On the upper bounds for the distance between zeros of solutions of a first-order linear neutral differential equation with several delays

1.   Introduction

Consider the first-order neutral differential equation with several delays

where $a, b_j \in C([t_0, \infty), [0, \infty))$, $0 < \sigma < \mu_1\leq \mu_2\leq\dots \leq \mu_n$, and the first-order differential equation with several delays

By a solution of Eq ($E_1$) on $[t^*+\mu_n, \, \infty)$, $t^*\geq t_0$, we mean a function $y\in C([t^*, \infty), \mathbb{R})$ such that $y(t)+a(t)y\left(t-\sigma\right)\in C^1([t^*+\mu_n-\sigma, \infty), \mathbb{R})$ and satisfies ($E_1$) for all $t \geq t^*+\mu_n$. The existence and uniqueness of a solution $y(t)$ of Eq ($E_1$) with an initial function $\phi\in C([t_{0}-\mu_n, \, t_{0}], \mathbb{R})$ can be proved using the method of steps as in [24, Thereoem 1.1.2]. Also, the existence of a positive solution for some neutral differential equations can be found in ^[25].

A solution is said to be oscillatory if it is neither eventually negative nor eventually positive; otherwise, it is called non-oscillatory. If all solutions of a differential equation are oscillatory, then it is called oscillatory; otherwise, it is called non-oscillatory.

Recently, neutral differential equations have arisen in many applications; see ^{[1,5,23,24,27,28,29,34,35,36]}. In this type of equation, the delays appear in both the unknown function and its derivatives. The qualitative properties of neutral differential equations have received a great deal of attention from many mathematicians; see ^{[1,2,5,6,7,9,12,14,15,16,19,20,21,22,23,24,26,27,28,29,34,35,36,37,38,40,41,42]}. In dynamical models, delay and oscillation effects are often formulated by means of external sources and/or nonlinear diffusion, perturbing the natural evolution of related systems; see, e.g., ^[30,31]. The oscillation of neutral and delay differential equations has been extensively developed and studied in many works; see ^{[2,5,6,7,9,12,13,14,15,16,19,20,21,22,24,26,37,38,40,41]}. On the other hand, the oscillation of first-order differential equations has numerous applications in the study of the oscillatory behavior of higher-order neutral differential equations; see ^[32,33]. However, only a few works have been interested in studying the distance between zeros of all solutions for first-order neutral and delay differential equations; see ^{[2,3,4,6,7,8,10,11,15,17,18,37,38,39,40,41]}. In this type of study, the interest is not only in the existence of zeros of solutions (i.e., proving the oscillation) but also in determining their locations. Therefore, studying the distance between zeros for first-order linear neutral and delay differential equations can give a deeper understanding of the dynamics of nonlinear systems of neutral differential equations that are used to model many real-life phenomena. This motivates us to study the distance between successive zeros for all solutions of Eqs ($E_1$) and ($E_2$).

In the following, we display some results for the distance between zeros for Eqs ($E_1$) and ($E_2$):

The distribution of zeros in first-order differential equations with one delay or several delays was studied by ^{[3,4,8,10,11,17,18,39]}. Further, many estimations for the upper bounds of the distance between zeros for first-order neutral differential equations were established by ^{[2,6,7,15,37,38,40,41]}. However, there are no results dealing with the distribution of zeros in Eq ($E_1$).

In this work, we obtain sufficient conditions ensuring that any solution of the first-order differential inequality with several delays

where $L_0\geq L+\alpha$, $L\geq t_0$, $\alpha\geq 0$ such that $0 < \eta_1 \leq \eta_2 \leq \dots \leq \eta_n$, and $B_j\in C([L+\alpha, \infty), [0, \infty))$, $j = 1, 2, ..., n$, cannot be positive on certain intervals. By using these results, many new estimations for the upper bounds of the distance between zeros for both Eqs ($E_1$) and ($E_2$) are obtained. A new approximation for the distance between successive zeros of a certain differential equation of the form ($E_2$) is given, while all previous results cannot give this approximation. Since the distribution of zeros in all solutions of Eq ($E_1$) has never been examined before, our results for Eq ($E_1$) are obviously new. An illustrative example is given to show the applicability of our results to Eq ($E_1$).

2.   First-order delay differential inequality

In this section, we study the properties of a positive solution to the first-order differential inequality with several delays (1.1).

Let $Y(t)$ be a solution of inequality (1.1) on $[L+\alpha, \, L_0]$, $L_0 \geq L+\alpha$ such that

where $\gamma, \delta\geq0$ and $\beta = \max \{\alpha, \eta_n+\gamma\}$.

Next, we prove some lemmas that play an important role in establishing the main results of this work.

Assume that $s\in \{1, 2, ..., n\}$ and the sequence of nonnegative real numbers $\{M_l^s\}_{l\geq -1}$ is defined by

Lemma 2.1. Assume that $l\in \mathbb{N}_0$, $Y(t)$ is a solution of inequality (1.1) on $[L+\alpha, \, L_0]$ such that (2.1) is satisfied with $L_0 \geq L+\max\{\delta, \, \alpha\}+(l+1)\eta_n$ and $Y(t) > 0$ on $[L+\alpha, L_0]$. Then

for $s = 1, 2, ..., n$.

Proof. In view of $Y'(t)\leq0$ for $t \in [L+\delta, L+\beta]$ and $Y(t) > 0$ for $t\in [L+\gamma, L_0]$, it follows from (1.1) that

Therefore,

Integrating (1.1) from $t-\eta_s$ to $t$, it follows that

Therefore,

for $t\in[L+\alpha+\eta_n, L_0]$. Dividing (1.1) by $Y(t)$, and integrating the resulting inequality from $\zeta$ to $\mu$, $L+\alpha+\eta_n \leq \zeta \leq \mu \leq L_0$, we obtain

Substituting into (2.6), we obtain

for $t\in[L+\alpha+\eta_n+\eta_s, L_0]$.

By (2.4), we have

for $t\in[L+\max\{\delta+\eta_s, \, \alpha+\eta_n\}+ \eta_s, L_0]$. This leads to

for $t\in[L+\max\{\delta+\eta_s, \, \alpha+\eta_n\}+ \eta_s, L_0]$. That is,

for $t\in[L+\max\{\delta+\eta_s, \, \alpha+\eta_n\}+ \eta_s, L_0]$. Therefore

In view of (2.5), we obtain

for $t\in[L+\max\{\delta+\eta_s, \, \alpha+\eta_n\}+ \eta_n, L_0] \subseteq [L+\max\{\delta, \, \alpha\}+ 2\eta_n, L_0]$. Therefor,

Substituting into (2.8), we have

for $t\in[L+\max\{\delta, \, \alpha\}+3\eta_n, L_0]$. By repeating this procedure, we obtain (2.3). The proof is complete. □

Lemma 2.2. Assume that $l\in \mathbb{N}_0$, $0 < \epsilon\leq1$, and $Y(t)$ is a solution of inequality (1.1) on $[L+\alpha, \, L_0]$ such that (2.1) is satisfied with $L_0 \geq L+\max\{\delta, \, \alpha\}+(l+1+\epsilon)\eta_n$. If,

then $Y(t)$ cannot be positive on $[L+\alpha, \, L_0 ]$.

Proof. Assume the contrary, and let $Y(t) > 0$ on $[L+\alpha, \, L_0 ]$. This together with (2.1) implies that $Y(t) > 0$ on $[L+\gamma, \, L_0 ]$. Form (2.4), we have

Integrating (1.1) from $t$ to $t- \epsilon \eta_s$, $s = 1, 2, \dots, n$, we obtain

Therefore,

for $t \in [L+\alpha+\epsilon \eta_s, \, L_0]$. By (2.10), we obtain

for $t \in [L+\max\{\delta, \, \alpha\}+\epsilon \eta_s+\eta_n, \, L_0]$. That is,

for $t \in [L+\max\{\delta, \, \alpha\}+\epsilon \eta_s+\eta_n, \, L_0]$.

From (2.7) and (2.11), we have

for $t \in [L+\alpha+\epsilon \eta_s+2\eta_n, \, L_0]$. This, together with (2.3) leads to

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_s, L_0]$. From this and (2.12), we have

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_s, L_0]$. Using the arithmetic-geometric mean, we obtain

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_s, L_0]$. Taking the product on both sides,

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1+\epsilon)\eta_n, L_0]$. Therefore,

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1+\epsilon)\eta_n, L_0]$. Then

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1+\epsilon)\eta_n, L_0]$. This contradicts with (2.9). The proof is complete. □

Lemma 2.3. Assume that $l\in \mathbb{N}_0$, $1\leq k_1\leq k_2 \leq n$, $0 < \epsilon\leq1$, and $Y(t)$ is a solution of inequality (1.1) on $[L+\alpha, \, L_0]$ such that (2.1) is satisfied with $L_0 \geq L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_{k_2}$. If,

then $Y(t)$ cannot be positive on $[L+\alpha, \, L_0 ]$.

Proof. As before, assume that $Y(t) > 0$ on $[L+\alpha, \, L_0 ]$. Integrating (1.1) from $t$ to $t- \epsilon \eta_s$, $s = 1, 2, \dots, k_2$, it follows that

Therefore,

for $t \in [L+\alpha+\epsilon \eta_s, \, L_0]$. The same reasoning as in Lemma 2.2 leads to

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_s, L_0]$. Consequently,

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_s, L_0]$. Taking the product on both sides,

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_{k_2}, L_0]$. That is,

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_{k_2}, L_0]$. Therefore,

for $t\in[L+\max\{\delta, \, \alpha\}+(l+1)\eta_n+\epsilon \eta_{k_2}, L_0]$, which contradicts (2.13). The proof is complete.□

3.   Main results

Assume that $D_{t\geq t_1 }\left(E\right)$, $t_1 \geq t_0$ is the upper bound of the distance between successive zeros of all solutions of a differential equation $E$ on $[t_1, \, \infty)$.

3.1.   The distribution of zeros in a first-order differential equation with several delays

Below, we obtain new approximations for the upper bound of the distance between successive zeros of all solutions of Eq ($E_2$).

Theorem 3.1. Let

Assume that $l\in \mathbb{N}_0$, $0 < \epsilon\leq1$. If,

then $D_{t_1}\left(E_2\right)\leq (l+2+\epsilon)\mu_n$ and every solution of Eq ($E_2$) is oscillatory.

Proof. Assume the contrary, and let $y(t)$ be a solution of Eq ($E_2$) such that $y(t) > 0$ on $[L, \, L_0]$, $L\geq t_0$, $L_0\geq L+(l+2+\epsilon)\mu_n$. Therefore, $y'(t)\leq0$ for $t\in[L+\mu_n, \, L_0]$. Then, all assumptions of Lemma 2.2 are satisfied with $\alpha = \gamma = 0$, $\delta = \beta = \mu_n$, $B_j(t) = b_j(t)$, and $\eta_j = \mu_j$, $j = 1, 2, \dots, n$. Therefore, $y(t)$ cannot be positive on $[L, \, L+\left(l+2+\epsilon\right) \mu_n ] \subseteq [L, \, L_0]$. This contradiction completes the proof. □

Using Lemma 2.3 instead of Lemma 2.2 in the proof of the preceding theorem, one can prove the following result, and hence the proof is omitted.

Theorem 3.2. Let

Assume that $l\in \mathbb{N}_0$, $1\leq k_1 \leq k_2\leq n$, $0 < \epsilon\leq1$. If,

then $D_{t_1}\left(E_2\right)\leq (l+2)\mu_n+\epsilon \mu_{k_2}$ and every solution of Eq ($E_2$) is oscillatory.

3.2.   The distribution of zeros in a first-order neutral differential equation with several delays

We obtain many upper bounds for the distance between zeros of all solutions of Eq ($E_1$) with the following assumptions:

(H1) Let $R \in C^1[[t_*+\sigma, \infty), \, [0, \infty)]$, $t_* \geq t_0$, $b_j(t)a(t-\mu_j)\leq b_j(t-\sigma) a(t-\sigma)$, $j = 1, 2, \dots, n$, and $R(t)\geq a(t-\sigma)$, $t\geq t_1+\sigma$ for some $t_1\geq t_*$.

(H2) Let $N \in C^1[[t_*+\mu_n, \infty), \, [0, \infty)]$, $t_* \geq t_0$, $b_j(t) > 0$, $j = 1, 2, \dots, n$, $N(t)\geq \sum_{j = 1}^n \frac{b_j(t) a(t-\mu_j)}{b_j(t-\sigma)}$, $t\geq t_1+\mu_n$ for some $t_1\geq t_*$.

Lemma 3.1. Assume that (H1) holds, $R'(t)\leq 0$ for $t\geq t_1+\sigma$. Let $y(t)$ be a solution of Eq ($E_1$) such that $y(t) > 0$ on $[L, \, L_0]$, $L\geq L_0+2\mu_n$, $L\geq t_1$. Then, there exists a function $z(t)$ that satisfies $z(t) > 0$ on $[L+2\sigma, \, L_0]$ and $z'(t)\leq 0$ on $[L+\sigma+\mu_n, \, L_0]$, and

Proof. Letting $u(t) = y(t)+a(t)y\left(t-\sigma\right)$, so $u(t) > 0$ for $[L+\sigma, \, L_0]$ and

Therefore $u'(t)\leq0$ for $t \in [L+\mu_n, \, L_0]$. Note that

Then

By (H1), we obtain

Clearly,

Therefore,

Consequently,

Assume that $v(t) = u(t)+R(t) u(t-\sigma)$, so $v(t) > 0$ on $[L+2\sigma, \, L_0]$. Then

for $t \in [L+\mu_n+\sigma, \, L_0]$, and so $v'(t)\leq0$ on $[L+\mu_n+\sigma, \, L_0]$. Consequently,

In view of $u'(t)\leq0$ on $[L+\mu_n, \, L_0]$, it follows that

Therefore,

and

Substituting into (3.5), we obtain

Let $z(t) = \frac{v(t)}{1+R(t)}$. Then $z(t) > 0$ on $[L+2\sigma, \, L_0]$ and

Also, by using (3.4) and (3.6), we obtain

for $t\in [L+\sigma+\mu_n, \, L_0]$. The proof is complete. □

Lemma 3.2. Assume that (H2) holds, and $N'(t)\leq 0$ for $t\geq t_1+\sigma$. Let $y(t)$ be a solution of Eq ($E_1$) such that $y(t) > 0$ on $[L, \, L_0]$, $L\geq L_0+2\mu_n$, $L\geq t_1$. Then, there exists a function $z(t)$ that satisfies $z(t) > 0$ on $[L+2\sigma, \, L_0]$ and $z'(t)\leq 0$ on $[L+\sigma+\mu_n, \, L_0]$, and

Proof. By the same method as in the proof of Lemma 2.1 we obtain (see (3.3))

where $u(t) = y(t)+a(t)y\left(t-\sigma\right)$, $u(t) > 0$ for $[L+\sigma, \, L_0]$ and $u'(t)\leq0$ for $t \in [L+\mu_n, \, L_0]$. Note that

Therefore,

This together with (3.8) implies that

In view of (H2), it follows that

By using the same method as in Lemma 2.1, we obtain

where $z(t) > 0$ on $[L+2\sigma, \, L_0]$, $z'(t) \leq 0$ on $t \in [L+\mu_n+\sigma, \, L_0]$, $z(t) = \frac{v(t)}{1+N(t)}$ and $v(t) = u(t)+N(t) u(t-\sigma)$, $u(t) = y(t)+a(t)y\left(t-\sigma\right)$. The proof is complete. □

Lemma 3.3. Assume that (H1) holds and $b_r(t) \geq |R'(t)|$, $r\in \{1, 2, \dots, n\}$. Let $y(t)$ be a solution of Eq ($E_1$) such that $y(t) > 0$ for $t\in[L, \, L_0]$, $L\geq L_0+\mu_n+\mu_r$, $L\geq t_1$. Then, there exists a function $v(t)$ that satisfies $v(t) > 0$ for $t\in [L+2\sigma, \, L_0]$ and $v'(t) \leq 0$ for $t \in [L+\mu_n+\mu_r, \, L_0]$, and

for $t \in [L+\mu_n+\mu_r, \, L_0]$.

Proof. Letting $u(t) = y(t)+a(t)y\left(t-\sigma\right)$ and $v(t) = u(t)+R(t) u(t-\sigma)$, so $u'(t)\leq0$ for $t \in [L+\mu_n, \, L_0]$ and $v(t) > 0$ for $[L+2\sigma, \, L_0]$. By (3.5), we have

Since $u'(t) \leq 0$ for $t \in [L+\mu_n, \, L_0]$, then

for $t \in [L+\mu_n+\mu_r, \, L_0]$. Therefore, $v'(t) \leq 0$ for $t \in [L+\mu_n+\mu_r, \, L_0]$, and

Using (3.7), we obtain

for $t \in [L+\mu_n+\mu_r, \, L_0]$, where $v(t) > 0$ for $[L+2\sigma, \, L_0]$ and $v'(t) \leq 0$ for $t \in [L+\mu_n+\mu_r, \, L_0]$. The proof is complete. □

Lemma 3.4. Assume that (H2) holds and $b_r(t) \geq |N'(t)|$, $r\in \{1, 2, \dots, n\}$. Let $y(t)$ be a solution of Eq ($E_1$) such that $y(t) > 0$ for $t\in[L, \, L_0]$, $L_0 \geq L+\mu_n+\mu_r$, $L\geq t_1$. Then, there is a function $v(t)$ that satisfies $v(t) > 0$ for $t\in [L+2\sigma, \, L_0]$ and $v'(t) \leq 0$ for $t \in [L+\mu_n+\mu_r, \, L_0]$, and

for $t \in [L+\mu_n+\mu_r, \, L_0]$.

Proof. Using the same method as in the proof of Lemma 3.2, we obtain (see (3.9))

where $u(t) = y(t)+a(t)y\left(t-\sigma\right)$, $u(t) > 0$ for $t \in [L+\sigma, \, L_0]$ and $u'(t)\leq0$ for $t \in [L+\mu_n, \, L_0]$. Letting $v(t) = u(t)+N(t) u(t-\sigma)$, then

for $t \in [L+\mu_n+\sigma, \, L_0]$. Therefore,

for $t \in [L+\mu_n+\sigma, \, L_0]$. That is,

By using $u'(t)\leq0$ for $t \in [L+\mu_n, \, L_0]$, we obtain

and

Therefore,

From this and (3.11), we have

for $t \in [L+\mu_n+\mu_r, \, L_0]$, where $v(t) > 0$ for $[L+2\sigma, \, L_0]$ and $v'(t) \leq 0$ for $t \in [L+\mu_n+\mu_r, \, L_0]$. The proof is complete. □

Theorem 3.3. Let

Assume that (H1) holds, $l\in \mathbb{N}_0$, $0 < \epsilon\leq1$, $R'(t)\leq 0$ for $t\geq t_1+\sigma$. If condition (2.9) is satisfied, then $D_{t_1}\left(E_1\right)\leq \left(l+3+\epsilon\right) \mu_n-\left(l+1+\epsilon\right) \sigma$ and every solution of Eq ($E_2$) is oscillatory.

Proof. Assume the contrary, and let $y(t)$ be a solution of Eq ($E_1$) such that $y(t) > 0$ on $[L, \, L_0]$, $L\geq t_0$, $L_0\geq L+\left(l+3+\epsilon\right) \mu_n-\left(l+1+\epsilon\right) \sigma$. By using Lemma 3.1, then there exists a solution $z(t)$ of (3.2) such that $z(t) > 0$ on $[L+2\sigma, \, L_0]$ and $z'(t)\leq 0$ on $[L+\sigma+\mu_n, \, L_0]$. Clearly

where $\alpha = 2 \mu_n$, $\delta = \sigma+\mu_n$, and $\eta_n = \mu_n-\sigma$. Applying Lemma 2.2 with $\gamma = 2 \sigma$, $\alpha = \beta = 2 \mu_n$, $\delta = \sigma+\mu_n$, so $z(t)$ cannot be positive on $[L+2 \mu_n, \, L_0]$. □

Theorem 3.4. Assume that (H1) holds, $l\in \mathbb{N}_0$, $1 \leq k_1 \leq k_2 \leq n$, $0 < \epsilon\leq1$, $R'(t)\leq 0$ for $t\geq t_1+\sigma$. If condition (2.13) is satisfied such that $B_j(t)$, $\eta_j$, $j = 1, 2, \dots, n$, are defined by (3.12), then $D_{t_1}\left(E_1\right)\leq \left(l+3\right) \mu_n+\epsilon \mu_{k_2}-(l+1+\epsilon) \sigma$ and every solution of Eq ($E_2$) is oscillatory.

Proof. Assume $y(t)$ is a solution of Eq ($E_1$) such that $y(t) > 0$ on $[L, \, L_0]$, $L\geq t_0$, $L_0\geq L+\left(l+3\right) \mu_n+\epsilon \mu_{k_2}-(l+1+\epsilon) \sigma$. By using Lemma 3.1, there exists a solution $z(t)$ of (3.2) such that $z(t) > 0$ on $[L+2\sigma, \, L_0]$ and $z'(t)\leq 0$ on $[L+\sigma+\mu_n, \, L_0]$. It is clear that

where $\alpha = 2 \mu_n$, $\delta = \sigma+\mu_n$, and $\eta_j = \mu_j-\sigma$. Applying Lemma 2.3 with $\gamma = 2 \sigma$, $\alpha = 2 \mu_n$, $\delta = \sigma+\mu_n$, so $z(t)$ cannot be positive on $[L+2 \mu_n, \, L_0]$. □

The following two theorems can be proven using Lemma 3.2 instead of Lemma 3.1 in the proofs of Theorems 3.3 and 3.4, respectively.

Theorem 3.5. Let

Assume that (H2) holds, $l\in \mathbb{N}_0$, $0 < \epsilon\leq1$, $N'(t)\leq 0$ for $t\geq t_1+\sigma$. If condition (2.9) is satisfied, then $D_{t_1}\left(E_1\right)\leq \left(l+3+\epsilon\right) \mu_n-\left(l+1+\epsilon\right) \sigma$ and every solution of Eq ($E_2$) is oscillatory.

Theorem 3.6. Assume that (H2) holds, $l\in \mathbb{N}_0$, $1 \leq k_1 \leq k_2 \leq n$, $0 < \epsilon\leq1$, $N'(t)\leq 0$ for $t\geq t_1+\sigma$. If condition (2.13) is satisfied such that $B_j(t)$, $\eta_j$, $j = 1, 2, \dots, n$, are defined by (3.13), then $D_{t_1}\left(E_1\right)\leq \left(l+3\right) \mu_n+\epsilon \mu_{k_2}-(l+1+\epsilon) \sigma$ and every solution of Eq ($E_2$) is oscillatory.

Theorem 3.7. Let $r \in \{1, 2, \dots, n\}$,

Assume that (H1) holds, $l\in \mathbb{N}_0$, $0 < \epsilon\leq1$, $b_r(t)\geq |R'(t)|$ for $t\geq t_1+\sigma$. If condition (2.9) is satisfied, then $D_{t_1}\left(E_1\right)\leq \left(l+2+\epsilon\right) \mu_n+\mu_r-\left(l+1+\epsilon\right) \sigma$ and every solution of Eq ($E_2$) is oscillatory.

Proof. Assume the contrary, and let $y(t)$ be a solution of Eq ($E_1$) such that $y(t) > 0$ on $[L, \, L_0]$, $L\geq t_0$, $L_0\geq L+\left(l+2+\epsilon\right) \mu_n+\mu_r-\left(l+1+\epsilon\right) \sigma$. By using Lemma 3.3, there exists a solution $v(t)$ of (3.10) on $[L_0+\mu_n+\mu_r, L]$ such that $v(t) > 0$ on $[L+2\sigma, \, L_0]$ and $v'(t)\leq 0$ on $[L+\mu_n+\mu_r, \, L_0]$. Clearly

where $\alpha = \delta = \mu_n+\mu_r$, and $\eta_n = \mu_n-\sigma$. Applying Lemma 2.2 with $\gamma = 2 \sigma$, $\alpha = \delta = \beta \mu_n+\mu_r$, so $v(t)$ cannot be positive on $[L+\mu_n+\mu_r, \, L_0]$. □

Theorem 3.8. Assume that (H1) holds, $l\in \mathbb{N}_0$, $1 \leq k_1 \leq k_2\leq n$, $0 < \epsilon\leq1$, $b_r(t)\geq |R'(t)|$ for $t\geq t_1+\sigma$, $r \in \{1, 2, \dots, n\}$. If condition (2.13) is satisfied with $B_j(t)$ and $\eta_j$, $j = 1, 2, \dots, n$, are defined by (3.14), then $D_{t_1}\left(E_1\right)\leq \left(l+2\right) \mu_n+\mu_r+\epsilon \mu_{k_2}-\left(l+1+\epsilon\right) \sigma$ and every solution of Eq ($E_2$) is oscillatory.

Proof. Assume the contrary, and let $y(t)$ be a solution of Eq ($E_1$) such that $y(t) > 0$ on $[L, \, L_0]$, $L\geq t_0$, $L_0\geq L+\left(l+2\right) \mu_n+\mu_r+\epsilon \mu_{k_2}-\left(l+1+\epsilon\right) \sigma$. By using Lemma 3.3, then there exists a solution $v(t)$ of (3.10) on $[L_0+\mu_n+\mu_r, L]$ such that $v(t) > 0$ on $[L+2\sigma, \, L_0]$ and $v'(t)\leq 0$ on $[L+\mu_n+\mu_r, \, L_0]$. Clearly

where $\alpha = \delta = \mu_n+\mu_r$, and $\eta_j = \mu_j-\sigma$, $j = 1, 2, \dots, n$. Applying Lemma 2.3 with $\gamma = 2 \sigma$, $\alpha = \delta = \beta = \mu_n+\mu_r$, so $v(t)$ cannot be positive on $[L+\mu_n+\mu_r, \, L_0]$. □

Using Lemma 3.4 instead of Lemma 3.3 in the proofs of the preceding two theorems, we obtain the following results:

Theorem 3.9. Let $r\in \{1, 2, \dots, n\}$,

Assume that (H2) holds, $l\in \mathbb{N}_0$, $0 < \epsilon\leq1$, $b_r(t)\geq |N'(t)|$ for $t\geq t_1+\sigma$. If condition (2.9) is satisfied, then $D_{t_1}\left(E_1\right)\leq \left(l+2+\epsilon\right) \mu_n+\mu_r-\left(l+1+\epsilon\right) \sigma$ and every solution of Eq ($E_2$) is oscillatory.

Theorem 3.10. Assume that (H2) holds, $l\in \mathbb{N}_0$, $1\leq k_1 \leq k_2\leq n$, $0 < \epsilon\leq1$, $b_r(t)\geq |N'(t)|$ for $t\geq t_1+\sigma$, $r\in \{1, 2, \dots, n\}$. If condition (2.13) is satisfied with $B_j(t)$ and $\eta_j = \mu_j-\sigma$, $j = 1, 2, \dots, n$, are defined by (3.15), then $D_{t_1}\left(E_1\right)\leq \left(l+2\right) \mu_n+\mu_r+\epsilon \eta_{k_2}-\left(l+1+\epsilon\right) \sigma$ and every solution of Eq ($E_2$) is oscillatory.

Remark 3.1. Following the same techniques used in the proof of our results, several sufficient criteria for the oscillation of both Eqs ($E_1$) and ($E_2$) can be obtained. For example, the conditions

where the sequence $\{M_l^s\}_{l\geq 0}$ is defined by (2.2) with $B_j(t) = \frac{b_j(t)}{1+R(t)}$ and $\eta_j = \mu_j-\sigma$, $j = 1, 2, \dots, n$, and

where the sequence $\{M_l^s\}_{l\geq 0}$ is defined by (2.2) with $B_j(t) = q(t)$ and $\eta_j = \mu_j$, $j = 1, 2, \dots, n$, can be proved using the same proofs of Theorems 3.2 and 3.3, respectively.

4.   Examples

Example 4.1. Consider the first-order differential equation with several delays

where $b_1(t) = \frac{18}{5}$ and $b_2(t) = \frac{1}{10}$, and $\mu_1 = \delta$, $0 < \delta < \frac{5}{36}$, and $\mu_2 = 1$. Clearly,

Then, condition (3.1) is satisfied with $l = 1$, $k_2 = k_3 = 3$, and $\epsilon = 1$, and hence $D_{t_1}$(4.1)$\leq 4\mu_2 = 4$, and every solution of Eq (4.1) is oscillatory for $0 < \delta < \frac{5}{36}$. However, $\underset{1\leq j\leq2}{\min }\mu_j = \mu_1 = \delta$. Then, $\delta$ can be chosen small enough such that all the results of ^[18] and [4,Theorem 2.2] fail to apply. Also, note that

and

for sufficiently small $\delta$. Therefore, [4,Theorem 2.1] cannot give an approximation better than $4\mu_2$.

Example 4.2. Consider the first-order neutral differential equation with several delays

where $a(t) = a^* = \max\{2\pi-\frac{99}{100}, \frac{b^* \pi}{2}-\frac{99}{100}\}$, $b_1(t) = b^* > 0$, and $b_2(t) = 2+\sin\left(2t\right)$, $\sigma = \pi$, $\mu_1 = \frac{2\pi}{3}$, and $\mu_2 = 2\pi$. It is clear that condition (H1) is satisfied with $R(t) = a^*$. Note that

for $b^* > \frac{1}{50} \frac{200 \pi+1}{\pi \left(200 \pi-199\right)}$. Then, condition (2.9) is satisfied with $B_1(t) = \frac{b^*}{1+a^*}$, $B_2(t) = \frac{2+\sin\left(2t\right)}{1+a^*}$, $\eta_1 = \mu_1-\sigma = \frac{\pi}{2}$, $\eta_2 = \mu_2-\sigma = \pi$, $l = 0$ and $\epsilon = 1$. Therefore, all requirements of Theorem 3.3 with $l = 0$ are satisfied, and hence $D_{\pi}$(4.2)$\leq 4\mu_2-2 \sigma = 6 \pi$.

5.   Conclusions

In this work, we obtained many new estimates for the upper bounds of the distance between adjacent zeros of Eqs ($E_1$) and ($E_2$). Our results are established in a general form (for inequality (1.1)), and hence they can be applied to any differential equation that can be transformed into an inequality of the form (1.1). Many new oscillation results for neutral differential equations with several not necessarily monotone delays can be obtained using the methods proposed in this work.

Use of AI tools declaration

The author declares that he has not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The author extends their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2024/01/29461).

Conflict of interest

The authors declare no conflict of interest.