Abstract
This paper deals with the asymptotic and oscillatory behaviour of third-order non-linear differential equations with mixed non-linear neutral terms and a canonical operator. The results are obtained via utilising integral conditions as well as comparison theorems with the oscillatory properties of first-order advanced and/or delay differential equations. The proposed theorems improve, extend, and simplify existing ones in the literature. The results are illustrated by two numerical examples.
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1 Introduction
In this work, we aim to investigate the asymptotic and oscillatory behaviour of all solutions of the non-linear third-order differential equations with mixed neutral terms of the form:
where \(y(\zeta )=x(\zeta )+p_1(\zeta )x^\nu (\sigma (\zeta ))-p_2(\zeta )x^\kappa (\sigma (\zeta ))\). Throughout the paper, we always assume that
-
(A1)
\(\alpha , \nu , \kappa , \gamma \) and \(\lambda \) are the ratios of positive odd integers with \(\alpha \ge 1\),
-
(A2)
\(a, p_{1}, p_{2}, p\) and \(q\in C([\zeta _0,\infty ),{\mathbb {R}}^{+})\) with \(a'(\zeta ) \ge 0\) for \(\zeta \ge \zeta _0\),
-
(A3)
\(\omega , \tau , \sigma \in C([\zeta _0, \infty ),{\mathbb {R}})\) such that \(\tau (\zeta ),\sigma (\zeta ) \le \zeta \), \(\omega (\zeta ) \ge \zeta \) and \(\tau (\zeta ),\sigma (\zeta ) ,\omega (\zeta ) \rightarrow \infty \) as \(\zeta \rightarrow \infty \),
-
(A4)
\(h(\zeta ) = \sigma ^{-1}(\tau (\zeta )) \le \zeta \), \(h^*(\zeta ) = \sigma ^{-1}(\omega (\zeta )) \ge \zeta \) with \(h(\zeta ) \rightarrow \infty \) as \(\zeta \rightarrow \infty \),
-
(A5)
\(A(\zeta ,\zeta _0)=\int _{\zeta _0}^\zeta \frac{1}{a^{1/\alpha }(s)} ds\) with \( A(\zeta ,\zeta _0) \rightarrow \infty \) as \(\zeta \rightarrow \infty \).
A solution of Eq. (1.1) is a function \(x(\zeta )\) which is continuous on \([T_x, \infty ), T_x \ge \zeta _0\) and satisfies Eq. (1.1) on \([T_x, \infty )\). The solutions which are vanishing identically in some neighborhood of infinity will be excluded from our consideration. Such a solution of Eq. (1.1) is said to be oscillatory if it has arbitrarily large zeros, and to be nonoscillatory otherwise. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
A variety of physical and technological issues raise the question of developing a mathematical model that describes a specific process or structure. It is known that most differential equations, such as those used to model real-life processes, may not have closed-form solutions. This led to a new branch of the theory of differential equations, namely, qualitative theory. In particular, we are especially interested in the study of the oscillatory behaviour of some classes of functional differential equations. Over past years, the oscillation theory of functional differential equations has received much attention since it has a great number of applications in engineering and natural sciences, see, e.g., [1,2,3, 7,8,9, 20, 30, 40, 42] and the references cited therein. For example, third-order differential equations appear in a variety of real-world problems, such as in the study of curved beam deflection, scattering cross-section, steam turbine regulation, control of a flying apparatus in cosmic space, entry-flow phenomenon, and so on; see, e.g., [22, 40]. Danziger and Elemergreen [22] discovered a class of third-order linear differential equations by observing the thyroid-pituitary interaction over time. The governing equations that describe the variation of thyroid hormone with time are as follows:
where \(x(\zeta )\) is the concentration of thyroid hormone at time \(\zeta \) and \(a_{1}, a_{2}, a_{3}, l\) and c are constants.
Apart from this, neutral delay differential equations arise when lossless transmission lines are employed to interconnect switching circuits in high-speed computers, see [30].
In the recent years, some authors considered the special cases of Eq. (1.1), that is, \(p(\zeta )=0\), or, \(\alpha =1\), or \(\nu =\kappa =1\), or \(\lambda =1\), or \(\gamma =1\), see, e.g., [11, 12, 14,15,16,17,18, 21, 23, 24, 26, 27, 29, 32, 38, 39, 43,44,45] and references cited therein. In particular, Grace and Jadlovska [25] established several oscillation theorems for the odd-order neutral delay differential equation
where \(n\ge 3\) is an odd natural number, \(0\le p_{1}(\zeta )<1\) and the delay terms \(\tau ,\sigma \) are non-decresaing. This paper’s contribution is that it employs the comparison technique to provide conditions that only ensure the oscillation of the aforementioned problem. In [14], Chatzarakis and Grace considered the coupled of third-order neutral differential equation
where \(y(\zeta )= x(\zeta )\pm p_{1}(\zeta )x^{\beta }(\sigma (\zeta ))\), \(0\le p_{1}(\zeta )<\infty \), \(\alpha \ge 1\) and \(\sigma (\zeta )\) is strictly increasing. By using the comparison method, their two main conclusions (Theorems 1 and 2) guarantee that every solution of the aforementioned equations either oscillates or converges to zero.
Therefore, we aim here to initiate the study of the oscillation problem of (1.1) with either \(\nu < \kappa \le 1\) or \(\nu <1\) and \(\kappa >1\), via comparison with the known oscillatory behaviour of first order equations. The method we employ here in this work has naturally a partial resemblance of the works [14, 24, 28], however the results and most arguments are quite different due to more general nature of Eq. (1.1). The obtained results improve and correlate many of the known oscillation criteria existing in the literature, even for the case of Eq. (1.1) with \(p_{1}(\zeta )=0\), or \(p_{2}(\zeta )=0\), or \( p_{1}(\zeta )=p_{2}(\zeta )=0\).
To make it easier to read, we simplify our notations here: for \(b\in C\left( [\zeta _0, \infty ),{\mathbb {R}}_{+}\right) \),
and
2 Some Preliminaries Lemmas
In order to prove our results later, we have replicated some lemmas that are required.
Lemma 2.1
Let \(q: [\zeta _0, \infty ) \rightarrow {\mathbb {R}}^+\), \(g: [\zeta _0, \infty ) \rightarrow {\mathbb {R}}\) and \(f: {\mathbb {R}} \rightarrow {\mathbb {R}}\) are continuous functions, f is non-decreasing and \(x f(x) > 0\) for \(x \ne 0\) and \(g(\zeta ) \rightarrow \infty \) as \(\zeta \rightarrow \infty \). If
-
(I)
the first order delay differential inequality (i.e., \(g(\zeta ) \le \zeta \))
$$\begin{aligned} y'(\zeta )+q(\zeta )f\big (y(g(\zeta ))\big ) \le 0 \end{aligned}$$has an eventually positive solution, then so does the corresponding delay differential equation.
-
(II)
the first order advanced differential inequality (i.e., \(g(\zeta ) \ge \zeta \))
$$\begin{aligned} y'(\zeta )-q(\zeta )f\big (y(g(\zeta ))\big ) \ge 0 \end{aligned}$$has an eventually positive solution, then so does the corresponding advanced differential equation.
Proof
This Lemma is an extension of known results in [10, Lemma 2.3] and [41, Corollary 1] and hence the proof is omitted. \(\square \)
Lemma 2.2
[31] If \({\mathcal {X}}\) and \({\mathcal {Y}}\) are non-negative, then
and
where equalities hold if and only if \({\mathcal {X}} = {\mathcal {Y}}\).
Lemma 2.3
(Young’s Inequality) [31] If \({\mathcal {X}}\), Y be nonnegative real numbers and if \(m, n > 1\) are real numbers such that \(\frac{1}{n} + \frac{1}{m} = 1\). Then
Equality holds if and only if \({\mathcal {X}}^n\)=\({\mathcal {Y}}^m\).
3 The Case When \(\nu <1\) and \(\kappa >1\)
In this section, we present some oscillation criteria for Eq. (1.1) when \(\nu <1\) and \(\kappa >1\).
Theorem 3.1
Let (A1) – (A5) hold with \(\nu <1\) and \(\kappa >1\). Furthermore, assume that
-
(A6)
there exists a function \(b \in C\left( [\zeta _0, \infty ),{\mathbb {R}}_{+}\right) \) such that
$$\begin{aligned} \lim _{\zeta \rightarrow \infty } [g_{1}(\zeta )+g_{2}(\zeta )] = 0 \end{aligned}$$(3.1)
and
-
(A7)
there exist non-decreasing functions \(\mu (\zeta ),\ \pi (\zeta ) \in C([\zeta _0, \infty ), {\mathbb {R}})\) such that \(\mu _{1}(\zeta ) = \mu (\zeta )<\zeta \), \(\mu _{2}(\zeta )=\mu \big (\mu (\zeta )\big )\) with \(\rho (\zeta ) = h^{*}\big (\mu _{2}(\zeta )\big ) >\zeta \),
$$\begin{aligned} \tau (\zeta ) \le \pi (\zeta ) \le \zeta \ \ for \ \zeta \ge \zeta _{0} \end{aligned}$$(3.2)
hold. If there exist numbers \(\theta _{1},\ \theta _{2} \in (0,1)\) such that the delay differential equations
and the advanced differential equation
are oscillatory, then every solution of Eq. (1.1) is oscillatory or converges to zero.
Proof
Suppose \(x(\zeta )\) is a non-oscillatory solution of Eq. (1.1) with \(x(\zeta ) > 0\) and \(\lim _{\zeta \rightarrow \infty } x(\zeta ) \ne 0\) for \(\zeta \ge \zeta _0\). Therefore, \(x\big (\tau (\zeta )\big )>0\), \(x\big (\sigma (\zeta )\big )>0\) and \(x\big (\omega (\zeta )\big )> 0\) for \(\zeta \ge \zeta _1\) for some \(\zeta _1 >\zeta _0\). It follows from Eq. (1.1) that
Hence, \(a(\zeta ) \big (y''(\zeta )\big )^\alpha \) is decreasing and of one sign, that is, there exists a \(\zeta _2 >\zeta _1\) such that \(y''(\zeta )> 0\) or \(y''(\zeta ) < 0\) for \(\zeta \ge \zeta _2\). We shall distinguish the following four cases:
Case 1: Since \(y'''(\zeta ) < 0\) and \(y''(\zeta ) < 0\), then a constant \(K>0\) exists such that \(y''(\zeta ) \le \frac{-K^{\frac{1}{\alpha }}}{a^{\frac{1}{\alpha }}(\zeta )}<0\) for \(\zeta \ge \zeta _{3}>\zeta _{2}\), which on integration from \(\zeta _{3}\) to \(\zeta \) gives
Letting \(\zeta \rightarrow \infty \) and using \((A_{5})\), we get \(\lim _{\zeta \rightarrow \infty }y'(\zeta )=-\infty \). Therefore, \(y'(\zeta )<0\). But conditions \(y''(\zeta )<0\) and \(y'(\zeta )<0\) imply that \(y(\zeta )<0\), which contradicts our assumption \(y(\zeta )>0\).
Case 2: For this case we have following two sub-cases:
Case \(2_1\): Let \(y'(\zeta )<0\) for \(\zeta \ge \zeta _2\). This case is excluded because of the choice \(\lim _{\zeta \rightarrow \infty } x(\zeta ) \ne 0\).
Case \(2_2\): Let \(y'(\zeta )>0\) for \(\zeta \ge \zeta _2\). From the associated function \(y(\zeta )\), we have
or,
If we apply (2.1) to \(\left[ b(\zeta )x(\sigma (\zeta ))-p_2(\zeta )x^\kappa (\sigma (\zeta ))\right] \) with \(\varphi =\kappa >1\), \({\mathcal {X}}=p_2^{\frac{1}{\kappa }}(\zeta )x(\sigma (\zeta ))\) and \({\mathcal {Y}}= \left( \frac{1}{\kappa } b(\zeta )p_2^{\frac{-1}{\kappa }}(\zeta )\right) ^{\frac{1}{\kappa -1}}\), we get
Similarly, if we apply (2.2) to \(\left[ p_1(\zeta )x^\nu (\sigma (\zeta ))-b(\zeta )x(\sigma (\zeta ))\right] \) with \(\varphi =\nu <1\), \({\mathcal {X}}=p_1^{\frac{1}{\nu }}(\zeta )x(\sigma (\zeta ))\) and \({\mathcal {Y}} = \left( \frac{1}{\nu } b(\zeta )p_1^{\frac{-1}{\nu }}(\zeta )\right) ^{\frac{1}{\nu -1}}\), we get
Thus, from (3.7), we see that
Since \(y(\zeta )>0\) is increasing, a constant \({\mathcal {C}} > 0\) for \(\zeta _{3}>\zeta _{2}\) exists such that \(y(\zeta ) \ge {\mathcal {C}}\) for \(\zeta \ge \zeta _{3}\) and so, we have
Now, in view of (3.1), a constant \(c \in (0,1)\) exists such that
Thus, we have
Since \(y'(\zeta )>0\), then the last inequality can be written as
Because \(y''(\zeta )>0\) and \(y'(\zeta ) > 0\) for \(\zeta \ge \zeta _3\), then following [1, Lemma 2.2.3], a constant \(k \in (0,1)\) exists such that
Integrating this inequality from \(\zeta _{3}\) to \(\zeta \), we get
Using this inequality in (3.12) and setting \(W(\zeta ) = a(\zeta )(y''(\zeta ))^\alpha \), we have
where \(\theta _1=(ck)^{\gamma } \in (0,1)\). It follows from Lemma 2.1 (I) that the corresponding differential Eq. (3.3) also has a positive solution, which is a contradiction.
Case 3: For \(y(\zeta )<0\), we consider
or,
or,
and so,
From \({\hat{y}}''(\zeta ) < 0\) and for \(\zeta _{2}\le u\le v\), it follows that
In the above inequality, we let \(u = \tau (\zeta )\) and \(v = \pi (\zeta )\), then
According to [1, Lemma 2.2.3], a constant \(\theta _3 \in (0,1)\) exists such that
Combining (3.16) in (3.15), we have
Using (3.17) in (3.14), we get
where \(Y(\zeta ):=-a(\zeta )\big ({\hat{y}}''(\zeta )\big )^\alpha \) and \(\theta _{2}=(\theta _{3})^{\gamma /\kappa }\). As a result of Lemma 2.1(I), the differential Eq. (3.4) also has a positive solution, which is a contradiction.
Case 4: Clearly, we see that \({\hat{y}}''(\zeta ) > 0\). In this case we have \({\hat{y}}'(\zeta ) >0\). From Case 3, it follows that
Integrating (3.19) from \(\mu (\zeta )\) to \(\zeta \), we have
Therefore,
An integration from \(\mu (\zeta )\) to \(\zeta \) yields
Consequently, \({\hat{y}}(\zeta )\) is a positive solution of the advanced differential inequality
As a result of Lemma 2.1(II), the differential Eq. (3.5) also has a positive solution, which is a contradiction. This completes the proof. \(\square \)
Next, we have the following corollary that follows immediately from Theorem 3.1.
Corollary 3.1
Let \((A1)-(A5)\) hold with \(\nu <1\) and \(\kappa >1\). Furthermore, assume that there exists a function \(b \in C([\zeta _{0}, \infty ),{\mathbb {R}}_{+})\) such that condition (3.1), and non-decreasing functions \(\mu (\zeta ),\eta (\zeta ) \in C([\zeta _{0}, \infty ),{\mathbb {R}})\) such that condition (3.2) are satisfied. If
and
then every solution of Eq. (1.1) is oscillatory or converges to zero.
Proof
Suppose \(x(\zeta )\) is a non-oscillatory solution of Eq. (1.1) with \(x(\zeta ) > 0\) and \(\lim _{\zeta \rightarrow \infty } x(\zeta ) \ne 0\) for \(\zeta \ge \zeta _0\). Therefore, \(x\big (\tau (\zeta )\big )>0\), \(x\big (\sigma (\zeta )\big )>0\) and \(x\big (\omega (\zeta )\big )> 0\) for \(\zeta \ge \zeta _1\) for some \(\zeta _{1} >\zeta _{0}\). Proceeding as in the proof of Theorem 3.1, we arrive at (3.13) for \(\zeta \ge \zeta _{3}\), (3.18) for \(\zeta >\zeta _{2}\), and (3.20) for \(\zeta >\zeta _{2}\) respectively. Upon using the fact that \(\tau (\zeta )\le \zeta \), and \(W(\zeta )= a(\zeta )(y''(\zeta ))^{\alpha }\) is positive and decreasing, we have \(W(\tau (\zeta ))\ge W(\zeta )\) and hence inequality (3.13) can be written as
that is,
which on integration from \(\zeta _{4}\) to \(\zeta \) gives
and letting \(\zeta \rightarrow \infty \), we get a contradiction to (3.21). The reminder of proof follows from the inequalities (3.18) and (3.20), and noting that \(\pi (\zeta )\le \zeta \) and \(\rho (\zeta )\ge \zeta \), respectively. Hence, we omit the details. \(\square \)
The following example illustrate the applicability of Corollary 3.1.
Example 3.1
Consider
where \(\alpha =1\), \(\nu =\frac{3}{7}\), \(\kappa =\frac{9}{7}\), \(\gamma =\frac{5}{7}\), \(\lambda >\frac{9}{7}\), \(a(\zeta ) =\zeta \), \(p_{1}(\zeta )=p(\zeta )=\frac{1}{\zeta }\), \(p_{2}(\zeta )=\zeta \), \(q(\zeta )=1\), \(\sigma (\zeta )=\frac{\zeta }{2}\), \(\tau (\zeta )=\frac{\zeta }{8}\) and \(\omega (\zeta )=2\zeta \). Also, \(h(\zeta )=\frac{\zeta }{4}\), \(h(\zeta )=4\zeta \) \(Q(\zeta )=\frac{1}{\left( \frac{\zeta }{4}\right) ^{5/9}}\) and \(P(\zeta )=\frac{(2\zeta )^{\frac{5}{9}}}{\zeta }\) . Letting \(b=1\), it is not difficult to see that (3.1) holds. We let \(\mu (\zeta )=\frac{3}{4} \zeta \), then \(\rho (\zeta ) = \frac{9}{4} \zeta \). Since \(A(\zeta , \zeta _{0})=\int _{1}^{\zeta }\frac{ds}{s}=\ln {\zeta }\), then all conditions of Corollary 3.1 are met. Indeed, from (3.21), (3.22) and (3.23), we have
and
respectively. Thus, every solution to (3.24) is oscillatory or else converges to zero.
Remark 3.1
We may note that [24, Theorem 2.1] is not applicable to (3.24) due to the restriction that \(p_{1}(\zeta )=0=p(\zeta )\) and \(0<\kappa \le 1\). Apart from this, suppose that \(p_{1}(\zeta )=0=p(\zeta )\) in (3.24), then it is not difficult to see that Theorem 3.1 generalised/improved the results reported in [24]. A similar observation can be made for the papers [11, 13, 23, 25, 39].
Now, we shall present some special cases of Theorem 3.1. First we consider the case when \(p_1(\zeta )=p_2(\zeta )=0\), i.e., for the non-neutral equation
Accordingly, Theorem 3.1 can be expressed in the following form:
Corollary 3.2
Let assumptions \((A1)-(A5)\) hold. If there exists a number \(\theta _{1} \in (0,1)\) such that the first order delay differential Eq. (3.3) is oscillatory, then every solution of Eq. (3.25) is oscillatory or converges to zero.
Following that, we consider the case when \(p_2(\zeta )=0\), i.e., the neutral equation
For the Eq. (3.26) with \(0 <\nu \le 1\), we have the following new result:
Corollary 3.3
In addition to the hypotheses of Corollary 3.2, assume that \(\lim _{\zeta \rightarrow \infty } p_{1}(\zeta )=0\). Then every solution of Eq. (3.26) is oscillatory or converges to zero.
For complete oscillation criteria of Eq. (3.25), we have the following result.
Theorem 3.2
Let conditions \((A1)-(A5)\) hold. Assume that there exists a non-decreasing function \(\eta (\zeta ) \in C([\zeta _{0}, \infty ),{\mathbb {R}})\) such that
If there exist numbers \(\theta _{1} \in (0,1)\) such that the first order delay differential Eq. (3.3), and
are oscillatory, then Eq. (3.25) is oscillatory.
Proof
Let \(x(\zeta )\) be a non-oscillatory solution of Eq. (3.25), say \(x(\zeta ) > 0\), \(x(\tau (\zeta ))>0\) and \(x(\omega (\zeta ))>0\) for \(\zeta \ge \zeta _1\) for some \(\zeta _1 >\zeta _0\). Hence, \(a(\zeta )\big (x''(\zeta )\big )^\alpha \) is of one sign, that is, there exists a \(\zeta _2 \ge \zeta _1\) such that \(x''(\zeta )> 0\) or \(x''(\zeta ) < 0\) for \(\zeta \ge \zeta _2\). We shall distinguish the following two cases:
Case 1: Since \(x''(\zeta )\) is non-increasing and negative, a constant \({\mathcal {C}}>0\) exists for \(\zeta \ge \zeta _{3}>\zeta _{2}\) such that
Integrating the last inequality from \(\zeta _{3}\) to \(\zeta \), we get
Letting \(\zeta \rightarrow \infty \) and then using (A5), we get \(x'(\zeta )\rightarrow -\infty \). Therefore, \(x'(\zeta )<0\) together with \(x''(\zeta )<0\) implies that \(x(\zeta )<0\), a contradiction.
Case 2: For this case, we have following two subcases:
Case \(2_{1}\ (x'(\zeta )>0)\): This case can be follows from the proof of Case \(2_{2}\) of Theorem 3.1 and hence we omit the details.
Case \(2_{2}\ (x'(\zeta )<0)\): One can easily see that \(x(\zeta )\) satisfies
Therefore,
which implies that
Integrate this inequality from \(\zeta \) to \(\eta (\zeta )\) yields
that is,
Using this inequality in (3.25), we have
where \(X(\zeta )= a(\zeta )\big (x''(\zeta )\big )^\alpha \). It follows that the rest of the proof is similar to those mentioned above, so it is omitted. This completes the proof. \(\square \)
Finally, we consider the case when \(p_1(\zeta )=0\), i.e., the neutral equation
Now, we have the following oscillation result for Eq. (3.29).
Theorem 3.3
Let the hypotheses of Theorem 3.1 hold with \(p_1(\zeta )=0\). Then every solution of Eq. (3.29) is oscillatory or converges to zero.
Proof
Suppose \(x(\zeta )\) is a non-oscillatory solution of (3.29) with \(x(\zeta ) > 0\) and \(\lim \nolimits _{\zeta \rightarrow \infty } x(\zeta ) \ne 0\) for \(\zeta \ge \zeta _0\). Therefore, \(x\big (\tau (\zeta )\big )>0\), \(x\big (\sigma (\zeta )\big )>0\) and \(x\big (\omega (\zeta )\big )> 0\) for \(\zeta \ge \zeta _1\) for some \(\zeta _{1} > \zeta _{0}\). Following the same procedure used for the proof of Theorem 3.1, we obtain Cases 1 through 4.
If \(y(\zeta ) =x(\zeta )-p_2(\zeta ) x^\kappa \big (\sigma (\zeta )\big )\) is positive, then \(x(\zeta ) \ge y(\zeta )\) and so, Eq. (3.29) becomes
and we may apply Corollary 3.2. For the case when \(y(\zeta ) < 0\), we apply the Theorem 3.1 when the two cases, Case 3 and Case 4 hold. Therefore, we omit the details. \(\square \)
Remark 3.2
Theorem 3.3 improved or generalised the results reportted in [11, 23,24,25].
4 The Case When \(\nu <\kappa \le 1\)
In this section, we present some oscillation criteria for Eq. (1.1) when \(\nu <\kappa \le 1\).
Theorem 4.1
Let \((A1)-(A5)\) hold with \(\nu <\kappa \le 1\). Assume that all the hypotheses of the Theorem 3.1 hold, and the condition (3.1) is replaced by
Then the conclusion of Theorem 3.1 holds.
Proof
Suppose \(x(\zeta )\) is a non-oscillatory solution of (1.1) with \(x(\zeta ) > 0\) and \(\lim _{\zeta \rightarrow \infty } x(\zeta ) \ne 0\) for \(\zeta \ge \zeta _0\). Therefore, \(x\big (\tau (\zeta )\big )>0\), \(x\big (\sigma (\zeta )\big )>0\) and \(x\big (\omega (\zeta )\big )> 0\) for \(\zeta \ge \zeta _1\) for some \(\zeta _{1} > \zeta _{0}\). Following the same procedure used for the proof of Theorem 3.1, we obtain Cases 1 through 4.
First, we consider Case 1 and 2. Clearly, we see that \(y'(\zeta )>0\) for \(\zeta \ge \zeta _2\). It is not difficult to see that
Setting \(n=\frac{\kappa }{\nu }>1\), \({\mathcal {X}}= x^\nu \big (\sigma (\zeta )\big )\), \({\mathcal {Y}}= \frac{\nu }{\kappa } \left( \frac{p_1(\zeta )}{p_2(\zeta )}\right) \) and \(m = \frac{\kappa }{\kappa -\nu },\) in \([p_1(\zeta )x^\nu \big (\sigma (\zeta )\big )-p_2(\zeta )x^\kappa \big (\sigma (\zeta )\big )]\), we have
Applying (2.3) to \([p_1(\zeta )x^\nu \big (\sigma (\zeta )\big )-p_2(\zeta )x^\kappa \big (\sigma (\zeta )\big )]\), we obtain
Thus, using the last inequality to \(x(\zeta )=y(\zeta )-p_1(\zeta )x^\nu \big (\sigma (\zeta )\big )+p_2(\zeta )x^\kappa \big (\sigma (\zeta )\big )\), we see that
Due to non-decreasing of \(y(\zeta )>0\), we can find a constant \({\mathcal {C}} > 0\) such that \(y(\zeta ) \ge {\mathcal {C}}\), therefore, we have
Now, in view of (4.1), we can find \(\epsilon \in (0,1)\) such that
It follows that the remainder of the proof is similar to Theorem 3.1. This completes the proof. \(\square \)
Remark 4.1
We may note that the results similar to Corollary 3.1-Corollary 3.2 can also be extracted from Theorem 4.1. The details are left to the reader.
The following example illustrate the applicability of Theorem 4.1.
Example 4.1
Consider
where \(\alpha =1\), \(\nu =\frac{3}{7}\), \(\kappa =\frac{5}{7}\), \(\gamma =\frac{5}{7}\), \(\lambda >1\), \(a(\zeta ) =\frac{1}{\zeta }\), \(p_{1}(\zeta )=\frac{1}{\zeta }\), \(q(\zeta )=\frac{1}{\zeta ^{2}}=p(\zeta )\), \(p_{2}(\zeta )=1\), \(\sigma (\zeta )=\frac{\zeta }{2}\), \(\tau (\zeta )=\frac{\zeta }{4}\) and \(\omega (\zeta )=2\zeta \). It is not difficult to see that (4.1) holds. We let \(\mu (\zeta )=\frac{3}{4} \zeta \), then \(\rho (\zeta ) = \frac{9}{8} \zeta \). Since \(A(\zeta , \zeta _{0})=\int _{1}^{\zeta } s\ ds \simeq \frac{\zeta ^{2}}{2}\), then all conditions of Theorem 4.1 are met, and thus every solution of (4.5) is either oscillatory or converges to zero.
Remark 4.2
We may note that [24, Theorem 2.1] is not applicable to (3.24) due to the restriction that \(p_{1}(\zeta )=0\) and \(p(\zeta )=0\). Apart from this, suppose that \(p_{1}(\zeta )=0=p(\zeta )\) in (3.24), then it is not difficult to see that Theorem 4.1 generalised the results reported in [24]. A similar observation can be made for the papers [11, 13, 23, 25, 39].
5 Concluding Remark
In this paper, with the help of a novel comparison technique with the behaviour of first order delay and/or advanced differential equations as well as an integral criterion, several results for the oscillation and asymptotic behaviour of solutions of Eq. (1.1) are presented. As an application of the main results, Corollary 3.1, as well as some examples, are then presented. Articles [13, 16, 17, 24, 27, 32, 33, 43,44,45] are concerned with the asymptotic behaviour and oscillation of solutions to third/odd order neutral differential equations, which is a topic very close to our investigations but does not compliment our findings. We present our findings in a way that is essentially new and has high generality. Our findings are also easily applicable to higher-order equations of the form
where \(n \in {\mathbb {N}}\) and \(y(\zeta )=x(\zeta )+p_1(\zeta ) x^\nu \big (\sigma (\zeta )\big )-p_2(\zeta ) x^\kappa \big (\sigma (\zeta )\big )\). The details are left to the reader.
Secondly, in this work, we have considered third-order non-linear differential equations with mixed neutral terms in the sense of non-linearity of function, that is, sublinear and superlinear neutral terms. Therefore, following the work [39], we raise the question of whether it would be interesting to extend this work to third-order non-linear differential equations with mixed neutral terms, that is, the neutral term contains both retarded and advanced arguments. The details are left to the reader.
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
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Acknowledgements
The authors would like to thank the editor and the five anonymous reviewers for their constructive comments and suggestions, which helped us to improve the manuscript considerably. J. Alzabut is thankful to Prince Sultan University and OSTİM Technical University for their endless support.
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Alzabut, J., Grace, S.R., Santra, S.S. et al. Asymptotic and Oscillatory Behaviour of Third Order Non-linear Differential Equations with Canonical Operator and Mixed Neutral Terms. Qual. Theory Dyn. Syst. 22, 15 (2023). https://doi.org/10.1007/s12346-022-00715-6
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DOI: https://doi.org/10.1007/s12346-022-00715-6
Keywords
- Non-linear differential equations
- Oscillation
- Asymptotic behavior
- Canonical operator
- Mixed neutral terms