Analysis of the Distance Between Successive Generalized Zeros of First-Order Linear Difference Equations with Several Variable Delays

Abstract

This study examines the distances between consecutive generalized zeros of first-order difference equations with several variable delays. We develop new techniques to obtain the upper bounds for these distances. The results are not only new, but they also extend and improve several earlier contributions in this area. In particular, our analysis covers the case of nonmonotonic delay sequences, which, to the best of our knowledge, has not been previously investigated, even for the corresponding delay differential equations. To illustrate the effectiveness of our findings, three numerical examples will be presented.

1. Introduction

Consider the first-order difference equation with several variable delays

$$\Delta \varkappa \left(n\right)+{\displaystyle \sum _{i=1}^m}{a}_i\left(n\right)\varkappa \left({\delta }_i\left(n\right)\right)=0,n\in {\mathbb{N}}_0,$$

(1)

and the following special cases:

$$\Delta \varkappa \left(n\right)+{\displaystyle \sum _{i=1}^m}{b}_i\varkappa (n-{\sigma }_i)=0,n\in {\mathbb{N}}_0$$

(2)

and

$$\Delta \varkappa \left(n\right)+b\varkappa (n-\sigma )=0,n\in {\mathbb{N}}_0,$$

(3)

where $\Delta \varkappa \left(n\right)=\varkappa (n+1)-\varkappa \left(n\right)$, $b,b_1,\dots ,b_m$ are nonnegative numbers; $\sigma ,{\sigma }_1,\dots ,{\sigma }_m$ are positive integers; ${\left(a_i\left(n\right)\right)}_{n\ge 0}$ for $1\le i\le m$ are sequences of nonnegative real numbers; and ${\left({\delta }_i\left(n\right)\right)}_{n\ge 0}$ are sequences of integers such that

$${\delta }_i\left(n\right)\le n-1,n\in {\mathbb{N}}_0\mathrm{and}\underset{n\to \infty }{\lim }{\delta }_i\left(n\right)=\infty ,1\le i\le m.$$

(4)

A solution to Equation (1) refers to a sequence of real numbers ${\left(\varkappa \left(n\right)\right)}_{n\ge -s}$, where $s=-{\displaystyle \underset{1\le i\le m,j\in {\mathbb{N}}_0}{\min }}{\delta }_i\left(j\right)$ that satisfies Equation (1) for all $n\ge 0$. It is evident that for any selection of real numbers $c_{-s},c_{-s+1},\dots ,c_{-1},c_0$, there exists a unique solution ${\left(\varkappa \left(n\right)\right)}_{n\ge -s}$ that meets the initial conditions $\varkappa (-s)=c_{-s},\varkappa (-s+1)=c_{-s+1},\dots ,\varkappa (-1)=c_{-1},\varkappa \left(0\right)=c_0$. This solution is termed oscillatory if all elements of the sequence $\varkappa \left(n\right)$ are neither eventually positive nor negative; otherwise, it is called non-oscillatory. Equation (1) is considered oscillatory if all its solutions are oscillatory, while it is termed non-oscillatory if it has at least one non-oscillatory solution.

We will rearrange Equations (1) and (2) to ensure that the following conditions are satisfied:

$$\underset{n\to \infty }{lim\; inf}{\displaystyle \sum _{i={\delta }_{k_1}\left(n\right)}^{n-1}}{a}_{k_1}\left(i\right)\ge \underset{n\to \infty }{lim\; inf}{\displaystyle \sum _{i={\delta }_{k_2}\left(n\right)}^{n-1}}{a}_{k_2}\left(i\right)\mathrm{for}1\le k_1\le k_2\le m$$

(5)

and

$$b_{k_1}{\sigma }_{k_1}\ge b_{k_2}{\sigma }_{k_2}\mathrm{for}1\le k_1\le k_2\le m,$$

respectively.

The study of delay difference equations has gained significant momentum due to their ability to describe complex phenomena in discrete time, which is the natural framework for digital computing, population dynamics with seasonal breeding, and sampled-data control systems (see [1,2,3,4,5,6,7,8,9]). While delay differential equations provide a continuous perspective, their discrete counterparts offer a more direct approach for numerical implementations and recursive modeling. Understanding the spacing between the zeros of these solutions is not only a theoretical challenge but also a practical necessity for assessing the stability and oscillatory frequency of systems subject to propagation delays.

Over the past few years, there has been a noticeable increase in the study of oscillations in delay difference equations, primarily because many natural models exhibit such behavior (see [10,11,12]). This interest parallels that in the study of the same property for the corresponding delay differential equations (see [13,14,15]). However, the study of the distance between generalized zeros of delay difference equations (DGZ-DDEs) has not received comparable attention.

In contrast, the distance between zeros of solutions to delay differential equations has attracted significant attention. For example, first-order differential equations with a single delay were examined in [16,17,18,19,20], while equations with several delays were studied in [21,22]. In addition, first-order neutral differential equations were investigated in [23,24,25,26].

Recently, the study of DGZ-DDEs has received significant attention, with numerous direct investigations devoted to this topic; see, for example [27,28,29,30]. In these works, the delays are typically assumed to be constant, that is, ${\delta }_i\left(n\right)=n-{\sigma }_i$. In more general settings, the delays are not necessarily monotone increasing, but they are still bounded above by constant delays, i.e., ${\delta }_i\left(n\right)\le n-{\sigma }_i$. On the other hand, several authors have examined related problems by studying the distribution of zeros for broader classes of equations that unify discrete and continuous frameworks. These equations are commonly referred to as dynamic equations on time scales; see, for instance [28,31,32,33]. However, to our knowledge, Equation (1), which incorporates general (nonconstant and unbounded) delays, has not yet been investigated in the literature.

In this paper, we introduce iterative functional techniques to establish lower and upper bounds for the ratio ${\displaystyle \frac{\varkappa \left({\delta }_i\left(n\right)\right)}{\varkappa \left(n\right)}}$, and we employ these bounds to investigate DGZ-DDE for all solutions of Equation (1). To this end, we extend and refine the existing methods used to study the distances between the zeros of first-order delay differential equations.

We establish several new upper bounds for DGZ-DDE of the form (1), which not only generalize but also improve previously known results for important special cases of this equation. To demonstrate the applicability of our results, we present two illustrative examples that highlight the effectiveness of our conditions in predicting oscillatory behavior and determining the locations of generalized zeros of the solutions. Moreover, a comparison with previously known results shows that they do not yield comparable approximations.

2. Main Results

Let $D_{n_1}\left(E\right)$ be the maximum positive integer such that for every $i_0\in {\mathbb{N}}_0$ with $i_0\ge n_1$, and for every solution $\left(\varkappa \right(n\left)\right)$ of a difference equation E, the elements $\varkappa \left(i_0\right),$$\varkappa (i_0+1)$, $\dots ,\varkappa (i_0+D_{n_1})$ must change their sign, or be equal to zero, at least two times.

By ${[Z,Z_1]}_{{\mathbb{N}}_0}$, we mean that all nonnegative integers belong to the interval $[Z,Z_1]$.

Let $k\in \{1,2,\dots ,m\}$. The following notation will be used:

$${\displaystyle \sum _{i=r}^{r-1}}I\left(i\right)=0\mathrm{and}{\displaystyle \prod _{i=r}^{r-1}}I\left(i\right)=1.$$

$$w_k\left(n\right)=\underset{1\le j\le k}{max}{\delta }_j\left(n\right),$$

$$M_k\left(n\right)=max\{i\in \mathbb{N}:w_k\left(i\right)\le n-1\},$$

$$M_k^1\left(n\right)=M_k\left(n\right)+1,M_k^l\left(n\right)=M_k\left(M_k^{l-1}\left(n\right)\right)+1,l=2,3,\dots ,$$

$$W\left(n\right)=max\{i\in \mathbb{N}:\underset{1\le j\le m}{min}{\delta }_j\left(i\right)\le n-1\},$$

$$W^1\left(n\right)=W\left(n\right)+1,W^l\left(n\right)=W\left(W^{l-1}\left(n\right)\right)+1,l=2,3,\dots$$

It is important to note that the sequences ${\left\{M_i\left(n\right)\right\}}_{i=1}^m$ satisfy the following inequality:

$$M_1\left(n\right)\ge M_2\left(n\right)\ge \cdots \ge M_m\left(n\right)\mathrm{for}n\ge n_1.$$

Furthermore, if $n_1\ge M_1^1\left(n\right)$, then it follows that $n_1>M_1\left(n\right)$. Consequently, we have $w_k\left(n_1\right)>n-1$, which implies that ${\delta }_i\left(n_1\right)\ge n$ for $i=1,2,\dots ,k$.

Similarly, if $n_1\ge W^1\left(n\right)$, it can be deduced that $n_1>W\left(n\right)$, leading to ${\delta }_i\left(n_1\right)\ge n$ for $i=1,2,\dots ,m$.

Also, for $k\in \{1,2,\dots ,m\}$, let the sequence ${\left\{{\Psi }_k^l\right\}}_{l\ge 0}$ be defined by ${\Psi }_k^0=1$ and

$${\displaystyle \prod _{s=w_k\left(n\right)}^{n-1}}{\displaystyle \frac{1}{1-{\sum }_{i=1}^m{a}_i\left(s\right){\Psi }_k^{l-1}}}\ge {\Psi }_k^l,l=1,2,\dots$$

Lemma 1. 

Let $l\in {\mathbb{N}}_0$, $k\in \{1,2,\dots ,m\}$ and $\varkappa \left(n\right)$ be a solution of Equation (1) such that $\varkappa \left(n\right)>0$ on ${[Z,Z_1]}_{{\mathbb{N}}_0}$. Then

$$\begin{array}{c}\hfill {\displaystyle \frac{\varkappa \left(w_k\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\Psi }_k^l>0,n\in {[M_k^1\left(W^{l+1}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0},Z_1\ge M_k^1\left(W^{l+1}\left(Z\right)\right)\end{array}$$

(6)

and

$$\begin{array}{c}\hfill {\displaystyle \frac{\varkappa \left({\delta }_i\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\Psi }_i^l>0,i=1,2,\dots ,m,n\in {[W^{l+2}\left(Z\right),Z_1]}_{{\mathbb{N}}_0},Z_1\ge W^{l+2}\left(Z\right).\end{array}$$

(7)

Proof. 

Since $\varkappa \left(n\right)>0$ on ${[Z,Z_1]}_{{\mathbb{N}}_0}$, it follows from (1) that

$$\Delta \varkappa \left(n\right)=-{\displaystyle \sum _{i=1}^m}{a}_i\left(n\right)\varkappa \left({\delta }_i\left(n\right)\right)<0,n\in {[W^1\left(Z\right),Z_1]}_{{\mathbb{N}}_0},$$

and hence $\varkappa \left(n\right)$ is nonincreasing on ${[W^1\left(Z\right),Z_1]}_{{\mathbb{N}}_0}$. Therefore,

$${\displaystyle \frac{\varkappa \left(w_k\left(n\right)\right)}{\varkappa \left(n\right)}}\ge 1={\Psi }_k^0,n\in {[M_k^1\left(W^1\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0},$$

and

$${\displaystyle \frac{\varkappa \left({\delta }_i\left(n\right)\right)}{\varkappa \left(n\right)}}\ge 1={\Psi }_i^0,i=1,2,\dots ,m,n\in {[W^2\left(Z\right),Z_1]}_{{\mathbb{N}}_0}.$$

(8)

Dividing Equation (1) by $\varkappa \left(n\right)$, and then taking the product, from u to $v-1$, $v\ge u$, we get

$$0<{\displaystyle \frac{\varkappa \left(v\right)}{\varkappa \left(u\right)}}={\displaystyle \prod _{s=u}^{v-1}}{\displaystyle \frac{\varkappa (s+1)}{\varkappa \left(s\right)}}={\displaystyle \prod _{s=u}^{v-1}}\left(1-{\displaystyle \sum _{i=1}^m}{a}_i\left(s\right){\displaystyle \frac{\varkappa \left({\delta }_i\left(s\right)\right)}{\varkappa \left(s\right)}}\right),$$

(9)

that is,

$$\varkappa \left(u\right)=\varkappa \left(v\right){\displaystyle \prod _{s=u}^{v-1}}{\displaystyle \frac{1}{1-{\sum }_{i=1}^m{a}_i\left(s\right)\frac{\varkappa \left({\delta }_i\left(s\right)\right)}{\varkappa \left(s\right)}}},v\ge u,u\in {[W^1\left(Z\right),Z_1]}_{{\mathbb{N}}_0}.$$

(10)

This together with (8) leads to

$${\displaystyle \frac{\varkappa \left(w_k\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\displaystyle \prod _{s=w_k\left(n\right)}^{n-1}}{\displaystyle \frac{1}{1-{\sum }_{i=1}^m{a}_i\left(s\right){\Psi }_i^0}}\ge {\Psi }_k^1>0,n\in {[M_k^1\left(W^2\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$$

and

$${\displaystyle \frac{\varkappa \left({\delta }_i\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\displaystyle \frac{\varkappa \left(w_i\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\displaystyle \prod _{s=w_i\left(n\right)}^{n-1}}{\displaystyle \frac{1}{1-{\sum }_{i_1=1}^m{a}_{i_1}\left(s\right){\Psi }_{i_1}^0}}\ge {\Psi }_i^1>0,i=1,2,\dots ,m$$

(11)

for $n\in {[W^3\left(Z\right),Z_1]}_{{\mathbb{N}}_0}$. From (10) and (11), we obtain

$$\varkappa \left(u\right)\ge \varkappa \left(v\right){\displaystyle \prod _{s=u}^{v-1}}{\displaystyle \frac{1}{1-{\sum }_{i=1}^m{a}_i\left(s\right){\Psi }_i^1}}\mathrm{for} \mathrm{all}v\ge u,u\in {[W^3\left(Z\right),Z_1]}_{{\mathbb{N}}_0}.$$

Then

$${\displaystyle \frac{\varkappa \left(w_k\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\displaystyle \prod _{s=w_k\left(n\right)}^{n-1}}{\displaystyle \frac{1}{1-{\sum }_{i=1}^m{a}_i\left(s\right){\Psi }_i^1}}\ge {\Psi }_k^2>0,n\in {[M_k^1\left(W^3\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$$

and

$${\displaystyle \frac{\varkappa \left({\delta }_i\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\displaystyle \prod _{s=w_i\left(n\right)}^{n-1}}{\displaystyle \frac{1}{1-{\sum }_{i_1=1}^m{a}_{i_1}\left(s\right){\Psi }_{i_1}^1}}\ge {\Psi }_i^2>0,i=1,2,\dots ,m,n\in {[W^4\left(Z\right),Z_1]}_{{\mathbb{N}}_0}.$$

By repeating the arguments stated above, we derive (6) and (7). □

Let

$$\rho \left(n\right)=min\{k\in \mathbb{N}:\underset{1\le i\le m}{max}{\delta }_i\left(k\right)\ge n\},$$

$$V\left(n\right)=\underset{1\le i\le m}{max}{\delta }_i\left(n\right),$$

and

$$V^1\left(n\right)=V\left(n\right)-1,V^l\left(n\right)=V\left(V^{l-1}\left(n\right)+1\right)-1,l=2,3,\dots .$$

To justify that $\rho \left(n\right)$ is well-defined, we note that by assumption

$$\underset{k\to \infty }{lim}\underset{1\le i\le m}{max}{\delta }_i\left(k\right)=\infty .$$

Hence, for any fixed $n\in \mathbb{N}$, there exists $k_0\in \mathbb{N}$ such that

$$\underset{1\le i\le m}{max}{\delta }_i\left(k_0\right)\ge n.$$

Therefore, the set

$$\{k\in \mathbb{N}:\underset{1\le i\le m}{max}{\delta }_i\left(k\right)\ge n\}$$

is nonempty, and since it is a subset of $\mathbb{N}$, it admits a minimum element. Consequently, $\rho \left(n\right)$ is well-defined. Moreover, since ${max}_{1\le i\le m}{\delta }_i\left(k\right)\le k-1$ for all $k\in \mathbb{N}$, it follows that

$$\rho \left(n\right)\ge n+1.$$

We also define the sequences ${\left\{{\Phi }_k^l\right\}}_{l\ge 0}$ for $k\in \{1,2,\dots ,m\}$ as follows:

$$\begin{array}{ccc}\hfill {\Phi }_k^1& \ge & {\displaystyle \frac{1-{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right)}{{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right){\sum }_{i_2={\delta }_{i_1}\left(i\right)}^n{\sum }_{i_3=1}^k{a}_{i_3}\left(i_2\right)}},\hfill \\ \hfill {\Phi }_k^l& \ge & {\displaystyle \frac{1-\frac{1}{{\Phi }_k^{l-1}}-{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right)}{{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right){\sum }_{i_2={\delta }_{i_1}\left(i\right)}^n{\sum }_{i_3=1}^k{a}_{i_3}\left(i_2\right)}},l=1,2,\dots \hfill \end{array}$$

Lemma 2. 

Let $l,l_1\in \mathbb{N}$, $k\in \{1,2,\dots ,m\}$ and $\varkappa \left(n\right)$ be a solution of Equation (1) such that $\varkappa \left(n\right)>0$ on ${[Z,W^{l_1}\left(Z\right)]}_{{\mathbb{N}}_0}$, $V^l\left(W^{l_1}\left(Z\right)\right)\ge W^3\left(Z\right)-1$. Then

$${\displaystyle \frac{\varkappa \left({\omega }_k\left(n\right)\right)}{\varkappa (n+1)}}<{\Phi }_k^l,n\in {[W^3\left(Z\right)-1,V^l\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$$

(12)

Proof. 

Summing up Equation (1) for $k=n+1,\dots ,\rho (n+1)-1$, we obtain

$$\varkappa \left(\rho (n+1)\right)-\varkappa (n+1)+{\displaystyle \sum _{i=n+1}^{\rho (n+1)-1}}{\displaystyle \sum _{i_1=1}^m}{a}_{i_1}\left(i\right)\varkappa \left({\delta }_{i_1}\left(i\right)\right)=0$$

(13)

for $n\in {[W^1\left(Z\right)-1,V^1\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}$. We will demonstrate the following:

1.

$\rho (n+1)\le W^{l_1}\left(Z\right)\mathrm{for}n\le V^1\left(W^{l_1}\left(Z\right)\right).$

2.

For $i_1\in \{1,2,\dots ,m\}$,

$$Z\le {\delta }_{i_1}\left(i\right)\le n\mathrm{for}n+1\le i\le \rho (n+1)-1,n\in {[W^1\left(Z\right)-1,V^1\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$$

To prove the first claim, assume that $n\le V^1\left(W^{l_1}\left(Z\right)\right)$, so $n\le V\left(W^{l_1}\left(Z\right)\right)-1$, and hence $n+1\le {max}_{1\le i\le m}{\delta }_i\left(W^{l_1}\left(Z\right)\right)$. This shows that $W^{l_1}\left(Z\right)$ meets the criteria for being one of the integers included in the definition of $\rho (n+1)$; therefore,

$$\rho (n+1)\le W^{l_1}\left(Z\right)\mathrm{for}n\le V^1\left(W^{l_1}\left(Z\right)\right).$$

Thus, the first statement has been established.

For the second claim, we first show that ${\delta }_{i_1}\left(i\right)\le n$. Suppose, on the contrary, that there exists $i_{*}\in \{1,2,\dots ,m\}$ such that ${\delta }_{i_{*}}\left(i\right)\ge n+1$. Therefore,

$$\underset{1\le i_1\le m}{\max }{\delta }_{i_1}\left(i\right)\ge n+1.$$

In view of the definition of $\rho (n+1)$, it follows that $i\ge \rho (n+1)$. This contradicts the assumption that $i\le \rho (n+1)-1$. Therefore, ${\delta }_{i_1}\left(i\right)\le n$. To complete the proof of the second statement, we still need to demonstrate that ${\delta }_{i_1}\left(i\right)\ge Z$. We will clarify this as follows:

Since $i\ge n+1\ge W^1\left(Z\right)$, it follows that $i\ge W^1\left(Z\right)=W\left(Z\right)+1$, which implies $i>W\left(Z\right)$. According to the definition of $W\left(Z\right)$, we have ${\min }_{1\le r\le m}{\delta }_r\left(i\right)\ge Z$; thus, ${\delta }_{i_1}\left(i\right)\ge Z$.

Summing up Equation (1) from ${\delta }_{i_1}\left(i\right)$ to n (where $i_1\in \{1,2,\dots ,m\}$), we obtain

$$\varkappa \left({\delta }_{i_1}\left(i\right)\right)=\varkappa (n+1)+{\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^n}{\displaystyle \sum _{i_3=1}^m}{a}_{i_3}\left(i_2\right)\varkappa \left({\delta }_{i_3}\left(i_2\right)\right)$$

for $n+1\le i\le \rho (n+1)-1,n\in {[W^1\left(Z\right)-1,V^1\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}$.

• This, along with (13), implies that

  $$\begin{array}{c}\hfill \begin{array}{c}\varkappa \left(\rho (n+1)\right)-\varkappa (n+1)+\varkappa (n+1){\displaystyle \sum _{i=n+1}^{\rho (n+1)-1}}{\displaystyle \sum _{i_1=1}^m}{a}_{i_1}\left(i\right)\\ +{\displaystyle \sum _{i=n+1}^{\rho (n+1)-1}}{\displaystyle \sum _{i_1=1}^m}{a}_{i_1}\left(i\right){\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^n}{\displaystyle \sum _{i_3=1}^m}{a}_{i_3}\left(i_2\right)\varkappa \left({\delta }_{i_3}\left(i_2\right)\right)=0\end{array}\end{array}$$

  (14)

  for $n\in {[W^1\left(Z\right)-1,V^1\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$ By employing the definition of $W^3\left(Z\right)$, it can be established that

  $${\delta }_{i_3}\left(i_2\right)\ge W^1\left(Z\right)\mathrm{for}{i}_2\in \{{\delta }_{i_1}\left(i\right),\dots ,n\},i\in \{n+1,\dots ,\rho (n+1)-1\}$$

  for $n\ge W^3\left(Z\right)-1$. Using the positivity and non-increasing nature of $\varkappa \left({\delta }_{i_3}\left(i_2\right)\right)$, Equation (14) leads to

  $$\begin{array}{c}\varkappa \left(\rho (n+1)\right)-\varkappa (n+1)+\varkappa (n+1){\displaystyle \sum _{i=n+1}^{\rho (n+1)-1}}{\displaystyle \sum _{i_1=1}^m}{a}_{i_1}\left(i\right)\hfill \\ +\varkappa \left({\omega }_k\left(n\right)\right){\displaystyle \sum _{i=n+1}^{\rho (n+1)-1}}{\displaystyle \sum _{i_1=1}^m}{a}_{i_1}\left(i\right){\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^n}{\displaystyle \sum _{i_3=1}^k}{a}_{i_3}\left(i_2\right)\le 0\hfill \end{array}$$

  (15)

  for $n\in [W^3\left(Z\right)-1,V^1\left(W^{l_1}\left(Z\right)\right)].$ Since $\varkappa \left(\rho \right(n+1\left)\right)>0$ in ${[W^3\left(Z\right)-1,V^1\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}$, then

  $${\displaystyle \frac{\varkappa \left({\omega }_k\left(n\right)\right)}{\varkappa (n+1)}}<{\displaystyle \frac{1-{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right)}{{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right){\sum }_{i_2={\delta }_{i_1}\left(i\right)}^n{\sum }_{i_3=1}^k{a}_{i_3}\left(i_2\right)}}\le {\Phi }_k^1$$

  (16)

  for $n\in {[W^3\left(Z\right)-1,V^1\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}$.

Using the fact that $\rho (n+1)-1\ge n$ and the definition of the sequences $W^l\left(n\right)$ and $\rho \left(n\right)$, we can prove that

$$\rho (n+1)-1\in {[W^3\left(Z\right)-1,V^1\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}\mathrm{for}n\in {[W^3\left(Z\right)-1,V^2\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$$

This, along with (16), leads to

$${\displaystyle \frac{\varkappa \left({\omega }_k(\rho (n+1)-1)\right)}{\varkappa \left(\rho \right(n+1\left)\right)}}<{\Phi }_k^1,n\in {[W^3\left(Z\right)-1,V^2\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$$

(17)

Next, we demonstrate that

$${\omega }_k(\rho (n+1)-1)<n+1.$$

(18)

Assuming the contrary, let us suppose that ${\omega }_k(\rho (n+1)-1)\ge n+1$. This implies that ${max}_{1\le i\le m}{\delta }_i(\rho (n+1)-1)\ge n+1$. According to the definition of $\rho \left(n\right)$, it follows that $\rho (n+1)-1\ge \rho (n+1)$. This leads to a contradiction, thereby proving that ${\omega }_k(\rho (n+1)-1)<n+1$.

Furthermore, for $n\ge W^3\left(Z\right)-1$, it can be readily established that $\rho (n+1)-1>W^2\left(Z\right)$. Therefore, $\varkappa \left({\omega }_k(\rho (n+1)-1)\right)$ is non-increasing. Using (18), we have

$$\varkappa \left({\omega }_k(\rho (n+1)-1)\right)\ge \varkappa (n+1),n\in {[W^3\left(Z\right)-1,V^2\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$$

From this and (17), we obtain

$$\varkappa \left(\rho (n+1)\right)>{\displaystyle \frac{1}{{\Phi }_k^1}}\varkappa (n+1),n\in {[W^3\left(Z\right)-1,V^2\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$$

Substituting into (15) and rearranging, it follows that

$${\displaystyle \frac{\varkappa \left({\omega }_k\left(n\right)\right)}{\varkappa (n+1)}}<{\displaystyle \frac{1-\frac{1}{{\Phi }_k^1}-{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right)}{{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right){\sum }_{i_2={\delta }_{i_1}\left(i\right)}^n{\sum }_{i_3=1}^k{a}_{i_3}\left(i_2\right)}}\le {\Phi }_k^2,$$

for $n\in {[W^3\left(Z\right)-1,V^2\left(W^{l_1}\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$

By iterating this procedure l times, we obtain (12). □

Theorem 1. 

Let $l*$ be the smallest positive integer such that ${\Psi }_k^{l*}$ is negative or undefined, and l be the smallest nonnegative integer that satisfies

$$V^{l_2}\left(W^l\left(n\right)\right)\ge \max \{M_k^1\left(W^{l_1+1}\left(n\right)\right),W^3\left(Z\right)-1\},$$

$l_1$ and $l_2$ are the smallest positive integers such that ${\Psi }_k^{l_1}\ge {\Phi }_k^{l_2}.$

Equation (1) is oscillatory, and every solution has at least one zero in

$$[n-1,W^l\left(n\right)]or[n-1,M_k^1\left(W^{l*+1}\left(n\right)\right)],n\ge n_1.$$

Moreover, if

$$\underset{n\ge n_1}{sup}\left(W^l\left(n\right)-(n-1)\right)<\infty and\underset{n\ge n_1}{sup}\left(M_k^1\left(W^{l*+1}\left(n\right)\right)-(n-1)\right)<\infty ,$$

then

$$D_{n_1}(\mathrm{Equation} \left(1\right))\le min\left\{\underset{n\ge n_1}{sup}\left(W^l\left(n\right)-(n-1)\right),\underset{n\ge n_1}{sup}\left(M_k^1\left(W^{l*+1}\left(n\right)\right)-(n-1)\right)\right\}.$$

Proof. 

Let $\varkappa \left(n\right)$ be a positive solution of Equation (1) on ${[Z,W^l\left(Z\right)]}_{{\mathbb{N}}_0}$. Then, Lemmas 1 and 2 imply, respectively,

$$\begin{array}{c}\hfill {\displaystyle \frac{\varkappa \left(w_k\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\Psi }_k^{l_1},n\in {[M_k^1\left(W^{l_1+1}\left(Z\right)\right),W^l\left(Z\right)]}_{{\mathbb{N}}_0}\end{array}$$

and

$${\displaystyle \frac{\varkappa \left({\omega }_k\left(n\right)\right)}{\varkappa (n+1)}}<{\Phi }_k^{l_2},n\in {[W^3\left(Z\right)-1,V^{l_2}\left(W^l\left(Z\right)\right)]}_{{\mathbb{N}}_0}.$$

Therefore, for $n*=V^{l_2}\left(W^l\left(Z\right)\right)$, we have

$${\Psi }_k^{l_1}\le {\displaystyle \frac{\varkappa \left({\omega }_k\left(V^{l_2}\left(W^l\left(Z\right)\right)\right)\right)}{\varkappa \left(V^{l_2}\left(W^l\left(Z\right)\right)\right)}}\le {\displaystyle \frac{\varkappa \left({\omega }_k\left(V^{l_2}\left(W^l\left(Z\right)\right)\right)\right)}{\varkappa (V^{l_2}\left(W^l\left(Z\right)\right)+1)}}<{\Phi }_k^{l_2}.$$

This contradiction concludes the proof of the first part. Now, let us assume that $\varkappa \left(n\right)$ is a positive solution of Equation (1) on the interval ${[M_k^1\left(W^{l*+1}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$, where $Z_1\ge M_k^1\left(W^{l*+1}\left(Z\right)\right)$. In this case, Lemma 1 implies that

$${\displaystyle \frac{\varkappa \left(w_k\left(n\right)\right)}{\varkappa \left(n\right)}}\ge {\Psi }_k^{l*}>0,n\in [M_k^1\left(W^{l*+1}\left(Z\right)\right),Z_1].$$

This contradicts the definition of $l*$. Therefore, any solution has at least one generalized zero in any interval of the forms ${[Z,W^l\left(Z\right)]}_{{\mathbb{N}}_0}$ and ${[Z,M_1^{l*}\left(W^l\left(Z\right)\right)]}_{{\mathbb{N}}_0}$. To complete the proof, let us assume the existence of a positive integer $n*$ such that $\varkappa (n*-1)\le 0$ and $\varkappa \left(n*\right)>0$. Using the same method of the proof as before, we can prove that $\varkappa \left(n\right)$ changes its sign (or is equal to zero) in ${[n*,M_k^{l*}\left(W^{l+1}\left(n*\right)\right)]}_{{\mathbb{N}}_0}$ and in ${[n*,W^l\left(n*\right)]}_{{\mathbb{N}}_0}$. Then, $\varkappa \left(n\right)$ changes its sign (or equal to zero) at least twice in ${[n*-1,M_1^{l*}\left(W^l\left(n*\right)\right)]}_{{\mathbb{N}}_0}$ and in ${[n*-1,W^l\left(n*\right)]}_{{\mathbb{N}}_0}$. □

In the following, we assume that ${\delta }_i\left(n\right)$ is non-decreasing for $i=1,2,\dots ,m$.

Let $k\in \{1,2,\dots ,m\}$ and the sequence ${\left\{R_{i_1,k}^j(i,n)\right\}}_{j\ge 1}$, be defined by

$$\begin{array}{ccc}\hfill R_{i_1,k}^1(i,n)& =& a_{i_1}\left(i\right),\hfill \\ \hfill R_{i_1,k}^j(i,n)& =& a_{i_1}\left(i\right){\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^{w_k\left(n\right)-1}}{\displaystyle \sum _{i_3=1}^k}{R}_{i_3,k}^{j-1}(i_2,w_k\left(n\right)),j=2,3,\dots \hfill \end{array}$$

Theorem 2. 

Let $l\in \mathbb{N}$, and $k\in \{1,2,\dots ,m\}$. If

$${\displaystyle \sum _{j=1}^l}\left({\displaystyle \prod _{i_2=2}^j}{\Psi }_k^{l+1-i_2}\right){\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{R}_{i_1,k}^j(i,n)\ge 1,$$

(19)

then Equation (1) oscillates, and every solution has at least one zero in

$$[n-1,M_k^1\left(W^{l+1}\left(n\right)\right)],n\ge n_1.$$

If

$$\underset{n\ge n_1}{sup}\left(M_k^1\left(W^{l+1}\left(n\right)\right)-(n-1)\right)<\infty ,$$

then

$$D_{n_1}\left(Equation \left(1\right)\right)\le \underset{n\ge n_1}{sup}\left(M_k^1\left(W^{l+1}\left(n\right)\right)-(n-1)\right).$$

Proof. 

Suppose the opposite, and let $\varkappa \left(n\right)>0$ for ${[Z-1,Z_1]}_{\mathbb{N}0}$, where $Z_1\ge M_k^1\left(W^{l+1}\left(Z\right)\right)$. We first demonstrate that $\varkappa \left(n\right)$ cannot remain positive on the interval $[Z,Z_1]{\mathbb{N}}_0$. It follows from Equation (1) that

$$\Delta \varkappa \left(n\right)+{\displaystyle \sum _{i=1}^k}{a}_i\left(n\right)\varkappa \left({\delta }_i\left(n\right)\right)\le 0,n\in {[W^1\left(Z\right),Z_1]}_{{\mathbb{N}}_0}.$$

(20)

Summing up from r to $s-1$, we get

$$\varkappa \left(s\right)-\varkappa \left(r\right)+{\displaystyle \sum _{i=r}^{s-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right)\varkappa \left({\delta }_{i_1}\left(i\right)\right)\le 0,s\ge r,s\le Z_1,r\in {[W^1\left(Z\right),Z_1]}_{{\mathbb{N}}_0}.$$

(21)

Then,

$$\varkappa \left(n\right)-\varkappa \left(w_k\left(n\right)\right)+{\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right)\varkappa \left({\delta }_{i_1}\left(i\right)\right)\le 0,n\in {[M_k^1\left(W^1\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.$$

(22)

In view of $w_k\left(n\right)\ge {\delta }_{i_1}\left(n\right),i_1=1,2,\dots k$, it follows from the non-decreasing natural of ${\delta }_{i_1}$ that $w_k\left(n\right)\ge {\delta }_{i_1}\left(i\right)$ for $w_k\left(n\right)\le i<n$, $i_1=1,2,\dots k$. Again, Equation (21), yields

$$\varkappa \left(w_k\left(n\right)\right)-\varkappa \left({\delta }_{i_1}\left(i\right)\right)+{\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^{w_k\left(n\right)-1}}{\displaystyle \sum _{i_3=1}^k}{a}_{i_3}\left(i_2\right)\varkappa \left({\delta }_{i_3}\left(i_2\right)\right)\le 0,n\in {[M_k^1\left(W^2\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.$$

(23)

From (22) and (23), we obtain

$$\varkappa \left(n\right)-\varkappa \left(w_k\left(n\right)\right)+{\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right)\left(\varkappa \left(w_k\left(n\right)\right)+{\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^{w_k\left(n\right)-1}}{\displaystyle \sum _{i_3=1}^k}{a}_{i_3}\left(i_2\right)\varkappa \left({\delta }_{i_3}\left(i_2\right)\right)\right)\le 0$$

for $n\in {[M_k^1\left(W^2\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$, that is,

$$\begin{array}{cc}\hfill \varkappa \left(n\right)& - \varkappa \left(w_k\left(n\right)\right)\left(1-{\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right)\right)\hfill \\ & + {\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right){\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^{w_k\left(n\right)-1}}{\displaystyle \sum _{i_3=1}^k}{a}_{i_3}\left(i_2\right)\varkappa \left({\delta }_{i_3}\left(i_2\right)\right)\le 0\hfill \end{array}$$

(24)

for $n\in {[M_k^1\left(W^2\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$. Since $w_k^2\left(n\right)\ge {\delta }_{i_3}\left(w_k\left(n\right)\right)\ge {\delta }_{i_3}\left(i_2\right)$, $i_3=1,2,\dots ,r$ for ${\delta }_{i_1}\left(i\right)\le i_2<w_k\left(n\right),w_k\left(n\right)\le i<n$. It follows from (21) that

$$\varkappa \left(w_k^2\left(n\right)\right)-\varkappa \left({\delta }_{i_3}\left(i_2\right)\right)+{\displaystyle \sum _{i_4={\delta }_{i_3}\left(i_2\right)}^{w_k^2\left(n\right)-1}}{\displaystyle \sum _{i_5=1}^k}{a}_{i_5}\left(i_4\right)\varkappa \left({\delta }_{i_5}\left(i_4\right)\right)\le 0,n\in {[M_k^1\left(W^3\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.$$

(25)

Substituting into (24), we have

$$\begin{array}{cc}\hfill \varkappa \left(n\right)& - \varkappa \left(w_k\left(n\right)\right)\left(1-{\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right)\right)\hfill \\ & + \varkappa \left(w_k^2\left(n\right)\right){\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right){\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^{w_k\left(n\right)-1}}{\displaystyle \sum _{i_3=1}^k}{a}_{i_3}\left(i_2\right)\hfill \\ & + {\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right){\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^{w_k\left(n\right)-1}}{\displaystyle \sum _{i_3=1}^k}{a}_{i_3}\left(i_2\right){\displaystyle \sum _{i_4={\delta }_{i_3}\left(i_2\right)}^{w_k^2\left(n\right)-1}}{\displaystyle \sum _{i_5=1}^k}{a}_{i_5}\left(i_4\right)\varkappa \left({\delta }_{i_5}\left(i_4\right)\right)\le 0\hfill \end{array}$$

(26)

for $n\in {[M_k^1\left(W^3\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.$ Therefore,

$$\begin{array}{cc}\hfill \varkappa \left(n\right)& - \varkappa \left(w_k\left(n\right)\right)+{\displaystyle \sum _{j=1}^2}\varkappa \left(w_k^j\left(n\right)\right){\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{R}_{i_1,k}^j(i,n)\hfill \\ & + {\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right){\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^{w_k\left(n\right)-1}}{\displaystyle \sum _{i_3=1}^k}{a}_{i_3}\left(i_2\right)\hfill \\ & \times {\displaystyle \sum _{i_4={\delta }_{i_3}\left(i_2\right)}^{w_k^2\left(n\right)-1}}{\displaystyle \sum _{i_5=1}^k}{a}_{i_5}\left(i_4\right)\varkappa \left({\delta }_{i_5}\left(i_4\right)\right)\le 0,\mathrm{for}n\in {[M_k^1\left(W^3\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.\hfill \end{array}$$

where $w_k^1\left(n\right)=w_k\left(n\right)$. By using the positivity of $\varkappa$, we obtain

$$\varkappa \left(n\right)-\varkappa \left(w_k\left(n\right)\right)+{\displaystyle \sum _{j=1}^2}\varkappa \left(w_k^j\left(n\right)\right){\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{R}_{i_1}^j(i,n)\le 0\mathrm{for}n\in {[M_k^1\left(W^3\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.$$

Following l iterations of this argument, we arrive at

$$\varkappa \left(n\right)-\varkappa \left(w_k\left(n\right)\right)+{\displaystyle \sum _{j=1}^l}\varkappa \left(w_k^j\left(n\right)\right){\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{R}_{i_1,k}^j(i,n)\le 0\mathrm{for}n\in {[M_k^1\left(W^{l+1}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.$$

(27)

Clearly

$$\varkappa \left(w_k^j\left(n\right)\right)=\left({\displaystyle \prod _{i_2=2}^j}{\displaystyle \frac{\varkappa \left(w_k^{i_2}\left(n\right)\right)}{\varkappa \left(w_k^{i_2-1}\left(n\right)\right)}}\right)\varkappa \left(w_k\left(n\right)\right),j=1,2,\dots ,l.$$

(28)

It can be inferred from $n\in {[M_k^1\left(W^{l+1}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$, that $w_k^{i_2-1}\left(n\right)\in {[M_k^1\left(W^{l+2-i_2}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$. This, in conjunction with Equation (6), results in

$${\displaystyle \frac{\varkappa \left(w_k^{i_2}\left(n\right)\right)}{\varkappa \left(w_k^{i_2-1}\left(n\right)\right)}}\ge {\Psi }_k^{l+1-i_2}\mathrm{for}n\in {[M_k^1\left(W^{l+1}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.$$

Substituting into (28), we obtain

$$\varkappa \left(w_k^j\left(n\right)\right)\ge \left({\displaystyle \prod _{i_2=2}^j}{\Psi }_k^{l+1-i_2}\right)\varkappa \left(w_k\left(n\right)\right)\mathrm{for}n\in {[M_k^1\left(W^{l+1}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0},j=1,2,\dots ,l.$$

This, together with (27), leads to

$$\varkappa \left(n\right)+\left({\displaystyle \sum _{j=1}^l}\left({\displaystyle \prod _{i_2=2}^j}{\Psi }_k^{l+1-i_2}\right){\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{R}_{i_1,k}^j(i,n)-1\right)\varkappa \left(w_k\left(n\right)\right)\le 0$$

(29)

for $n\in {[M_k^1\left(W^{l+1}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$.

This contradiction demonstrates that $\varkappa \left(n\right)$ cannot be positive on the interval ${[Z,Z_1]}_{{\mathbb{N}}_0}$. To finalize the proof, we can show that any solution $\varkappa \left(n\right)$ has at least two generalized zeros in the interval ${[Z-1,Z_1]}_{{\mathbb{N}}_0}$ by applying the same method used at the conclusion of the proof of Theorem 1. □

Theorem 3. 

Let $l\in \mathbb{N}$. If

$${\displaystyle \prod _{j=1}^m}{\left({\displaystyle \prod _{i=1}^m}{\displaystyle \sum _{i_1=w_j\left(n\right)}^{n-1}}{a}_i\left(i_1\right){\displaystyle \prod _{i_2={\delta }_i\left(i_1\right)}^{w_i\left(n\right)-1}}{\displaystyle \frac{1}{1-{\sum }_{i_3=1}^m{a}_{i_3}\left(i_2\right){\Psi }_{i_3}^{l-1}}}\right)}^{{\displaystyle \frac{1}{m}}}\ge {\displaystyle \frac{1}{m^m}},$$

(30)

then Equation (1) oscillates, and every solution has at least one zero in

$$[n-1,W^{l+3}\left(n\right)],n\ge n_1.$$

If

$$\underset{n\ge n_1}{sup}\left(W^{l+3}\left(n\right)-(n-1)\right)<\infty ,$$

then

$$D_{n_1}\left(Equation \left(1\right)\right)\le \underset{n\ge n_1}{sup}\left(W^{l+3}\left(n\right)-(n-1)\right).$$

Proof. 

Let $\varkappa \left(n\right)$ be a positive solution of Equation (1) on ${[Z-1,Z_1]}_{{\mathbb{N}}_0}$, $W^{l+3}\left(Z_1\right)\ge Z$. First, we clarify that $\varkappa \left(n\right)$ cannot remain positive over the interval ${[Z,Z_1]}_{{\mathbb{N}}_0}$. Summing up Equation (1) from $w_j\left(n\right)$ to $n-1$, where $j\in \{1,\dots ,m\}$, we get

$$\varkappa \left(n\right)-\varkappa \left(w_j\left(n\right)\right)+{\displaystyle \sum _{i_1=w_j\left(n\right)}^{n-1}}{\displaystyle \sum _{i=1}^m}{a}_i\left(i_1\right)\varkappa \left({\delta }_i\left(i_1\right)\right)=0,n\in {[M_j^1\left(W^1\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}.$$

Using (10) and $w_i\left(n\right)\ge {\delta }_i\left(i_1\right)$ for $w_j\left(n\right)\le i_1<n$, the last equation takes the form

$$\varkappa \left(n\right)-\varkappa \left(w_j\left(n\right)\right)+{\displaystyle \sum _{i_1=w_j\left(n\right)}^{n-1}}{\displaystyle \sum _{i=1}^m}{a}_i\left(i_1\right)\varkappa \left(w_i\left(n\right)\right){\displaystyle \prod _{i_2={\delta }_i\left(i_1\right)}^{w_i\left(n\right)-1}}{\displaystyle \frac{1}{1-{\sum }_{i_3=1}^m{a}_{i_3}\left(i_2\right)\frac{\varkappa \left({\delta }_{i_3}\left(i_2\right)\right)}{\varkappa \left(i_2\right)}}}=0$$

for $n\in {[M_j^1\left(W^2\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$. From (7) and the positivity of $\varkappa \left(n\right),$ we have

$$\varkappa \left(w_j\left(n\right)\right)>{\displaystyle \sum _{i=1}^m}\varkappa \left(w_i\left(n\right)\right){\displaystyle \sum _{i_1=w_j\left(n\right)}^{n-1}}{a}_i\left(i\right){\displaystyle \prod _{i_2={\delta }_i\left(i_1\right)}^{w_i\left(n\right)-1}}{\displaystyle \frac{1}{1-{\sum }_{i_3=1}^m{a}_{i_3}\left(i_2\right){\Psi }_{i_3}^{l-1}}}=0$$

for $n\in {[M_j^1\left(W^{l+2}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$. Using the arithmetic–geometric mean, we get

$$\varkappa \left(w_j\left(n\right)\right)>m{\left({\displaystyle \prod _{i=1}^m}\varkappa \left(w_i\left(n\right)\right)\right)}^{{\displaystyle \frac{1}{m}}}{\left({\displaystyle \prod _{i=1}^m}{\displaystyle \sum _{i_1=w_j\left(n\right)}^{n-1}}{a}_i\left(i_1\right){\displaystyle \prod _{i_2={\delta }_i\left(i_1\right)}^{w_i\left(n\right)-1}}{\displaystyle \frac{1}{1-{\sum }_{i_3=1}^m{a}_{i_3}\left(i_2\right){\Psi }_{i_3}^{l-1}}}\right)}^{{\displaystyle \frac{1}{m}}}$$

for $n\in {[M_j^1\left(W^{l+2}\left(Z\right)\right),Z_1]}_{{\mathbb{N}}_0}$. Taking the product of both sides

$${\displaystyle \prod _{j=1}^m}\varkappa \left(w_j\left(n\right)\right)>m^m\left({\displaystyle \prod _{j=1}^m}\varkappa \left(w_j\left(n\right)\right)\right){\displaystyle \prod _{j=1}^m}{\left({\displaystyle \prod _{i=1}^m}{\displaystyle \sum _{i_1=w_j\left(n\right)}^{n-1}}{a}_i\left(i_1\right){\displaystyle \prod _{i_2={\delta }_i\left(i_1\right)}^{w_i\left(n\right)-1}}{\displaystyle \frac{1}{1-{\sum }_{i_3=1}^m{a}_{i_3}\left(i_2\right){\Psi }_{i_3}^{l-1}}}\right)}^{{\displaystyle \frac{1}{m}}}$$

for $n\in {[W^{l+3}\left(Z\right),Z_1]}_{{\mathbb{N}}_0}$. That is,

$${\displaystyle \prod _{j=1}^m}{\left({\displaystyle \prod _{i=1}^m}{\displaystyle \sum _{i_1=w_j\left(n\right)}^{n-1}}{a}_i\left(i_1\right){\displaystyle \prod _{i_2={\delta }_i\left(i_1\right)}^{w_i\left(n\right)-1}}{\displaystyle \frac{1}{1-{\sum }_{i_3=1}^m{a}_{i_3}\left(i_2\right){\Psi }_{i_3}^{l-1}}}\right)}^{{\displaystyle \frac{1}{m}}}<{\displaystyle \frac{1}{m^m}},n\in {[W^{l+3}\left(Z\right),Z_1]}_{{\mathbb{N}}_0}.$$

Contradicting (30), the rest of the proof is similar to the ends of the proofs of Theorems 1 and 2. Thus, the proof is complete. □

Let $B_k={\sum }_{i=1}^k{b}_i$, ${\sigma }_{max}^k={\displaystyle \underset{1\le i\le k}{max}}{\sigma }_i$, and ${\sigma }_{min}^k={\displaystyle \underset{1\le i\le k}{min}}{\sigma }_i$, where $k=1,2,\dots ,m$.

Corollary 1. 

Assume that $k\in \{1,2,\dots ,m\}$.

(i) 

If $B_k\ge 1$, then $D\left(Equation \left(2\right)\right)\le 2{\sigma }_{max}^m+{\sigma }_{min}^k+1$.

(ii) 

If

$${\left({\displaystyle \frac{1}{1-B_m}}\right)}^{{\sigma }_{min}^k}\ge {\displaystyle \frac{1-{\sigma }_{min}^m{B}_m}{B_k\left({\sigma }_{min}^m{\sum }_{i=1}^m{b}_i+B_m\frac{({\sigma }_{min}^m-1){\sigma }_{min}^m}{2}\right)}},$$

(31)

then

$$D_{n_1}\left(Equation \left(2\right)\right)\le \left\{\begin{array}{cc}3{\sigma }_{max}^m+1\hfill & if{\sigma }_{max}^m\ge {\sigma }_{min}^k+{\sigma }_{min}^m,\hfill \\ 4{\sigma }_{max}^m+1\hfill & otherwise.\hfill \end{array}\right.$$

Proof. 

Assume that $a_i\left(n\right)=b_i$, ${\delta }_i\left(n\right)=n-{\sigma }_i$, $i=1,2,\dots ,m$. At the beginning, we will calculate some sequences that we need to obtain the result. Clearly

$$w_k\left(n\right)=\underset{1\le j\le k}{max}{\delta }_j\left(n\right)=\underset{1\le j\le k}{max}\left(n-{\sigma }_i\right)=n-{\sigma }_{min}^k,$$

$$\begin{array}{ccc}\hfill M_k\left(n\right)& =& max\{i\in \mathbb{N}:w_k\left(i\right)\le n-1\}\hfill \\ & =& max\{i\in \mathbb{N}:\left(i-{\sigma }_{min}^k\right)\le n-1\}=n+{\sigma }_{min}^k-1,\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill W\left(n\right)& =& max\{i\in \mathbb{N}:\underset{1\le r\le m}{min}{\delta }_r\left(i\right)\le n-1\}\hfill \\ & =& max\{i\in \mathbb{N}:\underset{1\le r\le m}{min}\left(i-{\sigma }_r\right)\le n-1\}=n+{\sigma }_{max}^m-1.\hfill \end{array}$$

$$W^1\left(n\right)=W\left(n\right)+1=n+{\sigma }_{max}^m,W^l\left(n\right)=W\left(W^{l-1}\left(n\right)\right)+1=n+l{\sigma }_{max}^m,l=2,3,\dots ,$$

$$M_k^1\left(n\right)=M_k\left(n\right)+1=n+{\sigma }_{min}^k{M}_k^l\left(n\right)=M_k\left(M_k^{l-1}\left(n\right)\right)+1=n+l{\sigma }_{min}^k,l=2,3,\dots$$

Furthermore,

$$\rho \left(n\right)=min\{i\in \mathbb{N}:\underset{1\le j\le m}{max}{\delta }_j\left(i\right)\ge n\}=min\{i\in \mathbb{N}:\underset{1\le j\le m}{max}\left(i-{\sigma }_j\right)\ge n\}=n+{\sigma }_{min}^m,$$

$$V\left(n\right)=\underset{1\le i\le m}{max}{\delta }_i\left(n\right)=n-{\sigma }_{min}^m,$$

and

$$V^1\left(n\right)=V\left(n\right)-1=n-{\sigma }_{min}^m-1,V^l\left(n\right)=V(V^{l-1}\left(n\right)+1)-1=n-l{\sigma }_{min}^m-1,l=2,3,\dots$$

In view of $B_m={\sum }_{i=1}^k{b}_i\ge 1$, it follows that ${\Psi }_k^1$ is not defined. By applying Theorem 1 with $l*=1$, we obtain

$$D_{n_1}\left(Equation \left(2\right)\right)\le \underset{n\ge n_1}{sup}\left(M_k^1\left(W^2\left(n\right)\right)-(n-1)\right)=2{\sigma }_{max}^m+{\sigma }_{min}^k+1.$$

Thus, the proof of (i) is complete.

Next, we show that (31) leads to ${\Psi }_k^1\ge {\Phi }_k^1$. Clearly

$${\displaystyle \prod _{s=w_k\left(n\right)}^{n-1}}{\displaystyle \frac{1}{1-{\sum }_{i=1}^m{a}_i\left(s\right)}}={\displaystyle \prod _{s=n-{\sigma }_{min}^k}^{n-1}}{\displaystyle \frac{1}{1-{\sum }_{i=1}^m{b}_i}}={\left({\displaystyle \frac{1}{1-B_m}}\right)}^{{\sigma }_{min}^k}.$$

Therefore, one can choose

$${\Psi }_k^1={\left({\displaystyle \frac{1}{1-B_m}}\right)}^{{\sigma }_{min}^k}.$$

(32)

Also,

$$\begin{array}{c}{\displaystyle \frac{1-{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right)}{{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^m{a}_{i_1}\left(i\right){\sum }_{i_2={\delta }_{i_1}\left(i\right)}^n{\sum }_{i_3=1}^k{a}_{i_3}\left(i_2\right)}}\hfill \\ ={\displaystyle \frac{1-{\sigma }_{min}^m{B}_m}{{\sum }_{i=n+1}^{n+{\sigma }_{min}^m}{\sum }_{i_1=1}^m{b}_{i_1}{\sum }_{i_2=i-{\sigma }_{i_1}}^n{B}_k}}\hfill \\ =\left(1-{\sigma }_{min}^m{B}_m\right)\left(B_k\left((b_1{\sigma }_1+\cdots +b_m{\sigma }_m)+(b_1({\sigma }_1-1)+\cdots +b_m({\sigma }_m-1))+\dots \right.\right.\hfill \\ {\left.\left.+(b_1({\sigma }_1-({\sigma }_{min}^m-1))+\cdots +b_m({\sigma }_m-({\sigma }_{min}^m-1))\right)\right)}^{-1}\hfill \\ =\left(1-{\sigma }_{min}^m{B}_m\right)B_k^{-1}{\left({\sigma }_{min}^m{\displaystyle \sum _{i=1}^m}{b}_i{\sigma }_i-B_m{\displaystyle \frac{({\sigma }_{min}^m-1){\sigma }_{min}^m}{2}}\right)}^{-1}.\hfill \end{array}$$

Then

$${\Phi }_k^1=\left(1-{\sigma }_{min}^m{B}_m\right)B_k^{-1}{\left({\sigma }_{min}^m{\displaystyle \sum _{i=1}^m}{b}_i{\sigma }_i-B_m{\displaystyle \frac{({\sigma }_{min}^m-1){\sigma }_{min}^m}{2}}\right)}^{-1}.$$

(33)

From Equations (31)–(33), we have ${\Psi }_k^1\ge {\Phi }_k^1$. It is clear that

$$V^1\left(W^l\left(n\right)\right)=n+l{\sigma }_{max}^m-{\sigma }_{min}^m-1\mathrm{and}{M}_k^1\left(W^2\left(n\right)\right)=n+2{\sigma }_{max}^m+{\sigma }_{min}^k-1.$$

Therefore, $V^1\left(W^l\left(n\right)\right)\ge M_k^1\left(W^2\left(n\right)\right)$, whenever

$$n+l{\sigma }_{max}^m-{\sigma }_{min}^m-1\ge n+2{\sigma }_{max}^m+{\sigma }_{min}^k,$$

that is equivalent to

$$l{\sigma }_{max}^m\ge 2{\sigma }_{max}^m+{\sigma }_{min}^k+{\sigma }_{min}^m+1.$$

Thus

$$\left\{\begin{array}{cc}{V}^1\left(W^3\left(n\right)\right)\ge M_k^1\left(W^2\left(n\right)\right)\hfill & \mathrm{if}{\sigma }_{max}^m\ge {\sigma }_{min}^k+{\sigma }_{min}^m+1,\hfill \\ V^1\left(W^4\left(n\right)\right)\ge M_k^1\left(W^2\left(n\right)\right)\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Applying Theorem 1 with $l_1=l_2=1$ and $l=3$ or $l=4$, we have

$$D_{n_1}\left(\mathrm{Equation} \left(2\right)\right)\le \left\{\begin{array}{cc}3{\sigma }_{max}^m+1\hfill & if{\sigma }_{max}^m\ge {\sigma }_{min}^k+{\sigma }_{min}^m+1,\hfill \\ 4{\sigma }_{max}^m+1\hfill & otherwise.\hfill \end{array}\right.$$

The proof is complete. □

Corollary 2. 

(i) 

If $b\ge 1$, then $D\left(Equation \left(3\right)\right)\le 3\sigma +1$.

(ii) 

If

$${\left({\displaystyle \frac{1}{1-b}}\right)}^{\sigma }\ge {\displaystyle \frac{2\left(1-\sigma b\right)}{b^2\sigma \left(1+\sigma \right)}},$$

(34)

then $D_{n_1}\left(Equation \left(3\right)\right)\le 4\sigma +1$.

Corollary 3. 

(i) 

If $B_k{\sigma }_{min}^k\ge 1$, then $D\left(Equation \left(2\right)\right)\le 2{\sigma }_{max}^m+{\sigma }_{min}^k+1$.

(ii) 

If

$$B_k{\sigma }_{min}^k+B_k{\left({\displaystyle \frac{1}{1-B_m}}\right)}^{{\sigma }_{min}^k}\left({\sigma }_{min}^k{\displaystyle \sum _{i=1}^k}{b}_i{\sigma }_i+B_k{\displaystyle \frac{\left({\sigma }_{min}^k-1\right){\sigma }_{min}^k}{2}}\right)\ge 1,$$

then $D_{n_1}\left(Equation \left(2\right)\right)\le 3{\sigma }_{max}^m+{\sigma }_{min}^k+1$.

Proof. 

Using the same method as in the proof of Corollary 1, we obtain

$${\displaystyle \sum _{j=1}^1}\left({\displaystyle \prod _{i_2=2}^j}{\Psi }_k^{2-i_2}\right){\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{R}_{i_1,k}^j(i,n)={\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right)=B_k{\sigma }_{min}^k.$$

Applying Theorem 2 with $l=1$, it follows that

$$D_{n_1}\left(Equation\left(2\right)\right)\le \underset{n\ge n_1}{sup}\left(M_k^1\left(W^2\left(n\right)\right)-(n-1)\right)=2{\sigma }_{max}^m+{\sigma }_{min}^k+1.$$

The proof of (i) is complete.

Let

$$I\left(n\right)={\displaystyle \sum _{j=1}^2}\left({\displaystyle \prod _{i=2}^j}{\Psi }_k^{3-i}\right){\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{R}_{i_1,k}^j(i,n).$$

Clearly,

$$\begin{array}{ccc}\hfill I\left(n\right)& =& {\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right)+{\Psi }_k^1{\displaystyle \sum _{i=w_k\left(n\right)}^{n-1}}{\displaystyle \sum _{i_1=1}^k}{a}_{i_1}\left(i\right){\displaystyle \sum _{i_2={\delta }_{i_1}\left(i\right)}^{w_k\left(n\right)-1}}{\displaystyle \sum _{i_3=1}^k}{a}_{i_3}\left(i_2\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill I\left(n\right)& \ge & B_k{\sigma }_{min}^k+{\left({\displaystyle \frac{1}{1-B_m}}\right)}^{{\sigma }_{min}^k}{B}_k\left((b_1{\sigma }_1+b_2{\sigma }_2+\cdots +b_k{\sigma }_k)\right.\hfill \\ & & +(b_1({\sigma }_1-1)+b_2({\sigma }_2-1)+\cdots +b_k({\sigma }_k-1))\hfill \\ & & \left.+\cdots +(b_1({\sigma }_1-({\sigma }_{min}^k-1))+b_2({\sigma }_2-({\sigma }_{min}^k-1))+\cdots +b_k({\sigma }_k-({\sigma }_{min}^k-1)))\right).\hfill \end{array}$$

Then

$$I\left(n\right)\ge B_k{\sigma }_{min}^k+{\left({\displaystyle \frac{1}{1-B_m}}\right)}^{{\sigma }_{min}^k}{B}_k\left({\sigma }_{min}^k{\displaystyle \sum _{i=1}^k}{b}_i{\sigma }_i+B_k{\displaystyle \frac{\left({\sigma }_{min}^k-1\right){\sigma }_{min}^k}{2}}\right).$$

Since all the requirements of Theorem 2 with $l=2$ are fulfilled, then $D_{n_1}\left(\mathrm{Equation} \left(2\right)\right)\le 3{\sigma }_{max}^m+{\sigma }_{min}^k+1$. The proof is complete. □

Corollary 4. 

(i) 

If $b\sigma \ge 1$, then $D\left(Equation \left(3\right)\right)\le 3\sigma +1$.

(ii) 

If

$$b\sigma +b{\left({\displaystyle \frac{1}{1-b}}\right)}^{\sigma }\left({\sigma }^{2}b+b{\displaystyle \frac{\left(\sigma -1\right)\sigma }{2}}\right)\ge 1,$$

then $D_{n_1}\left(Equation \left(3\right)\right)$$\le 4\sigma +1$.

Remark 1. 

The sequences ${\left\{{\Psi }_k^l\right\}}_{l\ge 0}$ and ${\left\{{\Phi }_k^l\right\}}_{l\ge 0}$ are defined iteratively through positive quantities under the standing assumptions on the coefficients $a_i\left(n\right)$.

First, since $a_i\left(n\right)\ge 0$ and ${\Psi }_k^0=1$, it follows that ${\Psi }_k^l>0$ for all $l\ge 0$, provided that

$${\displaystyle \sum _{i=1}^m}{a}_i\left(s\right){\Psi }_k^{l-1}<1,$$

which ensures that the product defining ${\Psi }_k^l$ is well-defined. Moreover, as l increases, the term ${\sum }_{i=1}^m{a}_i\left(s\right){\Psi }_k^{l-1}$ typically increases, which makes each factor

$${\displaystyle \frac{1}{1-{\sum }_{i=1}^m{a}_i\left(s\right){\Psi }_k^{l-1}}},$$

larger, and hence the lower bound on ${\Psi }_k^l$ becomes larger.

Similarly, the sequence $\left\{{\Phi }_k^l\right\}$ is well-defined and positive whenever

$$0<{\displaystyle \sum _{i=n+1}^{\rho (n+1)-1}}{\displaystyle \sum _{i_1=1}^m}{a}_{i_1}\left(i\right)<1.$$

Moreover, the sequence $\left\{{\Phi }_k^l\right\}$ is nonincreasing in l and hence bounded. Consequently, there exists an index $l*\in \mathbb{N}$ such that

$${\Psi }_k^{l*}\ge {\Phi }_k^{l*},$$

which shows that the sequence $\left\{{\Psi }_k^l\right\}$ eventually dominates $\left\{{\Phi }_k^l\right\}$ as l increases.

3. Numerical Examples

In this section, we present three numerical examples that illustrate the applicability of our results. The first example considers a case where the delays are neither monotonic nor comparable (i.e., ${\delta }_1\left(n\right)-{\delta }_2\left(n\right)$ is not consistently positive or negative). This scenario poses significant challenges in determining the distance between zeros. The second example focuses on analyzing the distance between zeros in a single-delay equation where the delay is nonmonotone and unbounded. In the final example, we show that the error in our estimate is less than twice the delay.

Example 1. 

Consider the first-order difference equation

$$\Delta \varkappa \left(n\right)+a_1\left(n\right)\varkappa \left({\delta }_1\left(n\right)\right)+a_2\left(n\right)\varkappa \left({\delta }_2\left(n\right)\right)=0,n\in {\mathbb{N}}_0,$$

(35)

where $a_1\left(n\right)=a_2\left(n\right)={\displaystyle \frac{1}{10}}$, and

$${\delta }_1\left(n\right)=\left\{\begin{array}{cc}n-3\hfill & \mathrm{if}n=2s,\hfill \\ n-5\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0\right.$$

and

$${\delta }_2\left(n\right)=\left\{\begin{array}{cc}n-2\hfill & \mathrm{if}n=2s,\hfill \\ n-6\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0.\right.$$

Figure 1 and Figure 2 depict the graphical representations of the sequences ${\delta }_1\left(n\right)$ and ${\delta }_2\left(n\right)$, respectively.

Figure 1. The graph of the sequence ${\delta }_1\left(n\right)$.

Figure 2. The graph of the sequence ${\delta }_2\left(n\right)$.

Clearly

$$\underset{n\to \infty }{lim\; inf}{\displaystyle \sum _{i={\delta }_1\left(n\right)}^{n-1}}{a}_1\left(i\right)=\underset{s\to \infty }{lim}{\displaystyle \sum _{i={\delta }_1\left(2s\right)}^{2s-1}}{a}_1\left(i\right)=\underset{s\to \infty }{lim}{\displaystyle \sum _{i=2s-3}^{2s-1}}{a}_1\left(i\right)={\displaystyle \frac{3}{10}}$$

and

$$\underset{n\to \infty }{lim\; inf}{\displaystyle \sum _{i={\delta }_2\left(n\right)}^{n-1}}{a}_1\left(i\right)=\underset{s\to \infty }{lim}{\displaystyle \sum _{i={\delta }_2\left(2s\right)}^{2s-1}}{a}_1\left(i\right)=\underset{s\to \infty }{lim}{\displaystyle \sum _{i=2s-2}^{2s-1}}{a}_1\left(i\right)={\displaystyle \frac{2}{10}}={\displaystyle \frac{1}{5}}.$$

Then, Condition (5) is satisfied. Clearly, $w_1\left(n\right)={\delta }_1\left(n\right)$, and

$$w_2\left(n\right)=\underset{1\le j\le 2}{max}{\delta }_j\left(n\right)=\left\{\begin{array}{cc}n-2\hfill & \mathrm{if}n=2s,\hfill \\ n-5\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0,\right.$$

$$M_1\left(n\right)=max\{i\in \mathbb{N}:w_1\left(i\right)\le n-1\}=\left\{\begin{array}{cc}n+3\hfill & \mathrm{if}n=2s,\hfill \\ n+4\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0,\right.$$

$$M_2\left(n\right)=max\{i\in \mathbb{N}:w_2\left(i\right)\le n-1\}=\left\{\begin{array}{cc}n+3\hfill & \mathrm{if}n=2s,\hfill \\ n+4\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0\right.$$

and

$$\begin{array}{ccc}\hfill W\left(n\right)& =& max\{i\in \mathbb{N}:\underset{1\le r\le 2}{min}{\delta }_r\left(i\right)\le n-1\}=\left\{\begin{array}{cc}n+5\hfill & \mathrm{if}n=2s,\hfill \\ n+4\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0.\right.\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}\hfill W^1\left(n\right)& =& \left\{\begin{array}{cc}n+6\hfill & \mathrm{if}n=2s,\hfill \\ n+5\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0\right.\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill W^2\left(n\right)& =& \left\{\begin{array}{cc}n+12\hfill & \mathrm{if}n=2s,\hfill \\ n+11\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0.\right.\hfill \end{array}$$

Consequently,

$$\begin{array}{ccc}\hfill W^l\left(n\right)& =& \left\{\begin{array}{cc}n+6l\hfill & \mathrm{if}n=2s,\hfill \\ n+6l-1\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0.\right.\hfill \end{array}$$

Furthermore,

$$\rho \left(n\right)=min\{i\in \mathbb{N}:\underset{1\le j\le m}{max}{\delta }_j\left(i\right)\ge n\}=\left\{\begin{array}{cc}n+2\hfill & \mathrm{if}n=2s,\hfill \\ n+3\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0\right.$$

and

$$V\left(n\right)=\underset{1\le i\le m}{max}{\delta }_i\left(n\right)=\left\{\begin{array}{cc}n-2\hfill & \mathrm{if}n=2s,\hfill \\ n-5\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0.\right.$$

Hence

$$V^1\left(n\right)=V\left(n\right)-1=\left\{\begin{array}{cc}n-3\hfill & \mathrm{if}n=2s,\hfill \\ n-6\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0.\right.$$

Assume that $a={\displaystyle \frac{1}{10}}$. Therefore,

$${\displaystyle \prod _{u=w_1\left(2s\right)}^{2s-1}}{\displaystyle \frac{1}{1-{\sum }_{i=1}^2{a}_i\left(u\right)}}={\displaystyle \prod _{u=2s-3}^{2s-1}}{\displaystyle \frac{1}{1-2a}}={\left({\displaystyle \frac{1}{1-2a}}\right)}^3.$$

Accordingly, it is feasible to choose

$${\Psi }_1^1={\left({\displaystyle \frac{1}{1-2a}}\right)}^3.$$

Let

$$I_1\left(n\right)={\displaystyle \frac{1-{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^2{a}_{i_1}\left(i\right)}{{\sum }_{i=n+1}^{\rho (n+1)-1}{\sum }_{i_1=1}^2{a}_{i_1}\left(i\right){\sum }_{i_2={\delta }_{i_1}\left(i\right)}^n{\sum }_{i_3=1}^1{a}_{i_3}\left(i_2\right)}}.$$

Then

$$\begin{array}{ccc}\hfill I_1\left(2s\right)& =& {\displaystyle \frac{1-{\sum }_{i=2s+1}^{\rho (2s+1)-1}{\sum }_{i_1=1}^2{a}_{i_1}\left(i\right)}{{\sum }_{i=2s+1}^{\rho (2s+1)-1}{\sum }_{i_1=1}^2{a}_{i_1}\left(i\right){\sum }_{i_2={\delta }_{i_1}\left(i\right)}^{2s}{\sum }_{i_3=1}^1{a}_{i_3}\left(i_2\right)}}\hfill \\ & =& {\displaystyle \frac{1-{\sum }_{i=2s+1}^{2s+3}{\sum }_{i_1=1}^{2}a}{{\sum }_{i=2s+1}^{2s+3}{\sum }_{i_1=1}^2{a}_{i_1}\left(i\right){\sum }_{i_2={\delta }_{i_1}\left(i\right)}^{2s}{\sum }_{i_3=1}^1{a}_{i_3}\left(i_2\right)}}={\displaystyle \frac{1-6a}{21a^2}}.\hfill \end{array}$$

As a result, one may select

$${\Phi }_1^1={\displaystyle \frac{1-6a}{21a^2}}.$$

Then

$${\Phi }_1^1={\displaystyle \frac{1-6a}{21a^2}}<{\Psi }_1^1={\left({\displaystyle \frac{1}{1-2a}}\right)}^3\mathrm{for}a={\displaystyle \frac{1}{10}}.$$

Since

$$V^1\left(W^4\left(n\right)\right)=\left\{\begin{array}{cc}n+21\hfill & \mathrm{if}n=2s,\hfill \\ n+20\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0,\right.,$$

and

$$M_1^1\left(W^2\left(n\right)\right)=\left\{\begin{array}{cc}n+15\hfill & \mathrm{if}n=2s,\hfill \\ n+14\hfill & \mathrm{if}n=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0,\right.,$$

it follows that $V^1\left(W^4\left(n\right)\right)>M_1^1\left(W^2\left(n\right)\right)$. Consequently, Theorem 1 with $l_1=l_2=1$ and $l=4$ implies that

$$D_{n_1}(Equation \left(35\right))\le \underset{n\ge n_1}{sup}\left(W^4\left(n\right)-(n-1)\right)\le 22.$$

It should be emphasized that, to the best of the authors’ knowledge, the DGZ-DDE of (35), have not been previously investigated, especially in the case of non-monotonic delays.

Example 2. 

Consider the first-order difference equation

$$\Delta \varkappa \left(n\right)+\varkappa \left({\delta }_1\left(n\right)\right)=0,n\in \mathbb{N},$$

(36)

where

$$\begin{array}{ccc}\hfill {\delta }_1\left(n\right)& =& \left\{\begin{array}{cc}\lfloor {\displaystyle \frac{n}{2}}\rfloor \hfill & \mathrm{if}n=2r,\hfill \\ \lfloor {\displaystyle \frac{n}{5}}\rfloor \hfill & \mathrm{if}n=2r+1,\hfill \end{array}r\in \mathbb{N},\right.\hfill \end{array}$$

where $\lfloor m\rfloor$ is the Floor function. This equation has the form of (1) with $m=1$ and $a_1\left(n\right)=1$.

To illustrate the non-monotone behavior of the sequence ${\delta }_1\left(n\right)$, we present its graph in Figure 3.

Figure 3. The graph of the sequence ${\delta }_1\left(n\right)$.

Clearly,

$$w_1\left(n\right)=\underset{1\le j\le 1}{max}{\delta }_j\left(n\right)={\delta }_1\left(n\right)\le {\displaystyle \frac{n}{2}}.$$

Next, we estimate $W\left(n\right)$. By definition,

$$W\left(n\right)=max\left\{i\in \mathbb{N}:{\delta }_1\left(i\right)\le n-1\right\},$$

it follows that

$$W\left(n\right)\le max\left\{i\in \mathbb{N}:{\delta }_1\left(i\right)\le n-1\right\}\le 5n.$$

Similarly,

$$M_1\left(n\right)=max\left\{i\in \mathbb{N}:w_1\left(i\right)\le n-1\right\}\le 5n.$$

Hence,

$$M_1^1\left(n\right)=M_1\left(n\right)+1\le 5n+1,$$

and, by induction,

$$M_1^l\left(n\right)=M_1\left(M_1^{l-1}\left(n\right)\right)+1\le 5^{l}n+{\displaystyle \frac{5^l-1}{4}},l\ge 2.$$

In the same way,

$$W^1\left(n\right)=W\left(n\right)+1\le 5n+1,$$

and

$$W^l\left(n\right)=W\left(W^{l-1}\left(n\right)\right)+1\le 5^{l}n+{\displaystyle \frac{5^l-1}{4}},l\ge 2.$$

In view of the fact that $a_1\left(n\right)=1$, it follows that ${\Psi }_1^1$ is undefined.

Consequently, by Theorem 1 with $l^{*}=1$ and $k=1$, every solution of (36) has at least one zero in the interval

$$[n-1,M_1^1\left(W^2\left(n\right)\right)]\subseteq [n-1,125n+62],n\ge 20.$$

It is worth noting that the previous criteria are not applicable to this example due to the non-monotonic nature of the delay, which is also not bounded above by a constant delay. More precisely, there does not exist $\sigma >0$ such that $\delta \left(n\right)\le n-\sigma$. Indeed, most existing methods rely essentially on these two assumptions. This highlights the significance of our results, which do not require monotonicity and are therefore capable of handling a broader class of delay difference equations.

Example 3. 

Consider the first-order difference equation

$$\Delta \varkappa \left(n\right)+a_1\left(n\right)\varkappa (n-1)=0,n\in {\mathbb{N}}_0,$$

(37)

where

$$\begin{array}{ccc}\hfill a_1\left(n\right)& =& \left\{\begin{array}{cc}a\hfill & ifn=2s,\hfill \\ b\hfill & ifn=2s+1,\hfill \end{array}s\in {\mathbb{N}}_0,\right.\hfill \end{array}$$

where $b\ge a>0$.

Below, we will calculate the left-hand side of Condition (19) for Equation (37) when $l=1,2,3$, and 4, respectively.

$${\displaystyle \sum _{i=n-1}^{n-1}}{R}_{1,1}^1(i,n)=a,$$

$${\displaystyle \sum _{j=1}^2}\left({\displaystyle \prod _{i_2=2}^j}{\Psi }_1^{2+1-i_2}\right){\displaystyle \sum _{i=n-1}^{n-1}}{R}_{1,1}^j(i,n)\ge {\displaystyle \sum _{i=n-1}^{n-1}}a+{\Psi }_1^1{\displaystyle \sum _{i=n-1}^{n-1}}a{\displaystyle \sum _{i_2=i-1}^{n-1-1}}a=a+{\displaystyle \frac{a^2}{1-a}},$$

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{j=1}^3}\left({\displaystyle \prod _{i_2=2}^j}{\Psi }_1^{3+1-i_2}\right){\displaystyle \sum _{i=n-1}^{n-1}}{R}_{1,1}^j(i,n)& \ge & {\displaystyle \sum _{i=n-1}^{n-1}}a+{\Psi }_1^2{\displaystyle \sum _{i=n-1}^{n-1}}a{\displaystyle \sum _{i_2=i-1}^{n-1-1}}a\hfill \\ & & +{\Psi }_1^2{\Psi }_1^1{\displaystyle \sum _{i=n-1}^{n-1}}a{\displaystyle \sum _{i_2=i-1}^{n-1-1}}a{\displaystyle \sum _{i_3=i_2-1}^{n-1-1-1}}a\hfill \\ & =& a+{\displaystyle \frac{a^2}{1-\frac{a}{1-a}}}+{\displaystyle \frac{a^3}{\left(1-a\right)\left(1-\frac{a}{1-a}\right)}}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{j=1}^4}\left({\displaystyle \prod _{i_2=2}^j}{\Psi }_1^{4+1-i_2}\right){\displaystyle \sum _{i=n-1}^{n-1}}{R}_{1,1}^j(i,n)& \ge & a+{\displaystyle \frac{a^2}{\left(1-\frac{a}{1-\frac{a}{1-a}}\right)}}+{\displaystyle \frac{a^3}{\left(1-\frac{a}{1-a}\right)\left(1-\frac{a}{1-\frac{a}{1-a}}\right)}}\hfill \\ & & +{\displaystyle \frac{a^4}{\left(1-a\right)\left(1-\frac{a}{1-a}\right)\left(1-\frac{a}{1-\frac{a}{1-a}}\right)}}.\hfill \end{array}$$

By applying Theorem 2, we obtain the distance between consecutive zeros for all solutions of Equation (37) for different values of the parameter a, as shown in Table 1.

Table 1. This table shows the relationship between the distance between consecutive zeros of all solutions of Equation (37) and the parameter a.

The Value of a	The Distance Between Generalized Zeros Is Not Grater Than
$a\ge 1$	4
${\displaystyle \frac{1}{2}}\le a<1$	5
${\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{2}}\sqrt{5}\le a<{\displaystyle \frac{1}{2}}$	6
${\displaystyle \frac{1}{3}}\le a<{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{2}}\sqrt{5}$	7

However, none of the results reported in [27,29] provide estimates that are sharper than those obtained in Table 1. A direct comparison based on Table 1 and Table 2 clearly illustrates this improvement.

Table 2. Upper bounds on the distance between consecutive generalized zeros of all solutions of Equation (37) obtained in earlier works for various ranges of the parameter a.

The Value of a	The Distance Between Generalized Zeros Is Not Grater Than
	[Corollary 3.7] in [27]	[Theorem 1] in [29]
$a\ge 1$	6	4
$a\ge 0.5897545123$	6	5
$a\ge {\displaystyle \frac{1}{2}}$	6	6
$a\ge 0.468$	7	7
$a\ge 0.407$	7	8

For instance, for $a\ge 1$, our result gives an upper bound equal to 4, which improves the bound 6 in [27] and coincides with the best bound 4 in [29]. Moreover, for ${\displaystyle \frac{1}{2}}\le a<1$, we obtain the bound 5, which is strictly better than the bound 6 in [27,29].

In addition, for $a\ge {\displaystyle \frac{1}{2}}$, our estimate does not exceed 5, which improves the bound 6 in [27,29]. Furthermore, for smaller values of a, our results yield bounds such as 6 and 7, whereas the existing results in [27,29] are unable to attain such bounds for the same ranges of the parameter a.

These comparisons confirm that the bounds obtained in this work are consistently smaller or equal to those in the existing literature. Since a smaller upper bound corresponds to a more accurate estimation of the distance between consecutive generalized zeros, this demonstrates the effectiveness and superiority of our results.

It is noteworthy that the specific case of Equation (37), given by

$$\Delta \varkappa \left(n\right)+\varkappa (n-1)=0,n\in {\mathbb{N}}_0,$$

(38)

with the initial conditions $\varkappa (-1)=\varkappa \left(0\right)=1$, possesses the solutions $\varkappa \left(1\right)=0$, $\varkappa \left(2\right)=1$, and $\varkappa \left(3\right)=1$. Consequently, the upper bound on the distance between the generalized zeros of all solutions of Equation (38) is greater than 2. However, as indicated in Table 1, this upper bound is not greater than 4. Therefore, we can conclude that our estimation error is less than 2.

4. Conclusions

In this work, we studied DGZ-DDE with several variable delays. We not only improved and generalized some methods used to examine the distance between zeros, but we also investigated this property in cases of non-monotone delays (as far as the authors are aware, the distribution of zeros for first-order differential equations with non-monotone delay has not been studied before). We obtained several new approximations for the distance between zeros. The methods employed in this work can also be used to study the oscillation and the distance between the zeros of many other equations, such as first-order neutral difference equations and higher-order delay differential equations, which are commonly employed to model various natural phenomena.