Asymptotic behavior of solutions of third-order neutral differential equations with discrete and distributed delay

1.   Introduction

It is prudential to say that mathematical modeling with delay differential equations have drawn clear significance because of their potential applications in diverse fields, which includes biological sciences, physical sciences, gas and fluid mechanics, signal processing, robotics and traffic system, engineering, population dynamics, medicine and the like (see for example ^[9,16,17]). It is now realized that the oscillation and asymptotic solutions of various classes of differential equation are an important field of investigation and its theory is a lot richer than the qualitative theory of differential equations (see for example ^[8,10,22]). The problem of oscillatory and nonoscillatory of solutions of various classes of second/third order differential equations with delayed and mixed arguments has been widely investigated in the literature (see for example ^{[2,4,5,6,7,11,12,18,23,24,25,26,27,28,29,30,31,32,33,34]}). Various types of techniques appeared for investigations of such equations.

The purpose of this work, we are concerned with third-order neutral differential equations with discrete and distributed delay

and

where $z(t) = y(t)+p_{1}(t)y(t-\tau_{1})+p_{2}(t)y(t+\tau_{2})$, $c < d$ and $\lambda\geq1$. Now onwards, we assume that, $a_{i}(t), p_{i}(t) \in C([t_{0}, +\infty))$, $a_{i}(t) > 0$, $p_{i}(t) > 0$ for $i = 1, 2$ and $0 \leq p_{i}(t) \leq \mu_{i}$, $\mu_{1}+\mu_{2} < 1$ where $\mu_{i}$ are constants, $q_{i} \;\; \in \;\; C\big([t_{0}, +\infty), \mathbb{R^{+}}\big)$, $\tilde{q_{i}}(t, \xi) \;\; \in \;\; C\big([t_{0}, +\infty) \;\; \times \;\; [c, d]$, $\mathbb{R^{+}}\big)$ for $i = 1, 2$, and not identically zero on $[t_{*}, +\infty) \times [c, d]$, $t_{*}\geq t$, constants $\tau_{i} \geq 0$, for $i = 1, 2$, and the integral of $\left({{E_2}} \right)$ is take in the sense of Riemann–Stieltjes.

Let us recall that, a solution $y(t)\in C([T_{y}, \infty), \mathbb{R})$ of $\left({{E_1}} \right)$ (or $\left({{E_2}} \right)$) is a non-trivial or $y(t)\neq 0$ with $T_{y}\geq t_{0}$, if the functions $z \;\; \in \;\; C^{1}([T_{y}, \infty), \mathbb{R})$, $a_{1}z^{\prime} \;\; \in \;\; C^{2}([T_{y}, \infty), \mathbb{R})$ and $a_{2}[(a_{1}z^{\prime})^{\prime}]^{\lambda} \;\; \in \;\; C^{1}([T_{y}, \infty), \mathbb{R})$ for certain $T_{y}\geq t_{0}$ which satisfies $\left({{E_1}} \right)$ (or $\left({{E_2}} \right)$). Our attention is restricted to those solutions of $\left({{E_1}} \right)$ (or $\left({{E_2}} \right)$) which exist on half-line $[T_{y}, \infty)$ and the condition $\sup\{|y(t)|: t > T_{*}\} > 0$ satisfies for any $T_{*} \geq t_{y}$. A solution of $\left({{E_1}} \right)$ (or $\left({{E_2}} \right)$), which is nontrivial (proper) for all large $t$, is called oscillatory if it has no last zero, otherwise, termed nonoscillatory.

We define the operators,

We shall consider the two cases,

and

Recently, Candan ^[24] investigated the oscillatory behavior of solutions of $\left({{E_1}} \right)$ and $\left({{E_2}} \right)$ by using the Riccati substitution techniques, he presented some new oscillation criteria for $\left({{E_1}} \right)$ and $\left({{E_2}} \right)$ by the assumption of condition (1.1). We notice that in ^[24], no criteria were found for $\left({{E_1}} \right)$ (or $\left({{E_2}} \right)$) to be oscillatory for the assumption of condition (1.2). It would be interesting to improve and extend them in the condition (1.2).

However, the corresponding result for $\left({{E_1}} \right)$ (or $\left({{E_2}} \right)$) under (1.2) is still missing. In this work, we fill up this gap, also we strengthen and extend the main results of Candan ^[24] under the condition (1.1) and (1.2) respectively. We present several oscillatory criteria for $\left({{E_1}} \right)$ and $\left({{E_2}} \right)$, by applying three Riccati substitution techniques, integral averaging techniques and comparison principles. We present two examples in order to illustrate the main results at the end.

2.   Preliminary

In this section, we present some basic Lemmas for helping to prove the main results. We use throughout this paper the following notations for convenience and for shortening the equations:

Lemma 2.1. Let $\lambda \geq 1$, assume $u \geq 0$. Then

Lemma 2.2. Let $\lambda \leq 1$, assume $u \geq 0$. Then

Lemma 2.3. If $\lambda > 0$ and $X, Y > 0$, then

Lemma 2.4. Assume that (1.1) holds. Furthermore, assume that $y$ is an eventually positive solution of $\left({{E_1}} \right)$ (or $\left({{E_2}} \right)$). Then $z$ for $t_{1}\in[t_{0}, \infty)$ satisfies, eventually of the following cases:

and if (1.2) holds, then also

Lemma 2.5. Assume that $z$ satisfies $(C_{1})$ for $t \geq t_{0}$. Then

and

Proof. Since $L^{[4]}z(t)\leq 0$, $L^{[3]}z(t)$ is nondecreasing. Then we have

Again integrate, we get

Lemma 2.6 (See ^[24]). Assume that $z$ is a solution of $\left({{E_1}} \right)$ which satisfies $(C_{2})$ in Lemma 2.4. Furthermore,

Then, there is $\lim_{t\to\infty} z(t) = 0$.

Lemma 2.7 (See ^[24]). Assume that $z$ is a solution of $\left({{E_2}} \right)$ which satisfies $(C_{2})$ in Lemma 2.4. Furthermore,

Then, there is $\lim_{t\to\infty} z(t) = 0$.

3.   Oscillation results for $\left( {{E_1}} \right)$

In this section, we will establish several oscillation criteria for $\left({{E_1}} \right)$. The following notations for convenience and for shortening the equations:

Let $\mathbb{S}_{0} = \{(t, s): a \leq s < t < +\infty \}$, $\mathbb{S} = \{(t, s): a \leq s \leq t < +\infty \}$ the continuous function $H(t, s)$, $H \, : \, \mathbb{S} \to \mathbb{R}$ belongs to the class function $\Re$

(ⅰ) $H(t, t) = 0$ for $t\geq t_{0}$ and $H(t, s) > 0$ for $(t, s)\in \mathbb{S}_0$,

(ⅱ) $\frac{\partial H(t, s)}{\partial s}\leq 0$, $(t, s)\in \mathbb{S}_0$ and some locally integrable function $h(t, s)$ such that

Theorem 3.1. Let (1.1) hold and $\sigma_{1} \geq \tau_{1}$. If there exists an $m(t) \in C^{1}([t_{0}, \infty), \mathbb{R}^{+})$ such that (2.6) and

then every solution $y(t)$ of $\left({{E_1}} \right)$ is either oscillatory or tends to $0$.

Proof. Suppose that $\left({{E_1}} \right)$ has a nonoscillatory solution $y$. Without loss of generality, we may take $y(t) > 0$, $y(t-\tau_{1}) > 0$, $y(t+\tau_{2}) > 0$, $y(t-\sigma_{1}) > 0$ and $y(t+\sigma_{1}) > 0$ for $t\geq t_{1}\geq t_{0}$. Since $y(t) > 0$ for all $t\geq t_{1}$, in view of $\left({{E_1}} \right)$, we have

Assumption of (1.1), by Lemma 2.4 there exists two cases $(C_{1})$ and $(C_{2})$. If ($C_{2}$) holds, then by Lemma 2.6, $\lim_{t\to\infty} z(t) = 0$. If $(C_{1})$ holds.

Furthermore, from Lemma 2.1, we get

Similarly, we get

Substituting (3.4), (3.5) into (3.3), we have

Using the fact of $L^{[1]}z(t) > 0$, we obtain

Define

We obtain $w_{1}(t) > 0$, then

By Lemma (2.5), one gets $z^{\prime}(t-\sigma_{1}) \geq \frac{a_{2}^{1/\lambda}(t)}{a_{1}(t-\sigma_{1})} \pi_{1}[t_{0}, t-\sigma_{1}]\, L^{[2]}z(t)$. Therefore

Using (3.8) in (3.10), we obtain

Next, define

We obtain $w_{2}(t) > 0$, then

By Lemma (2.5), one gets $z^{\prime}(t-\sigma_{1}) \geq \frac{a_{2}^{1/\lambda}(t-\tau_{1})}{a_{1}(t-\sigma_{1})} \pi_{1}[t_{0}, t-\sigma_{1}]\, L^{[2]}_{-\tau_{1}}z(t)$ and using (3.12) in (3.13), we have

Finally, define

We obtain $w_{3}(t) > 0$, then

By Lemma 2.5, one gets $z^{\prime}(t-\sigma_{1}) \geq \frac{a_{2}^{1/\lambda}(t+\tau_{2})}{a_{1}(t-\sigma_{1})} \pi_{1}[t_{0}, t-\sigma_{1}]\, L^{[2]}_{\tau_{2}}z(t)$ and using (3.15) in (3.16), we get

From (3.8), (3.10) and (3.15), we have

Using (3.7) in (3.18), we have

that is,

Multiply $H(t, s)$ and integrate (3.20) from $t_{3}$ to $t$, one can get that

Thus, we obtain

Then

Setting $Y = \frac{|h(t, s)|(H(t, s))^{\frac{\lambda}{\lambda+1}}}{m(s)}$, $X = \frac{ H(t, s) \, \lambda \pi_{1}[t_{0}, s-\sigma_{1}]}{(m(s))^{1/\lambda}a_{1}(s-\sigma_{1})}$ and $u = w_{i}(t)$ for $i = 1, 2, 3$. By using the Lemma 2.3, we conclude that

which contradicts condition (3.20).

Theorem 3.2. Let (1.1) hold and $\tau_{1} \geq \sigma_{1}$. If there exists an $m(t) \in C^{1}([t_{0}, \infty), \mathbb{R}^{+})$ such that (2.6) and

then every solution $y(t)$ of $\left({{E_1}} \right)$ is either oscillatory or tends to $0$.

Proof. Suppose that $\left({{E_1}} \right)$ has a nonoscillatory solution $y$. Without loss of generality, we may take $y(t) > 0$, $y(t-\tau_{1}) > 0$, $y(t+\tau_{2}) > 0$, $y(t-\sigma_{1}) > 0$ and $y(t+\sigma_{1}) > 0$ for $t\geq t_{1}\geq t_{0}$. Assumption of (1.1), by Lemma 2.4 there exists two cases $(C_{1})$ and $(C_{2})$. If ($C_{2}$) holds, then by Lemma 2.6, $\lim_{t\to\infty} z(t) = 0$. We only consider ($C_{1}$), by using the fact that $z^{\prime}(t) > 0$ and $\tau_{1} \geq \sigma_{1}$, we obtain that Using the fact of $L^{[1]}z(t) > 0$, we obtain

Next, we categorize the functions as $w_{1}(t) = m(t) \frac{L^{[3]}z(t)}{z^{\lambda}(t-\tau_{1})}$, $w_{2}(t) = m(t) \frac{L^{[3]}_{-\tau_{1}}z(t)}{z^{\lambda}(t-\tau_{1})}$ and $w_{3}(t) = m(t) \frac{L^{[3]}_{\tau_{2}}z(t)}{z^{\lambda}(t-\tau_{1})}$ respectively. The rest of the proof is similar to that of Theorem 3.1, therefore, it is omitted.

Theorem 3.3. Let (1.2) hold and $\sigma_{1} \geq \tau_{1}$. If there exists an $m(t) \in C^{1}([t_{0}, \infty), \mathbb{R}^{+})$ such that (2.6),

and

where $(m^{\prime}(t))_{+} = \max\{0, m^{\prime}(t)\}$, $\pi_{*}(t) = \int_{t+\sigma_{1}}^{\infty} a_{2}^{-1/\lambda}(s) ds$, then every solution $y(t)$ of $\left({{E_1}} \right)$ is either oscillatory or tends to $0$.

Proof. Suppose that $\left({{E_1}} \right)$ has a nonoscillatory solution $y$. Without loss of generality, we may take $y(t) > 0$, $y(t-\tau_{1}) > 0$, $y(t+\tau_{2}) > 0$, $y(t-\sigma_{1}) > 0$ and $y(t+\sigma_{1}) > 0$ for $t\geq t_{1}\geq t_{0}$. Since $y(t) > 0$ for all $t\geq t_{1}$. Assumption of (1.2), by Lemma 2.4 there exists three cases $(C_{1})$, $(C_{2})$ and $(C_{3})$. If case $(C_{1})$ and $(C_{2})$ holds, using the similar proof of (^[24], Theorem 2.1) by using Lemma 2.1, we get the conclusion of Theorem 3.3.

If case $(C_{3})$ holds, $z^{\prime}(t-\sigma_{1}) < 0$ for $t\geq t_{1}$. The facts that $z^{\prime}(t) < 0$, $c+d\geq0$ and (3.6), we obtain

Define

We obtain $w_{*}(t) < 0$ for $t \geq t_{2}$. Noting that $L^{[3]}z(t)$ is decreasing, we obtain

for $s\geq t\geq t_{2}$. Dividing (3.31) by $a_{2}(s)$ and integrating from $t+\sigma_{1}$ to $l \;\; (l\geq t)$, we get

letting $l \to \infty$, we get

for $t \geq t_{2}$. From (3.30), we have

By (3.2) we have $a_{1}(t+\sigma_{1})z^{\prime}(t+\sigma_{1}) \leq a_{1}(t)z^{\prime}(t)$. Differentiating (3.30) gives,

Using (3.30) in (3.34), we have

Again, we define

We obtain $w_{**}(t) < 0$ and $w_{**}(t)\geq w_{*}(t)$ for $t\geq t_{2}$. By (3.33), we obtain

By (3.2) we have $a_{1}(t+\sigma_{1})z^{\prime}(t+\sigma_{1}) \leq a_{1}(t-\tau_{1})z^{\prime}(t-\tau_{1})$. Differentiating (3.36) gives,

Using (3.36) in (3.38), we have

Finally, we define a function

We obtain $w_{***}(t) < 0$ and $w_{***}(t) = w_{*}(t+\tau_{2})$ for $t\geq t_{2}$. By (3.33), we obtain

By (3.2) we have $a_{1}(t+\tau_{2}+\sigma_{1})z^{\prime}(t+\tau_{2}+\sigma_{1}) \leq a_{1}(t+\tau_{2})z^{\prime}(t+\tau_{2})$. Differentiating (3.40) gives,

Using (3.40) in (3.42), we have

From (3.35), (3.39), (3.43) and (3.29) which implies

In case $(C_{3})$, $(a_{1}(t)z^{\prime}(t))^{\prime} < 0$ we seen that

Using (3.45) in (3.44), we get

Multiplying $\pi^{\lambda}_{*}(t+\tau_{2})$ and integrating from $t_{3} \;\; (t_{3} > t_{2})$ to $t$, yields

Applying Lemma 2.3, we conclude that

Using the fact of $\pi^{\lambda}_{*}(t+\tau_{2}) \leq \pi^{\lambda}_{*}(t)$ in (3.33), (3.37), (3.41) and (3.48) imply that

a contradiction to (3.28).

Finally, we establish new comparison theorems for $\left({{E_1}} \right)$ under the case when (1.2) holds.

Theorem 3.4. Let (1.2), (2.6) hold and $\sigma_{1} > \tau_{1}$, $\sigma_{1} > \tau_{2}$. If the first-order differential inequality

for $t\geq t_{0}$, has no positive nonincreasing solution and the first-order differential inequality

for $t\geq t_{0}$, has no positive nondecreasing solution. Then Eq. $\left({{E_1}} \right)$ oscillatory.

Proof. Suppose that $\left({{E_1}} \right)$ has a nonoscillatory solution $y$. Without loss of generality, we may take $y(t) > 0$, $y(t-\tau_{1}) > 0$, $y(t+\tau_{2}) > 0$, $y(t-\sigma_{1}) > 0$ and $y(t+\sigma_{1}) > 0$ for $t\geq t_{1}\geq t_{0}$. Since $y(t) > 0$ for all $t\geq t_{1}$. Assumption of (1.2), by Lemma 2.4, there exists three cases $(C_{1})$, $(C_{2})$ and $(C_{3})$. If case $(C_{2})$ hold, the proof is follows from Lemma 2.6.

If case $(C_{1})$ holds, we have $L^{[2]}z(t) > 0$, from (3.6), we obtain

By Lemma 2.5, one gets $z(t-\sigma_{1})\geq (L_{-\sigma_{1}}^{[3]}z(t))^{1/\lambda}A(t-\sigma_{1})$ and using in (3.52), we have

Now, set

Then $\psi(t) > 0$ and the fact that $L^{[3]}z(t)$ is nonincreasing, we have

Using (3.54) and (3.53), we see that $\psi(t)$ is a nonincreasing positive solution of the first order differential inequality

which is contradiction to (3.50).

If case $(C_{3})$ holds, we have $L^{[2]}z(t) < 0$, from (3.6), we obtain

Since $L^{[3]}z(t)$ is nondecreasing. Then we get

Integrating above inequality from $t$ to $l$, we get

Letting $l \to \infty$, we get

Again integrating, we get

From 3.57, one gets $z(t+\sigma_{1})\geq -(L_{\sigma_{1}}^{[3]}z(t))^{1/\lambda}B(t+\sigma_{1})$ and using in (3.56), we have

Now, set

Then $\psi(t) > 0$, $\psi^{\prime}(t)\geq 0$ and the fact that $L^{[3]}z(t)$ is nondecreasing, we have

Using (3.59) and (3.58), we see that $\psi(t)$ is a nonincreasing positive solution of the first order differential inequality

which is contradiction to (3.51).

Corollary 3.5. Let (1.2), (2.6) hold and $\sigma_{1} > \tau_{1}$, $\sigma_{1} > \tau_{2}$. If

and

hold, then Eq. $\left({{E_1}} \right)$ oscillatory.

Proof. The proof follows from Theorem 3.4 and (^[10], Theorem 2.1.1), and the details are omitted.

Example 3.6. Consider the third order differential equation

Compared with $\left({{E_1}} \right)$, we can see that $a_{1}(t) = a_{2}(t) = 1$, $p_{1}(t) = \frac{e^{-2}}{3}$, $p_{2}(t) = \frac{e^{1}}{3}$, $q_{1}(t) = \frac{3 e^{-3}}{4} \Bigl(\frac{5}{3}\Bigr)^{3/2}$, $q_{2}(t) = \frac{3 e^{3}}{4} \Bigl(\frac{5}{3}\Bigr)^{3/2}$, $\lambda = 3/2$, $\tau_{1} = 2$, $\tau_{2} = 1$ and $\sigma_{1} = 2$. By taking $m(t) = 1$, $H(t, s) = (t-s)^2$, we obtain $h(t, s) = (3s-t)(t-s)^{-1/5}$. It is easy to verify that all conditions of Theorem 3.1 are satisfied. Therefore, all the solutions of (3.63) is either oscillates or tends to $0$ and $y(t) = e^{-t}$ is a such solution of (3.63).

Example 3.7. Consider the third order differential equation

Compared with $\left({{E_1}} \right)$, we can see that $a_{1}(t) = 1$, $a_{2}(t) = t^{2}$, $p_{1}(t) = k_{1}$, $p_{2}(t) = k_{2}$, $q_{1}(t) = k_{3}t$, $q_{2}(t) = k_{4}$, $\lambda = 1$ and $k_{1}$, $k_{2}$, $k_{3}$, $k_{4}$ are nonnegative constants. It is easy to verify that all conditions of Corollary 3.5 are satisfied and hence all solutions of equation (3.64) are oscillatory.

4.   Oscillation results for $\left({{E_2}} \right)$

In this section, we will establish several oscillation criteria for $\left({{E_2}} \right)$. For convenience, we define,

Theorem 4.1. Let (1.1) holds and $c+d\geq0$, $b \geq \tau_{1}$. If there exists an $m(t) \in C^{1}([t_{0}, \infty), \mathbb{R}^{+})$ such that (2.7) and

then every solution $y(t)$ of $\left({{E_2}} \right)$ is either oscillatory or tends to $0$.

Proof. Suppose that $\left({{E_2}} \right)$ has a nonoscillatory solution $y$. Without loss of generality, we may take $y(t) > 0$, $y(t-\tau_{1}) > 0$, $y(t+\tau_{2}) > 0$, $y(t-\xi) > 0$ and $y(t+\xi) > 0$ for $t\geq t_{1}\geq t_{0}$ and $\xi\in[c, d]$. Since $y(t) > 0$ for all $t\geq t_{1}$, in view of $\left({{E_2}} \right)$, we have

Assumption of (1.1), by Lemma 2.4 there exists two cases $(C_{1})$ and $(C_{2})$. If ($C_{2}$) holds, then by Lemma 2.7, $\lim_{t\to\infty} z(t) = 0$. If ($C_{1}$) holds.

Furthermore, from Lemma 2.1, we have

Similarly, we get

Substituting (4.4), (4.5) into (4.3), we have

Using the fact of $L^{[1]}z(t) > 0$ and $c+d\geq0$, we have

Define a function

We obtain $w_{1}(t) > 0$, then

By Lemma (2.5), one gets $z^{\prime}(t-d) \geq \frac{a_{2}^{1/\lambda}(t)}{a_{1}(t-d)} \pi_{1}[t_{0}, t-d]\, L^{[2]}z(t)$. Therefore

Using (4.8) in (4.10), we have

Next, define

We obtain $w_{2}(t) > 0$, then

By Lemma (2.5), one gets $z^{\prime}(t-d) \geq \frac{a_{2}^{1/\lambda}(t-\tau_{1})}{a_{1}(t-d)} \pi_{1}[t_{0}, t-d]\, L^{[2]}_{-\tau_{1}}z(t)$ and using (4.12) in (4.13), we have

Finally, define

We obtain $w_{3}(t) > 0$, then

By Lemma 2.5, one gets $z^{\prime}(t-d) \geq \frac{a_{2}^{1/\lambda}(t+\tau_{2})}{a_{1}(t-d)} \pi_{1}[t_{0}, t-d]\, L^{[2]}_{\tau_{2}}z(t)$ and using (4.15) in (4.16), we have

From (4.8), (4.10) and (4.15), we have

Using (4.7) in (4.18), we have

that is,

Multiply both sides $H(t, s)$ and integrate (4.51) from $t_{3}$ to $t$, one can get that

Thus, we obtain

Then

Setting $Y = \frac{|h(t, s)|(H(t, s))^{\frac{\lambda}{\lambda+1}}}{m(s)}$, $X = \frac{ H(t, s) \, \lambda \pi_{1}[t_{0}, s-d]}{(m(s))^{1/\lambda}a_{1}(s-d)}$ and $u = w_{i}(t)$ for $i = 1, 2, 3$. By using the Lemma 2.3, we conclude that

which contradicts condition (4.51).

Theorem 4.2. Let (1.1) holds and $c+d\geq0$, $-c \geq \tau_{1}$. If there exists an $m(t) \in C^{1}([t_{0}, \infty), \mathbb{R}^{+})$ such that (2.7) and

then every solution $y(t)$ of $\left({{E_2}} \right)$ is either oscillatory or tends to $0$.

Proof. Suppose that $\left({{E_2}} \right)$ has a nonoscillatory solution $y$. Without loss of generality, we may take $y(t) > 0$, $y(t-\tau_{1}) > 0$, $y(t+\tau_{2}) > 0$, $y(t-\xi) > 0$ and $y(t+\xi) > 0$ for $t\geq t_{1}\geq t_{0}$ and $\xi\in[c, d]$. Assumption of (1.1), by Lemma 2.4 there exists two cases $(C_{1})$ and $(C_{2})$. If ($C_{2}$) holds, then by Lemma 2.7, $\lim_{t\to\infty} z(t) = 0$. We only consider ($C_{1}$), by using the fact that $z^{\prime}(t) > 0$ and $-c \geq \tau_{1}$, we obtain that Using the fact of $L^{[1]}z(t) > 0$, we obtain

Next, we categorize the functions as $w_{1}(t) = m(t) \frac{L^{[3]}z(t)}{z^{\lambda}(t+c)}$, $w_{2}(t) = m(t) \frac{L^{[3]}_{-\tau_{1}}z(t)}{z^{\lambda}(t+c)}$ and $w_{3}(t) = m(t) \frac{L^{[3]}_{\tau_{2}}z(t)}{z^{\lambda}(t+c)}$ respectively. The rest of the proof is similar to that of Theorem 4.1, therefore, it is omitted.

Theorem 4.3. Let (1.2) holds and $b\geq \tau_{1}$ (or $b\leq \tau_{1}$). If there exists an $m(t) \in C^{1}([t_{0}, \infty), \mathbb{R}^{+})$ such that (2.7),

and

where $\beta(t) = \int_{t+d}^{\infty} a_{2}^{-1/\lambda}(s) ds$, then every solution $y(t)$ of $\left({{E_2}} \right)$ is either oscillatory or tends to $0$.

Proof. Suppose that $\left({{E_1}} \right)$ has a nonoscillatory solution $y$. Without loss of generality, we may take $y(t) > 0$, $y(t-\tau_{1}) > 0$, $y(t+\tau_{2}) > 0$, $y(t-\xi) > 0$ and $y(t+\xi) > 0$ for $t\geq t_{1}\geq t_{0}$ and $\xi\in[c, d]$. Since $y(t) > 0$ for all $t\geq t_{1}$. Assumption of (1.2), by Lemma 2.4 there exists three cases $(C_{1})$, $(C_{2})$ and $(C_{3})$. If case $(C_{1})$ and $(C_{2})$ holds, using the similar proof of (^[24], Theorem 2.3) by using Lemma 2.1, we get the conclusion of Theorem 4.3

If case $(C_{3})$ holds, $z^{\prime}(t-d) < 0$ for $t\geq t_{1}$. The facts that $z^{\prime}(t) < 0$, $c+d\geq0$ and (4.6), we obtain

Define

We obtain $w_{*}(t) < 0$ for $t \geq t_{2}$. Noting that $L^{[3]}z(t)$ is decreasing, we obtain

for $s\geq t\geq t_{2}$. Dividing (4.31) by $a_{2}(s)$ and integrating from $t+d$ to $l (l\geq t)$, we get

letting $l \to \infty$, we get

From (4.30), we have

By (4.2) we have $a_{1}(t+d)z^{\prime}(t+d) \leq a_{1}(t)z^{\prime}(t)$. Differentiating (4.30) gives,

Using (4.30) in (4.34), we have

Next, we define

We obtain $w_{**}(t) < 0$ and $w_{**}(t)\geq w_{*}(t)$ for $t\geq t_{2}$. By (4.33), we obtain

By (3.2) we have $a_{1}(t+d)z^{\prime}(t+d) \leq a_{1}(t-\tau_{1})z^{\prime}(t-\tau_{1})$. Differentiating (4.36) gives,

Using (4.36) in (4.38), we have

Finally, We define a function

We obtain $w_{***}(t) < 0$ and $w_{***}(t) = w_{*}(t+\tau_{2})$ for $t\geq t_{2}$. By (4.33), we obtain

By (4.2) we have $a_{1}(t+\tau_{2}+d)z^{\prime}(t+\tau_{2}+d) \leq a_{1}(t+\tau_{2})z^{\prime}(t+\tau_{2})$. Differentiating (4.40) gives,

Using (4.40) in (4.42), we have

From (4.35), (4.39), (4.43) and (4.29) which implies

In case $(C_{3})$, $(a_{1}(t)z^{\prime}(t))^{\prime} < 0$ we seen that

Using (4.45) in (4.44), we get

Multiplying $\beta^{\lambda}(t+\tau_{2})$ and integrating from $t_{3} \;\; (t_{3} > t_{2})$ to $t$, yields

Applying Lemma 2.3, we conclude that

Using the fact of $\beta^{\lambda}(t+\tau_{2}) \leq \beta^{\lambda}(t)$ in (4.33), (4.37), (4.41) and (4.48) imply that

a contradiction to (4.28).

Example 4.4. Consider a third-order differential equation

Compared with $\left({{E_2}} \right)$, we can see that $c = 0$, $d = \pi$, $a_{1}(t) = 1/2$, $a_{2}(t) = 1$, $p_{1}(t) = \frac{1}{3}$, $p_{2}(t) = \frac{2}{3}$, $\tilde{q}_{1}(t, \xi) = \tilde{q}_{2}(t, \xi) = 1$, $\lambda = 1$, $\tau_{1} = \pi/4$ and $\tau_{2} = \pi/2$. By taking $m(t) = 1$, we obtain

and we take $H(t, s) = (t-s)^2$ then $h(t, s) = (3s-t)(t-s)^{-1/5}$ and $0 < \mu_{1} +\mu_{2} < 1$, we see that

Since all the conditions of Theorem 4.1 hold, (4.50) is either oscillates or tends to $0$.

5.   Conclusion

In this paper, we have used Riccati substitution techniques, integral averaging technique and some new oscillation and asymptotic theorems for $\left({{E_1}} \right)$ and $\left({{E_2}} \right)$ under the conditions (1.1) and (1.2) have been established. Additionally, we established new comparison theorem that permit to study properties of $\left({{E_1}} \right)$ regardless under the conditions (1.2). The results obtained indicated that it improved theorems reported by Candan ^[24]. Similar results can be presented under the assumption that $\lambda \leq 1$. In this case, using Lemma 2.2, one has to simply replace $3^{\lambda-1}$ by $1$ and proceed as above. In literature, very few works has been paid in the research activities related to qualitative behavior of solutions of various types of stochastic differential equations, see the recent works ^{[1,3,13,14,15,19,20,21]}. The results of this paper could be extended to the stochastic differential equations with time delay in further research.

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable suggestions on improving the content of this article.

Conflict of interest

The authors declare there are no conflicts of interest.