,27,28,29,30,31,32,33]}. Also some applications related to biomathematics can be found in ^{[4,17,22,23,24]}. However, third-order differential equations get less attention from researchers compared to second-order differential equations. Now, we recall some studies related to the content of this paper.

In ^[1] Agarwal et al. obtained some new oscillation criteria for the third-order non-linear differential equation of the form

under

and

In ^[5] Alzabut et al. produced several results for the asymptotic and oscillatory behaviour of third order differential equations

under $\int_{t_0}^{t}\dfrac{1}{a^{\frac{1}{\alpha}}(s)}ds = \infty$ where $y(t) = x(t)+P(t)x^{\nu}(\omega(t))-q(t)x^{\kappa}(\omega(t))$.

In ^[6], Baculicova and Dzurina considered the equation (E) and they derived new techniques for studying the oscillatory and asymptotic properties of (E) by property $A$ by considering the canonical case

In ^[26], Saranya et al. developed new technique to study the oscillation criteria for the third order quasi-linear delay differential equation

under

In ^[27], Saranya et al. attained new criteria for the oscillatory and asymptotic behavior of the solution of the equation

under the semi-canonical type

In ^[30], Santra et al. studied the oscillatory behavior of half-linear neutral conformable differential equations

where $x(t) = y(t)+p(t)x(\delta(t))$ under the canonical case

However, most of the research work is done under the canonical case

as the investigation of the oscillatory properties of the canonical equation is much easier than the noncanonical case. Therefore, in this paper, we aim to investigate the oscillatory properties of solutions of (E) in noncanonical form, which is under $(H5)$ using the canonical representation of a strongly noncanonical operator. Thus all the results are new in this paper and complement to those results arrived in ^[1,2,21,26]. Furthermore, we assume that

$(H6)$ $\int_{t_0}^{\infty}\dfrac{1}{p_1 (t)}\int_{t}^{\infty}\dfrac{1}{p_2 (\zeta)}\int_{\zeta}^{\infty}b(u) du\; d\zeta\; dt < \infty.$

Instead of the above condition, one can easily show that the effect of the negative term is reduced and allows us to study property $\bar{A}$. Property $\bar{A}$ refers to the condition under which every nonoscillatory solution $x(t)$ of (E) satisfies $\lim_{t\to \infty}\dfrac{x(t)}{\Omega(t)} = 0$.

This paper is organized as follows. In Section 2, we present some basic notations and lemmas. We present the main result for the equation (E) in Section 3. In Section 4, we illustrate few examples for our main result.

2.   Preliminaries

In this section, initially we provide some lemmas to easily verify the conditions for property $\bar{A}$ of (E).

To simplify our notation, let us denote

and

Lemma 2.1. Assume that $(H1)-(H6)$ hold, and $x(t)$ is an eventually positive solution of (E) then, the function

is a positive solution of the equation

where $\mathbb{L}(z) = (p_2(t)(p_1(t)z'(t))')'.$

Proof. It follows from (H6) and the boundedness of $h(u)$ that the definition of function $z(t)$ defined by (2.1) is well defined for all $t \ge T_x\ge t_0$. We write that $z(t) > x(t)$, $z'(t) < x'(t)$ and

which completes the proof. □

Lemma 2.2. Let $(H1)-(H6)$ hold, and the strongly noncanonical operator

has the following unique canonical representation

Proof. The proof can be found in Theorem 3.2 of ^[8] and is hence omitted.

By Lemma 2.2, $(E_1)$ can be written in the equivalent canonical form as

Setting $\mu(t) = \dfrac{z(t)}{\Omega(t)}$, we immediately obtain the follows results (see ^[8]). □

Lemma 2.3. Let $(H1)-(H6)$ hold, and the strongly noncanonical differential equation $(E_1)$ can be written as the equivalent canonical equation

3.   Main Result

Corollary 3.1. Let $(H1)-(H6)$ hold, and the strongly noncanonical equation (E) has an eventually positive solution if and only if the canonical equation $(E_2)$ has an eventually positive solution. Corollary 3.1 significantly simplifies the examination of (E) since for $(E_2)$ we deal with only two classes of eventually positive solutions existing of four classes of eventually positive solutions. Thus, $\mu(t)$ satisfies either

and in this case we say $\mu \in S_0$ or

and for this property, we denote that $\mu \in S_2$.

Theorem 3.2. Assume that $(H1)-(H6)$ hold, and for all $t_1 \ge t_0$ large enough

and

If

then (E) holds property $\bar{A}$.

Proof. Let $x(t)$ be an eventually positive solution of (E) on $[T_x, \infty), T_x\ge t_0$. Then the function $z(t)$ defined on 2.1 satisfies $(E_1)$ with $z(t) > x(t) > 0$ and $z'(t) < x'(t)$ for all $t \ge t_1 \ge T_x$. Setting $\mu(t) = \dfrac{z(t)}{\omega(t)}$, we see that $\mu(t) > 0$ satisfies the equation $(E_2)$ and therefore it satisfies $\mu \in S_0$ or $\mu \in S_2$ for all $t \ge t_1$. First assume that $\mu \in S_2$. Using the fact that $\eta_2(t)(\eta_1(t)\mu'(t))$ is decreasing, we have

It follows from (3.4) that $\dfrac{\eta_1(t)\mu'(t)}{Q_2(t)}$ is decreasing. Then

Setting the last estimate into $(E_2)$, we see that $w(t) = \eta_1(t)\bigg(\dfrac{z(t)}{\Omega(t)}\bigg)'$ is a positive solution of the differential inequality

and we have that $\dfrac{w(t)}{Q_2(t)}$ is decreasing and $\eta_2(t)w'(t) > 0$. By integrating (3.5), we get

Hence,

In view of (H3) and given that $w(t)$ is non-decreasing and $\dfrac{w(t)}{Q_2(t)}$ is non-increasing, we have

Therefore, letting $u = \dfrac{w(\tau(t))}{Q_2(\tau(t))}$, we obtain

and condition (3.2) implies that $\dfrac{w(t)}{Q_2(t)}\to0\; a\zeta\; t\to\infty$. Indeed, if we assume $\dfrac{w(t)}{Q_2(t)}\to L > 0$, then $\dfrac{w(t)}{Q_2(t)}\ge L$ and substituting in (3.5), we have

Integrating from $t_1$ to $\infty$ yields

which contradicts (3.2). Taking $\limsup$ in (3.7) contradicts (3.3).

Now assume that $\mu(t)\in S_0$. Since $\mu(t)$ is positive and decreasing, there exists $\lim_{t\to\infty}\mu(t) = 2l\ge 0$. It follows from (2.1) that

If we assume that $l > 0$, then $\dfrac{x(\tau(t))}{\Omega(\tau(t))}\ge l > 0$, eventually. Intergrating $(E_2)$ yields

Integrating from $t$ to $\infty$ and then from $t_1\; to \; \infty$, we get

which contradicts (3.1). □

Corollary 3.3. Let $(H1)-(H6)$ and (3.1) hold, and for all $t_1$ large enough

If

then (E) holds property $\bar{A}$.

Theorem 3.4. Let $(H1)-(H6)$ and (3.1) hold, and

for all $t_1$ large enough. If

then (E) holds the property $\bar{A}$.

Proof. Let the positive solution of (E) be $x(t)$ and proceed as in Theorem 3.2; we verified that $\mu(t)$ belongs to either $S_0$ or $S_2$. If $\mu(t) \in S_2$, then $w(t) = \eta_1(t)\mu'(t)$ satisfies (3.7). The condition (3.10) imply, $w(t)\to\infty$ as $t\to\infty$. If not, then $w(t)\to M$ as $t\to \infty$. Integrating (3.5), we get

Integrating once more, we have

which contradicts with (3.10) and we conclude that $w(t)\to \infty$ as $t\to\infty$. Therefore, by setting $v = w(\tau(t))$, we get

Taking $\limsup$ on both sides contradicts (3.11). □

Next, if $\mu(t)\in S_0$, then proceeding by the proof of Theorem 3.2, we verify that $\lim_{t\to\infty}\dfrac{y(t)}{\Omega(t)} = 0$. □

Remark 3.5. Theorems 3.2 and 3.4 are applicable for

with $0\le \alpha\le 1 \; \& \; \alpha > 1$.

Remark 3.6. The integral criteria (3.3) and (3.11) of Theorems 3.2 and 3.4 provide better results than the one term integral criteria that are usually used.

4.   Example

Example 4.1. Consider the equation

with $a > 0$ and $b > 0$.

The condition (H6) holds. Moreover $\Omega_1(t) = \Omega_2(t) = \dfrac{1}{t}, \Omega(t) = \Omega_*(t) = \dfrac{1}{2t^2}, \tau(t) = \dfrac{t}{2}, \eta_1(t) = \eta_2(t) = \dfrac{1}{2}$, $Q_1(t) = Q_2(t) = 2t, Q(t) = 2t^2, g(u) = u$ and $h(u) = \arctan(u)$. The condition (3.1) becomes

Therefore, condition (3.1) holds. The condition (3.8) becomes

Therefore, condition (3.8) holds. The condition (3.9) becomes

By integrating the above equation and applying the limits, we get

and

Therefore, condition (3.8) holds if $a\bigg(\dfrac{1}{2}+\dfrac{1}{4}\log 2\bigg) > 1$. Therefore by Corollary 3.3, any nonoscillatory solution $x(t)$ of (4.1) satisfies

provided that

The results were also applicable when $\tau(t)\equiv t$.

Example 4.2. Consider the equation

with $a > 0$ and $b > 0$.

Clearly the condition (3.1) holds and (3.9) becomes

that is,

Integrating, we get

which implies

Therefore, the condition (3.9) holds if $a > \dfrac{1}{2}$. Hence, any nonoscillatory solution $x(t)$ of (4.2) satisfies

provided that $a > \dfrac{1}{2}$.

5.   Summary

In this paper, we looked at the oscillatory and asymptotic properties of third-order differential equations with positive and negative terms in the noncanonical case. Because the noncanonical case is hard, we turned (E) into a canonical representation by using a strongly noncanonical operator that only works with two classes of eventually positive solutions out of four classes of eventually positive solutions. This paper has better results than the ones that are already out there. The main results are illustrated through the examples.

Acknowledgments

The authors are thankful for the reviewer's valuable comments and suggestions that improved the quality of the paper.

Conflict of interest

The authors do not have any conflict of interest.