Oscillatory and asymptotic behavior of third-order neutral delay differential equations with distributed deviating arguments

1.   Introduction

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

where $z(t) = x(t)+p(t)x(\tau(t))$, $t\geq t_0 > 0$, $0\leq a < b$. We also assume that the following conditions are satisfied:

● (H1) $r_1(t)$, $r_2(t)$, $p(t)\in C([t_0, \infty), \mathbb{R})$, $q(t, \xi)\in C([t_0, \infty)\times[a, b], [0, \infty))$, $r_1(t) > 0$, $r_2(t) > 0$ and $p(t)\geq1$ with $p(t)\not\equiv 1$;

● (H2) $\tau(t)\in C([t_0, \infty), \mathbb{R})$ is invertible, $\tau(t)\leq t$ and $\lim_{t\rightarrow\infty}\tau(t) = \infty$;

● (H3) $\sigma(t, \xi)\in C([t_0, \infty)\times[a, b], \mathbb{R})$ is non-increasing for $\xi$ and $\lim\inf_{t\rightarrow\infty}\sigma(t, \xi) = \infty\ \text{for}\ \xi\in[a, b]$;

● (H4) $f(x)\in C(\mathbb{R}, \mathbb{R})$ is assumed to satisfy $xf(x) > 0$ and there exists a positive constant $K$ such that

● (H5) $\alpha_i$, $i = 1, \ 2, \ 3$ are ratios of positive odd integers.

By a solution of Eq. (1.1) we mean a function $x(t)\in C([T_y, \infty), \mathbb{R})$, $T_y\geq t_0$, which has $z(t)$, $r_2(t)(z'(t))^{\alpha_2}$, $r_1(t)((r_2(t)(z'(t))^{\alpha_2})')^{\alpha_1}\in C^1([T_y, \infty), \mathbb{R})$ and satisfies (1.1) on $[T_y, \infty)$. A solution $x(t)$ of (1.1) is said to be proper if it exists on the interval $[T_y, \infty)$ and satisfies the condition

Our attention is restricted to these solutions and we make the standing hypothesis that (1.1) admits such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on $[T_y, \infty)$ and otherwise it is called non-oscillatory, i.e., the solution is positive or negative eventually. Eq. (1.1) is said to be oscillatory if all its solutions are oscillatory.

Main results of this paper are organized into three parts in accordance with different assumptions on the coefficients $r_1(t)$ and $r_2(t)$. In Section 2, oscillation results of (1.1) are established in the case

In Section 3, some new oscillation criteria for (1.1) are obtained in the case

By assuming that

Some oscillation theorems of (1.1) are given in Section 4. In order to illustrate the results reported in Sections 2, 3 and 4, we present some examples in Section 5.

During the last few decades, analysis of the oscillation and asymptotic behavior of solutions of third-order differential equations, difference equations and dynamic equations on time scales have experienced long-term interest and we refer the reader to the papers ^{[1,2,3,4,5,6,7,8,9,10,11,12]}. Due to the huge advantage of neutral differential equations in describing several neutral phenomena, there is of great scientific and academic values theoretically and practically for studying neutral differential equations. Hence, a large amount of research attention has been focused on the oscillation problem of third-order linear and nonlinear neutral differential equations in resent years; see, for example ^{[13,14,15,16,17]}, and the references are cited therein.

The third-order neutral differential equation

and its special cases have been studied by Şenel and Utku ^[8], Baculíková and Džurina ^[13], Jiang et al. ^[14,15], where $\int^\infty_{t_0}r^{-\frac{1}{\gamma}}(t)dt = \infty$, $0\leq p(t)\leq P < 1$. Candan ^[11], Došlá and Liška ^[16], and Li et al. ^[17] established some sufficient conditions for oscillation of the following class of third-order neutral differential equations

where ^[11] and ^[16] only considered the conditions $0\leq p(t)\leq P < 1$ and

and ^[17] also studied the cases

Recently, there has been an increasing interest in studying the oscillatory and asymptotic behavior of third-order neutral differential equations with distributed deviating arguments ^{[18,19,20,21,22,23,24]}. Elabbasy and Moaaz ^[19] considered the special cases of (1.1) and obtained several oscillation results under the assumption (1.2) and $0\leq p(t)\leq P < 1$. By using a new method different from the existing results, Tunç ^[22] established some new oscillation criteria for

where $p(t)\geq 1$, and the obtained results greatly enriched the oscillation theory.

It is clear that the above introduced equations are the special cases of (1.1), i.e., (1.1) can be transformed into these equations by letting the corresponding parameters being 1. To the best of our knowledge, there are few results in the literature which ensure that all solutions are either oscillatory or tends to zero monotonically for the third-order neutral differential equations with distributed deviating arguments under the conditions $p(t)\geq 1$, (1.3) or (1.4) holds, and the above mentioned results are inapplicable to these conditions. Motivated by Li et al. ^[17], Elabbasy and Moaaz ^[19] and Tunç ^[22], we consider Eq. (1.1) which is not studied in the past and utilize the Riccati transformation technique to establish several oscillation criteria for (1.1) by assume that $p(t)\geq 1$, (1.2), (1.3) or (1.4) holds. The results obtained in this paper improve and complement the related criteria reported in ^{[13,16,17,19,22]}. All functional inequalities considered here are assumed to hold eventually, that is, they are satisfied for all $t$ large enough.

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

where $\rho(t)$ will be explained later and $t_1$ is sufficiently large with $t_1\geq t_0$. Furthermore, assume that

where $\tau^{-1}(t)$ is the inverse function of $\tau(t)$. Then let

We will present the main contribution of this paper as follows.

2.   Oscillation criteria for the case (1.2)

In this section, we respectively consider the following two cases

and

We now begin with the case when (2.1) holds.

Theorem 2.1. Assume that conditions (H1)–(H5), (1.2), (1.5), (1.6) and (2.1) hold. Furthermore, assume that there exists a function $\rho(t)\in C^1([t_0, \infty), (0, \infty))$ such that for sufficiently large $t_* > t_2 > t_1\geq t_0$,

and

where

Then every solution of (1.1) is either oscillatory or tends to zero as $t\rightarrow\infty$.

Proof. Suppose that (1.1) has a non-oscillatory solution $x(t)$. Without loss of generality, we assume that there exists $t_1\geq t_0$ such that $x(t) > 0$, $x(\tau(t)) > 0$ and $x(\sigma(t, \xi)) > 0$ for $\xi\in[a, b]$ and $t\geq t_1$. Then from the definition of $z(t)$, we have $z(t) > 0$. Based on the condition (1.2), $z(t)$ satisfies the following two cases (see, for example ^[18,19]):

(Ⅰ) $z(t) > 0$, $z'(t) > 0$, $(r_2(t)(z'(t))^{\alpha_2})' > 0$ and $\big[r_1(t)((r_2(t)(z'(t))^{\alpha_2})')^{\alpha_1}\big]'\leq0$;

(Ⅱ) $z(t) > 0$, $z'(t) < 0$, $(r_2(t)(z'(t))^{\alpha_2})' > 0$ and $\big[r_1(t)((r_2(t)(z'(t))^{\alpha_2})')^{\alpha_1}\big]'\leq0$, for $t\geq t_1$.

Assume first that Case (Ⅰ) holds. Then we get

(2.5) can also be seen in ^[22]. Since $r_1(t)((r_2(t)(z'(t))^{\alpha_2})')^{\alpha_1}$ is non-increasing for $t\geq t_1$, it indicates that

We deduce from (2.6) that

and $z'(t)/\delta_2(t, t_1)$ is non-increasing for $t\geq t_1$. Therefore, we obtain

and

which yields that

for $\tau(t)\leq t$. Substituting (2.9) into (2.5), we have

Then there exists $t_2 > t_1$ such that $\sigma(t, \xi)\geq t_1$ and

Combining (1.1), (H3), (H4) and (2.10), we conclude that

Define a Riccati transformation $\omega(t)$ by

Clearly, $\omega(t) > 0$, and

Since $\sigma(t, \xi)\leq\tau(t)$ and $\big[r_1(t)((r_2(t)(z'(t))^{\alpha_2})')^{\alpha_1}\big]'\leq0$, we get $\tau^{-1}(\sigma_2(t))\leq t$ and

Combining (2.11), (2.13) and (2.14), we have

From (2.7), we get

Combining (2.12), (2.15) and (2.16), we obtain

In order to compute $z^{\alpha_3-\alpha_1\alpha_2}(\tau^{-1}(\sigma_2(t)))$, we consider the following two cases:

(1) $\alpha_1\alpha_2 > \alpha_3$. From (2.8), there exist $t_3 > t_2$ and $h_1 > 0$ such that $\tau^{-1}(\sigma_2(t))\geq t_2$ and

which implies that

where $m_1 = h^{\alpha_3-\alpha_1\alpha_2}_1$.

(2) $\alpha_1\alpha_2\leq\alpha_3$. Based on the fact that $z'(t) > 0$, there exists $h_2 > 0$ such that

which yields that

where $m_2 = h^{\alpha_3-\alpha_1\alpha_2}_2$.

Substituting (2.18) and (2.19) into (2.17), we have

Let

Applying the inequality (see ^[13])

together with (2.20), we get

Integrating (2.22) from $t_3$ to $t$, we obtain

for all sufficiently large $t$, which contradicts (2.3).

Secondly, assume that Case (Ⅱ) holds. Since $z'(t) < 0$ and $\tau(t)\leq t$, (2.5) yields that

which indicates that

for $t\geq t_2$. Using (1.1), (H4) and (2.23), we conclude that

By using a similar proof of [19,Lemma 2.2], we can obtain $\lim_{t\rightarrow\infty}x(t) = 0$ due to the condition (2.4). This completes the proof of Theorem 2.1.

Next, we turn our attention to the case when (2.2) holds.

Theorem 2.2. Assume that conditions (H1)–(H5), (1.2), (1.5), (1.6), (2.2) and (2.4) hold. Moreover, assume that there exists a function $\rho(t)\in C^1([t_0, \infty), (0, \infty))$ such that for sufficiently large $t_* > t_2 > t_1\geq t_0$,

where $\gamma(t)$ is defined as in Theorem 2.1. Then every solution of (1.1) is either oscillatory or tends to zero as $t\rightarrow\infty$.

Proof. Suppose that (1.1) has a non-oscillatory solution $x(t)$. Without loss of generality, we assume that there exists $t_1\geq t_0$ such that $x(t) > 0$, $x(\tau(t)) > 0$ and $x(\sigma(t, \xi)) > 0$ for $\xi\in[a, b]$ and $t\geq t_1$. As in the proof of Theorem 2.1, we have (2.11) and (2.16), and there exist two possible cases (Ⅰ) and (Ⅱ) for $z(t)$ (as those in the proof of Theorem 2.1).

Assume that Case (Ⅰ) holds. Define a Riccati transformation $\tilde{\omega}(t)$ by

Clearly, $\tilde{\omega}(t) > 0$, and since $\tau(t)\leq t$,

From (2.2), we get $\tau^{-1}(\sigma_2(t))\geq t\geq\tau(t)$, and then $z(\tau^{-1}(\sigma_2(t)))\geq z(\tau(t))$. Therefore, based on (2.16), we obtain

Substituting the later inequality, (2.18) and (2.19) into (2.27), we deduce that

By using the inequality (2.21) and (2.28), we conclude that

An integration of (2.29) from $t_3\ (t_3 > t_2)$ to $t$ leads to a contradiction to (2.25).

Secondly, assume that Case (Ⅱ) holds. Proceeding as in the proof of Case (Ⅱ) in Theorem 2.1, we arrive at the conclusion $\lim_{t\rightarrow\infty}x(t) = 0$. This completes the proof of Theorem 2.2.

Remark 2.1. With different choices of the function $\rho(t)$, one can derive a number of oscillation criteria for (1.1) from Theorems 2.1 and 2.2.

Remark 2.2. Our results in this section extend and improve those obtained by Tunç ^[22], and we can get some relevant results by using the technique presented in ^[22]. The established results here also complement and improve those in ^{[13,16,17,19]}, since the considered equations in these papers are special cases of (1.1) and our results can be applied to (1.1) in the case where $p(t)\geq1$.

3.   Oscillation criteria for the case (1.3)

In this section, we will establish some oscillation criteria for (1.1) under the assumption that (1.3) holds. Similarly as in Section 2, we start with the case when (2.1) is satisfied. Firstly, we define the following notations:

Furthermore, assume that

Then let

Theorem 3.1. Assume that conditions (H1)–(H5), (1.3), (1.5), (1.6), (2.1), (2.4) and (3.1) hold. Furthermore, assume that there exists $\rho(t)\in C^1([t_0, \infty), (0, \infty))$ such that (2.3) is satisfied for sufficiently large $t_* > t_2 > t_1\geq t_0$. If

where

then every solution of (1.1) is either oscillatory or tends to zero as $t\rightarrow\infty$.

Proof. Suppose that (1.1) has a non-oscillatory solution $x(t)$. Without loss of generality, we assume that there exists $t_1\geq t_0$ such that $x(t) > 0$, $x(\tau(t)) > 0$ and $x(\sigma(t, \xi)) > 0$ for $\xi\in[a, b]$ and $t\geq t_1$. Based on the condition (1.3), there exist three possible cases (Ⅰ), (Ⅱ) (as those in the proof of Theorem 2.1) and

(Ⅲ) $z(t) > 0$, $z'(t) > 0$, $(r_2(t)(z'(t))^{\alpha_2})' < 0$ and $\big[r_1(t)((r_2(t)(z'(t))^{\alpha_2})')^{\alpha_1}\big]'\leq0$, for $t\geq t_1$.

We firstly prove that Case (Ⅲ) holds. It is easy to verify that

holds under the condition

and in the proof of Theorem 2.1, we can see that $z(t)$ has properties (Ⅰ) and (Ⅱ). If $(r_2(t)(z'(t))^{\alpha_2})' < 0$, then we claim that $z'(t) > 0$. Otherwise, there exists a constant $M_1$ such that

Integrating the last inequality from $t_1$ to $t$, we have

Letting $t\rightarrow\infty$, we get $z(t)\rightarrow-\infty$, which contradicts the fact that $z(t) > 0$. Hence we conclude that $z'(t) > 0$ here.

Assume now that Cases (Ⅰ) and (Ⅱ) hold. Then we can obtain the conclusion of Theorem 3.1 by using the proof of Theorem 2.1.

Assume that Case (Ⅲ) holds. Since $r_1(t)((r_2(t)(z'(t))^{\alpha_2})')^{\alpha_1}$ is nonincreasing for $t\geq t_1$ and $(r_2(t)(z'(t))^{\alpha_2})' < 0$, we have

Letting $l\rightarrow\infty$, we get

that is

In view of $r^{\frac{1}{\alpha_2}}_2(t)z'(t)$ is non-increasing for $t\geq t_1$, we see that

and

which yields that

Substituting (3.6) into (2.5), we get

Then there exists $t_2 > t_1$ such that $\sigma(t, \xi)\geq t_1$ and

Define the function $\nu(t)$ by

Then $\nu(t) < 0$ for $t\geq t_2$. From (3.3) and (3.8), we obtain

Differentiating (3.8) and using (3.7) and (3.8), we get

From (1.5) and (3.5), we obtain

(3.4) implies that

Combining (3.10), (3.11) and (3.12), we get

due to $\tilde{\delta}_2(\tau^{-1}(\sigma_2(t)), t_1)\leq\tilde{\delta}_2(t, t_1)$. In order to compute $z^{\alpha_3-\alpha_1\alpha_2}(t)$, the case $\alpha_1\alpha_2\leq\alpha_3$ is the same as that in the proof of Theorem 2.1. We now compute the case $\alpha_1\alpha_2 > \alpha_3$. Applying the monotonicity of $z(t)/\tilde{\delta}_2(t, t_1)$ for $t\geq t_2$ derived from (3.5), there exists $h_3 > 0$ such that

which implies that

where $m_3 = h^{\alpha_3-\alpha_1\alpha_2}_3$. Combining (2.19), (3.13) and (3.14), we conclude that

Multiplying (3.15) by $\tilde{\delta}^{\alpha_1}_1(t)$ and integrating it from $t_2$ to $t$, we obtain

By using the inequality (2.21) and (3.16) with

we conclude that

due to (3.9), which contradicts (3.2). This completes the proof of Theorem 3.1.

With a similar proof to that of Theorems 2.2 and 3.1, we can obtain the following criteria for (1.1) by assuming that (2.2) is satisfied.

Theorem 3.2. Assume that conditions (H1)–(H5), (1.3), (1.5), (1.6), (2.2), (2.4) and (3.1) hold. Furthermore, assume that there exists $\rho(t)\in C^1([t_0, \infty), (0, \infty))$ such that (2.25) is satisfied for sufficiently large $t_* > t_2 > t_1\geq t_0$. If

where $\tilde{\gamma}(t)$ is defined as in Theorem 3.1, then every solution of (1.1) is either oscillatory or tends to zero as $t\rightarrow\infty$.

Proof. Suppose that (1.1) has a non-oscillatory solution $x(t)$. Without loss of generality, we assume that there exists $t_1\geq t_0$ such that $x(t) > 0$, $x(\tau(t)) > 0$ and $x(\sigma(t, \xi)) > 0$ for $\xi\in[a, b]$ and $t\geq t_1$. As in the proof of Theorem 3.1, we have (3.3) and (3.7), and there exist three possible cases (Ⅰ), (Ⅱ) and (Ⅲ) for $z(t)$.

Assume first that Cases (Ⅰ) and (Ⅱ) hold. We can obtain the conclusion of Theorem 3.2 by using the proof of Theorem 2.2.

Assume that Case (Ⅲ) holds. Define the function $\tilde{\nu}(t)$ by

Then $\tilde{\nu}(t) < 0$ for $t\geq t_1$. Since $\sigma_2(t)\geq\tau(t)$ and $(r_2(t)(z'(t))^{\alpha_2})' < 0$, we have

From (3.3), (3.18) and (3.19), we get

Since $r^{\frac{1}{\alpha_1}}_1(t)(r_2(t)(z'(t))^{\alpha_2})'$ is non-increasing for $t\geq t_1$, we obtain

Differentiating (3.18) and using (3.7), (3.18) and (3.20), we deduce that

Furthermore, substituting (3.12) and the definition of $\tilde{\gamma}(t)$ into (3.21), we conclude that

The rest of the proof is similar to that of Theorem 3.1 and we can get a contradiction to (3.17). So we omit it here. This completes the proof of Theorem 3.2.

4.   Oscillation criteria for the case (1.4)

In this section, we will establish some oscillation criteria for (1.1) under the assumption that (1.4) holds. Similarly as in the previous sections, we start with the case when (2.1) is satisfied.

Theorem 4.1. Assume that conditions (H1)–(H5), (1.4), (1.5), (1.6), (2.1), (2.4), (3.1) and (3.2) hold. Furthermore, assume that there exists $\rho(t)\in C^1([t_0, \infty), (0, \infty))$ such that for sufficiently large $t_* > t_2 > t_1\geq t_0$, one has (2.3). If

then every solution of (1.1) is either oscillatory or tends to zero as $t\rightarrow\infty$.

Proof. Suppose that (1.1) has a non-oscillatory solution $x(t)$. Without loss of generality, we assume that there exists $t_1\geq t_0$ such that $x(t) > 0$, $x(\tau(t)) > 0$ and $x(\sigma(t, \xi)) > 0$ for $\xi\in[a, b]$ and $t\geq t_1$. Based on the condition (1.4), there exist three possible cases (Ⅰ), (Ⅱ), (Ⅲ) (as those in the proof of Theorem 2.1) and (Ⅳ) $z(t) > 0$, $z'(t) < 0$, $(r_2(t)(z'(t))^{\alpha_2})' < 0$ and $\big[r_1(t)((r_2(t)(z'(t))^{\alpha_2})')^{\alpha_1}\big]'\leq0$, for $t\geq t_1$.

Assume first that Cases (Ⅰ), (Ⅱ) and (Ⅲ) hold. We can obtain the conclusion of Theorem 4.1 by using the proof of Theorem 3.1.

Assume that Case (Ⅳ) holds. Then there exists a constant $l\geq0$ such that

We claim that $l = 0$. Otherwise, assume that $l > 0$. We see that there exists $t_2 > t_1$ such that $\tau^{-1}(\sigma_1(t))\geq t_1$ and $z(\tau^{-1}(\sigma_1(t)))\geq l$, $t\geq t_2$. From (2.24), we obtain

Integrating (4.2) from $t_2$ to $t$, we have

which can be rewritten as

Integrating again from $t_2$ to $t$, we get

which yields that

Integrating the last inequality from $t_2$ to $t$, we obtain

which contradicts (4.1). This completes the proof of Theorem 4.1.

With a similar proof to that of Theorems 3.2 and 4.1, we can obtain the following criteria for (1.1) assuming that (2.2) is satisfied.

Theorem 4.2. Assume that conditions (H1)–(H5), (1.4), (1.5), (1.6), (2.2), (2.4), (3.1), (3.17) and (4.1) hold. Furthermore, assume that there exists a function $\rho(t)\in C^1([t_0, \infty), (0, \infty))$ such that for sufficiently large $t_* > t_2 > t_1\geq t_0$, one has (2.25). Then every solution of (1.1) is either oscillatory or tends to zero as $t\rightarrow\infty$.

Remark 4.1. The results in ^{[13,16,19,22]} are obtained only in the case (1.2) and they are inapplicable to (1.3) and (1.4). Hence, the main results of this paper complement and improve those in the literature.

5.   Examples

In this section, we will present some examples to illustrate the main results.

Example 4.1 For $t > k_1\geq1$, consider the third-order neutral delay differential equation with distributed deviating arguments

where $\alpha_1 = 1/3$, $\alpha_2 = 1$, $\alpha_3 = 5$, $a = k_1$, $b = k_1+1$, $f(x) = x^5$, $r_1(t) = t$, $r_2(t) = t^{\frac{1}{2}}$,

Choose $t_0 = t_1 = k_1$. Then we get $\alpha_1\alpha_2 < \alpha_3$, $5\leq p(t) < 6$,

Furthermore, we deduce that

It is easy to verify that

and

as $t\rightarrow\infty$, where we set $t_* = 4k_1+3$. Therefore, conditions (H1)–(H5), (1.3), (1.5), (1.6), (2.1), (2.4), (3.1) and (3.2) hold. We choose $\rho(t) = 1$. Applying Theorem 3.1, it remains to check (2.3), and we see that

as $t\rightarrow\infty$. Hence, every solution of (5.1) is either oscillatory or converges to zero by Theorem 3.1.

Example 4.2 Consider the third-order neutral delay differential equation with distributed deviating arguments

where $\lambda$ is a positive constant. Choose $t_0 = t_1 = 1$. Then we get

Furthermore, we deduce that

It is easy to verify that

and

as $t\rightarrow\infty$, if $\lambda\geq 1$, where we set $t_* = 2$. Therefore, conditions (H1)–(H5), (1.4), (1.5), (1.6), (2.2), (2.4), (3.1), (3.17) and (4.1) hold. We choose $\rho(t) = t^2$. Applying Theorem 4.2, it remans to check (2.25), and we get

as $t\rightarrow\infty$. Hence, every solution of (5.2) is either oscillatory or converges to zero by Theorem 4.2.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant No. 61803176, 61703180), and A Project of Shandong Province Higher Educational Science and Technology Program (Grant No. J18KA230).

Conflict of interest

The authors declare that they have no conflict of interest.