Oscillation Results for Difference Equations with Several Non-Monotone Advanced Arguments

Abstract

We investigate the oscillatory behavior of a first-order linear difference equation with several advanced arguments. New sufficient conditions for oscillation are established, and we show, through carefully constructed counterexamples, that many well-known oscillation criteria for equations with a single advanced argument fail to generalize to the several-argument setting, even when each advanced argument is increasing. Several illustrative examples are also provided to demonstrate the sharpness and practical effectiveness of the obtained conditions and to highlight their clear improvements over all existing results in the literature.

1. Introduction

Consider the first-order difference equations with several advanced arguments

$$\nabla z\left(n\right)-\sum _{r=1}^m{c}_r\left(n\right)z\left({\theta }_r\left(n\right)\right)=0,n\in \mathbb{N},$$

(1)

where $\mathbb{N}$ denotes the set of positive integers and $\nabla z\left(n\right)=z\left(n\right)-z(n-1)$ is the backward difference operator. Throughout the paper, the coefficient sequences $c_r\left(n\right)$, $r=1,2,\dots ,m$, are assumed to be nonnegative real-valued for all sufficiently large n. Moreover, the integer-valued sequences ${\theta }_r\left(n\right)$ represent advanced arguments and satisfy

$${\theta }_r\left(n\right)\ge n+1,n\in \mathbb{N},r=1,2,\dots ,m.$$

A sequence ${\left(z\left(n\right)\right)}_{n\ge n_0}$, for some $n_0\in \mathbb{N}$, is said to be a solution of Equation (1) if it satisfies the equation for all $n\ge n_0$.

A solution ${\left(z\left(n\right)\right)}_{n\ge n_0}$ of Equation (1) is called oscillatory if its terms $z\left(n\right)$ are neither eventually positive nor eventually negative. Otherwise, the solution is said to be nonoscillatory [1]. Equation (1) is called oscillatory if every solution ${\left(z\left(n\right)\right)}_{n\ge n_0}$ is oscillatory. Otherwise, it is said to be nonoscillatory.

The oscillation theory of differential and difference equations provides fundamental insights into the qualitative behavior of dynamical systems across diverse domains, including population dynamics, control systems, biological feedback or feedforward mechanisms, and economics; see [2,3,4,5]. The analysis of oscillatory behavior enables researchers to determine whether solutions fluctuate around equilibrium states or converge monotonically, thereby revealing underlying stability properties, periodic tendencies, or chaotic dynamics in physical and biological processes. In discrete systems, oscillation analysis assumes particular significance, as many real-world models evolve through discrete time steps and incorporate delay or advance arguments to capture memory effects or anticipation mechanisms; see [2,3].

The study of oscillation of difference equations plays an essential role in discrete mathematics, and its development has been addressed in a wide range of contributions; see [2,3,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. While classical results typically assume a single monotone argument, realistic systems often involve several advanced arguments that may vary non-monotonically over time. Such complex structures arise naturally in predictive control systems, biological population models with anticipation mechanisms, and signal propagation through feedback networks [2,3].

Ladas [19] investigated the oscillatory behavior of the first-order differential equation with non-decreasing delay

$$y^{\prime }\left(t\right)+a\left(t\right)y\left(g\left(t\right)\right)=0,a,g\in C([t_0,\infty ),[0,\infty )),g\left(t\right)\le t\underset{t\to \infty }{lim}g\left(t\right)=\infty ,t\ge t_0,$$

and established the well-known lim sup-type oscillation criterion

$$\underset{t\to \infty }{\mathrm{lim\; sup}}{\int }_{g\left(t\right)}^{t}a\left(s\right)ds>1.$$

(2)

Motivated by the continuous case, Chatzarakis and Stavroulakis [12] extended Ladas’ condition to the discrete setting. In particular, they obtained the following discrete analogue of Equation (2):

$$\underset{n\to \infty }{\mathrm{lim\; sup}}\sum _{r=n}^{h\left(n\right)}c\left(r\right)>1,$$

(3)

for the advanced difference equation

$$\nabla z\left(n\right)-c\left(n\right)z\left(h\left(n\right)\right)=0,n\in \mathbb{N},$$

(4)

where ${\left\{c\left(n\right)\right\}}_{n\ge 1}$ is a sequence of nonnegative real numbers and ${\left\{h\left(n\right)\right\}}_{n\ge 1}$ is an integer-valued non-decreasing sequence satisfying

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

Subsequently, Chatzarakis, Pinelas, and Stavroulakis [10] extended condition (3) to Equation (1) and obtained

$$\underset{n\to \infty }{\mathrm{lim\; sup}}\sum _{r=1}^m\sum _{r_1=n}^{{\theta }_{\min }\left(n\right)}{c}_r\left(r_1\right)>1.$$

(5)

where ${\theta }_{\min }\left(n\right)={\min }_{1\le r\le m}{\theta }_r\left(n\right)$ and each ${\theta }_r\left(n\right)$ represents a non-decreasing sequence.

Braverman and Karpuz [8] demonstrated that condition (2) is not valid for both the continuous and the corresponding discrete cases when the assumption of monotonicity is relaxed to allow for general, not necessarily monotone, delay arguments. This finding highlights the limitations of directly extending results from monotone to non-monotone delays and shows that more specific methods are required for equations with non-monotone arguments [16,17].

Despite notable progress in the oscillation theory of delay difference equations, the case involving several advanced arguments remains insufficiently explored. Existing studies [8,34] present only sufficient conditions for the oscillation of all solutions; however, they do not reveal the qualitative differences between equations with one or with several advanced arguments. Following the approach in [8], we construct a counterexample showing that some extensions of condition (3) cannot hold for Equation (1), even when the advanced arguments are non-decreasing and satisfy ${\theta }_r\left(n\right)\ge n+1$. In this work, we show that there is no constant $B>0$ such that any of the following conditions guarantees the oscillation of all solutions of Equation (1):

$$\underset{n\to \infty }{lim\; inf}\sum _{r=1}^m\sum _{r_1=n+1}^{{\theta }_{\max }\left(n\right)}{c}_r\left(r_1\right)>B,{\theta }_{\max }\left(n\right)=\underset{1\le i\le m}{\max }{\theta }_i\left(n\right),$$

or

$$\underset{n\to \infty }{lim\; inf}\sum _{r=n+1}^{{\theta }_k\left(n\right)}{c}_l\left(r\right)>B,k\ne l,k,l=1,2,\dots ,m.$$

This paper has several additional goals. First, we develop a new framework for studying oscillatory behavior and derive new sufficient conditions that extend and unify earlier results (see Table 1). Second, we show that, for a certain class of equations of the form (1), some of our criteria ensure oscillation, whereas all previously known conditions, whether iterative or non-iterative, fail for this class. Finally, we introduce a new concept for difference equations with advanced arguments, namely, the distance between generalized successive zeros of solutions, and illustrate its relevance through numerical simulations.

In summary, we establish new oscillation criteria and demonstrate their effectiveness relative to all existing results for a certain class of Equation (1). Furthermore, we show that several advanced arguments have a significant impact on the behavior of solutions, and that many conditions valid in the single advanced argument case cannot be extended to the several advanced argument case, even under the assumption that the advanced arguments are non-decreasing.

2. Main Results

The following result shows that several classical oscillation tests for difference equations with a single advanced argument fail when extended to equations with several advanced arguments. This highlights a fundamental limitation of such extensions and emphasizes that the interaction among several advanced arguments requires genuinely new criteria.

Theorem 1.

None of the following conditions, for any constant $B>0$, guarantees the oscillation of Equation (1) when ${\theta }_r\left(n\right)$ is non-decreasing for all $r=1,2,\dots ,m$ and $m\ge 2$:

(i)

$$\underset{n\to \infty }{lim\; inf}\sum _{r=1}^m\sum _{r_1=n+1}^{{\theta }_{\max }\left(n\right)}{c}_r\left(r_1\right)>B,{\theta }_{\max }\left(n\right)=\underset{1\le r\le m}{\max }{\theta }_r\left(n\right),$$

(ii)

$$\underset{n\to \infty }{lim\; inf}\sum _{r=n+1}^{{\theta }_k\left(n\right)}{c}_l\left(r\right)>B,k\ne l,k,l=1,2,\dots ,m.$$

Proof. 

Consider the first–order difference equation with several advanced arguments

$$\nabla z\left(n\right)-{\displaystyle \frac{1-e^{-1}}{2e}}z(n+1)-{\displaystyle \frac{1-e^{-1}}{2e^M}}z(n+M)=0,M>2,n\in \mathbb{N}.$$

(6)

It is evident that this equation is a particular case of (1), with

$${\theta }_1\left(n\right)=n+1,{\theta }_2\left(n\right)=n+M,c_1\left(n\right)={\displaystyle \frac{1-e^{-1}}{2e}},c_2\left(n\right)={\displaystyle \frac{1-e^{-1}}{2e^M}}.$$

The above equation possesses a non-oscillatory solution

$$z\left(n\right)=e^n,n\in \mathbb{N}.$$

Furthermore, we obtain

$$\sum _{r=1}^2\sum _{r_1=n+1}^{{\theta }_{\max }\left(n\right)}{c}_r\left(r_1\right)=\sum _{r=1}^2\sum _{r_1=n+1}^{n+M}{c}_r\left(r_1\right)=M(1-e^{-1})\left({\displaystyle \frac{1}{2e}}+{\displaystyle \frac{1}{2e^M}}\right)>{\displaystyle \frac{M(1-e^{-1})}{2e}},$$

and similarly,

$$\sum _{r=n+1}^{{\theta }_2\left(n\right)}{c}_1\left(r\right)={\displaystyle \frac{M(1-e^{-1})}{2e}}.$$

As the integer M may be taken arbitrarily large, the proof of the theorem is complete. □

In what follows, and throughout the remainder of this work, we assume that the sequences ${\left\{{\theta }_r\left(n\right)\right\}}_{n\ge 1}$, for $r=1,2,\dots ,m$, are integer-valued, with no monotonicity assumption. The analysis of this general setting is more complex and presents several challenges. To overcome these difficulties, we introduce the following associated non-decreasing sequences.

Specifically, we define the non-decreasing sequences

$${\varphi }_r\left(n\right)=\underset{r_1\ge n}{\min }{\theta }_r\left(r_1\right),r=1,2,\dots ,m.$$

Furthermore, for integers $p\ge q$, we define

$$\begin{array}{cc}\hfill C_1(p,q)& =\prod _{r_1=q+1}^p\left(1-\sum _{r=1}^m{c}_r\left(r_1\right)\right),\hfill \\ \hfill C_k(p,q)& =\prod _{r_1=q+1}^p\left(1-\sum _{r=1}^m{c}_r\left(r_1\right)C_{k-1}^{-1}\left({\theta }_r\left(r_1\right),r_1\right)\right),k=2,3,\dots .\hfill \end{array}$$

The following lemma establishes an iterative sequence $C_k(p,q)$ of lower bounds for the ratio ${\displaystyle \frac{z\left(p\right)}{z\left(q\right)}}$, $p\ge q$, which increases progressively with k. Although this lemma is stated without proof in ([9], Theorem 2.4), we give a proof here by following the approach of ([9], Lemma 2.1).

Lemma 1.

Assume that $\left(z\right(n\left)\right)$ is a positive solution of Equation (1). Then for all $p\ge q$, $k\in \mathbb{N}$, we have

$$z\left(p\right)\ge z\left(q\right)C_k^{-1}(p,q).$$

(7)

Proof. 

Dividing Equation (1) by $z\left(n\right)$ and rearranging, we obtain

$${\displaystyle \frac{z(n-1)}{z\left(n\right)}}=1-\sum _{r=1}^m{c}_r\left(n\right){\displaystyle \frac{z\left({\theta }_r\left(n\right)\right)}{z\left(n\right)}}.$$

Taking the product from $q+1$ to p, we have

$$0<{\displaystyle \frac{z\left(q\right)}{z\left(p\right)}}=\prod _{r_1=q+1}^p{\displaystyle \frac{z(r_1-1)}{z\left(r_1\right)}}=\prod _{r_1=q+1}^p\left(1-\sum _{r=1}^m{c}_r\left(r_1\right){\displaystyle \frac{z\left({\theta }_r\left(r_1\right)\right)}{z\left(r_1\right)}}\right).$$

(8)

Since the sequence $\left(z\right(n\left)\right)$ is positive, it follows from Equation (1) that $z\left(n\right)$ is eventually non-decreasing. Consequently,

$${\displaystyle \frac{z\left({\theta }_r\left(r_1\right)\right)}{z\left(r_1\right)}}\ge 1.$$

Hence, from (8) we have

$$0<{\displaystyle \frac{z\left(q\right)}{z\left(p\right)}}=\prod _{r_1=q+1}^p\left(1-\sum _{r=1}^m{c}_r\left(r_1\right){\displaystyle \frac{z\left({\theta }_r\left(r_1\right)\right)}{z\left(r_1\right)}}\right)\le \prod _{r_1=q+1}^p\left(1-\sum _{r=1}^m{c}_r\left(r_1\right)\right).$$

That is,

$${\displaystyle \frac{z\left(p\right)}{z\left(q\right)}}\ge \prod _{r_1=q+1}^p{\left(1-\sum _{r=1}^m{c}_r\left(r_1\right)\right)}^{-1}=C_1^{-1}(p,q).$$

Moreover, since ${\theta }_r\left(r_1\right)>r_1$, it follows that

$${\displaystyle \frac{z\left({\theta }_r\left(r_1\right)\right)}{z\left(r_1\right)}}\ge C_1^{-1}({\theta }_r\left(r_1\right),r_1).$$

Substituting this estimate into (8) and rearranging yields

$${\displaystyle \frac{z\left(p\right)}{z\left(q\right)}}\ge \prod _{r_1=q+1}^p{\left(1-\sum _{r=1}^m{c}_r\left(r_1\right)C_1^{-1}({\theta }_r\left(r_1\right),r_1)\right)}^{-1}=C_2^{-1}(p,q).$$

Proceeding inductively in this manner, we finally arrive at inequality (7). □

We are now in a position to state the main oscillation criterion for Equation (1). This result provides sufficient conditions ensuring that all solutions oscillate, even without assuming monotonicity of the advanced arguments. Moreover, the theorem introduces a novel approach to the oscillation problem, distinct from existing treatments in the literature.

Theorem 2

(Main oscillation criterion for several non-monotone advanced arguments). Let $k\in \mathbb{N}$. Suppose that there exists an unbounded sequence of positive integers ${\left\{{\alpha }_i\right\}}_{i\ge 1}$ such that

$$\underset{i\to \infty }{lim}\left(\sum _{r=1}^m\sum _{r_1={\alpha }_i}^{{\varphi }_r\left({\alpha }_i\right)}{C}_k^{-1}\left({\theta }_{j^{*}}\left(r_1\right),{\varphi }_{j^{*}}\left({\alpha }_i\right)\right)c_{j^{*}}\left(r_1\right)\right)>1\mathit{for}\mathit{every}{j}^{*}\in \{1,2,\dots ,m\}.$$

(9)

Then Equation (1) is oscillatory.

Proof. 

Assume, for the sake of contradiction, that Equation (1) possesses a non-oscillatory solution $\left(z\right(n\left)\right)$. Due to the linearity of the equation, we may, without loss of generality, assume that $z\left(n\right)>0$ for all sufficiently large n. It follows from Equation (1) that $z\left(n\right)$ is eventually nondecreasing for all sufficiently large n.

Summing Equation (1) from n to ${\varphi }_r\left(n\right)$, $r=1,2,\dots ,m$, yields

$$\begin{array}{c}\hfill z\left({\varphi }_r\left(n\right)\right)-z(n-1)-\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{c}_1\left(r_1\right)z\left({\theta }_1\left(r_1\right)\right)-\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{c}_2\left(r_1\right)z\left({\theta }_2\left(r_1\right)\right)\\ \hfill -\cdots -\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{c}_m\left(r_1\right)z\left({\theta }_m\left(r_1\right)\right)=0.\end{array}$$

(10)

By virtue of inequality (7) and the fact that ${\theta }_j\left(r_1\right)\ge {\varphi }_j\left(n\right)$ for all $r_1\in \{n,n+1,\dots ,{\varphi }_r\left(n\right)\}$, it follows that

$$z\left({\theta }_j\left(r_1\right)\right)\ge z\left({\varphi }_j\left(n\right)\right)C_k^{-1}({\theta }_j\left(r_1\right),{\varphi }_j\left(n\right)).$$

Substituting this estimate into the previous equality gives

$$\begin{array}{ccc}\hfill & & z\left({\varphi }_r\left(n\right)\right)-z(n-1)-z\left({\varphi }_1\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_1\left(r_1\right),{\varphi }_1\left(n\right))c_1\left(r_1\right)\hfill \\ & & -z\left({\varphi }_2\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_2\left(r_1\right),{\varphi }_2\left(n\right))c_2\left(r_1\right)-\cdots \hfill \\ & & -z\left({\varphi }_m\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_m\left(r_1\right),{\varphi }_m\left(n\right))c_m\left(r_1\right)\ge 0.\hfill \end{array}$$

(11)

Summing this inequality for $r=1,2,\dots ,m$, we obtain

$$\begin{array}{ccc}\hfill & & -mz(n-1)-z\left({\varphi }_1\left(n\right)\right)\left(\sum _{r=1}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_1\left(r_1\right),{\varphi }_1\left(n\right))c_1\left(r_1\right)-1\right)\hfill \\ & & -z\left({\varphi }_2\left(n\right)\right)\left(\sum _{r=1}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_2\left(r_1\right),{\varphi }_2\left(n\right))c_2\left(r_1\right)-1\right)\hfill \\ & & -\cdots -z\left({\varphi }_m\left(n\right)\right)\left(\sum _{r=1}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_m\left(r_1\right),{\varphi }_m\left(n\right))c_m\left(r_1\right)-1\right)\ge 0.\hfill \end{array}$$

(12)

Now, according to condition (9), there exists a sufficiently large i such that

$$\sum _{r=1}^m\sum _{r_1={\alpha }_i}^{{\varphi }_r\left({\alpha }_i\right)}{C}_k^{-1}({\theta }_{j^{*}}\left(r_1\right),{\varphi }_{j^{*}}\left({\alpha }_i\right))c_{j^{*}}\left(r_1\right)>1\mathrm{for}\mathrm{every}{j}^{*}\in \{1,2,\dots ,m\}.$$

In view of this fact and the positivity of $z\left(n\right)$, inequality (12) implies

$$0>-mz(n-1)\ge 0,$$

which is a contradiction and thus completes the proof. □

Under a common lower bound on the coefficient sequences, the oscillatory behavior of Equation (1) can be characterized by a simpler lim sup condition. The following theorem establishes an effective sufficient criterion for oscillation that is independent of the detailed structure of the coefficients.

Theorem 3.

Assume that $c_i\left(n\right)\ge c\left(n\right)$ for $n\ge n_0$, $n_0\in \mathbb{N}$, $i\in \{1,2,\dots ,m\}$. If

$$\underset{n\to \infty }{\mathrm{lim\; sup}}\left(\sum _{r=1}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}c\left(r_1\right)\right)>1,$$

(13)

then, Equation (1) is oscillatory.

Proof. 

Following the argument used in the proof of Theorem 2, together with the nondecreasing property of $z\left(n\right)$, we obtain

$$\begin{array}{ccc}& & z\left({\varphi }_r\left(n\right)\right)-z(n-1)-z\left({\varphi }_1\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{c}_1\left(r_1\right)-z\left({\varphi }_2\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{c}_2\left(r_1\right)-\cdots \hfill \\ & & -z\left({\varphi }_m\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{c}_m\left(r_1\right)\ge 0,\hfill \end{array}$$

where $r=1,2,\dots ,m$ and $\left(z\right(n\left)\right)$ is a positive solution of Equation (1). Using $c_i\left(n\right)\ge c\left(n\right)$ for all $i\in \{1,2,\dots ,m\}$, and summing over $r=1,2,\dots ,m$, we get

$$\sum _{r=1}^{m}z\left({\varphi }_r\left(n\right)\right)-mz(n-1)-\left(z({\varphi }_1\left(n\right)+z\left({\varphi }_2\left(n\right)\right)+\cdots +z\left({\varphi }_m\left(n\right)\right))\right)\sum _{r=1}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}c\left(r_1\right)\ge 0,$$

that is,

$$-mz(n-1)-\left(\sum _{r=1}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}c\left(r_1\right)-1\right)\sum _{r=1}^{m}z\left({\varphi }_r\left(n\right)\right)\ge 0.$$

Consequently,

$$\sum _{r=1}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}c\left(r_1\right)<1.$$

Therefore,

$$\underset{n\to \infty }{\mathrm{lim\; sup}}\left(\sum _{r=1}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}c\left(r_1\right)\right)\le 1,$$

contradicts (13). □

In some cases, certain terms of the difference equation restrict the applicability of existing oscillation criteria. To overcome this difficulty, we reorder the advanced arguments together with their associated coefficient sequences, thereby identifying the source of the obstruction and eliminating it. We then proceed to study the resulting reduced equation, as illustrated by the following result.

More precisely, the sequences $\left\{c_r\left(n\right)\right\}$ and $\left\{{\theta }_r\left(n\right)\right\}$, for $r=1,2,\dots ,m$, associated with Equation (1), can be rearranged so that the following inequality is eventually satisfied:

$$\sum _{r_1=n}^{{\varphi }_{j^{*}}\left(n\right)}{c}_{j^{*}}\left(r_1\right)\ge \sum _{r_1=n}^{{\varphi }_{i^{*}}\left(n\right)}{c}_{i^{*}}\left(r_1\right),j^{*}\ge i^{*},1\le i^{*},j^{*}\le m.$$

(14)

Theorem 4.

Let $k\in \mathbb{N}$ and $\mathcal{l}\in \{1,2,\dots ,m\}$. Suppose that condition (14) holds and that there exists an unbounded sequence of positive integers ${\left\{{\alpha }_i\right\}}_{i\ge 1}$ such that

$$\underset{i\to \infty }{lim}\left(\sum _{r=\mathcal{l}}^m\sum _{r_1={\alpha }_i}^{{\varphi }_r\left({\alpha }_i\right)}{C}_k^{-1}\left({\theta }_{j^{*}}\left(r_1\right),{\varphi }_{j^{*}}\left({\alpha }_i\right)\right)c_{j^{*}}\left(r_1\right)\right)>1\mathit{for}\mathit{every}{j}^{*}\in \{\mathcal{l},\mathcal{l}+1,\dots ,m\}.$$

(15)

Then Equation (1) is oscillatory.

Proof. 

As before, let $z\left(n\right)$ denote a positive solution of Equation (1). Following the same arguments as in the proof of Theorem 2, we have (inequaity (11))

$$\begin{array}{ccc}& & z\left({\varphi }_r\left(n\right)\right)-z(n-1)-z\left({\varphi }_1\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_1\left(r_1\right),{\varphi }_1\left(n\right))c_1\left(r_1\right)\hfill \\ & & -z\left({\varphi }_2\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_2\left(r_1\right),{\varphi }_2\left(n\right))c_2\left(r_1\right)-\cdots \hfill \\ & & -z\left({\varphi }_m\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_m\left(r_1\right),{\varphi }_m\left(n\right))c_m\left(r_1\right)\ge 0,r\in \{1,2,\dots ,m\}.\hfill \end{array}$$

The positivity of $z\left(n\right)$ implies that

$$\begin{array}{ccc}& & z\left({\varphi }_r\left(n\right)\right)-z(n-1)-z\left({\varphi }_{\mathcal{l}}\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_{\mathcal{l}}\left(r_1\right),{\varphi }_{\mathcal{l}}\left(n\right))c_{\mathcal{l}}\left(r_1\right)\hfill \\ & & -z\left({\varphi }_{\mathcal{l}+1}\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_{\mathcal{l}+1}\left(r_1\right),{\varphi }_{\mathcal{l}+1}\left(n\right))c_{\mathcal{l}+1}\left(r_1\right)-\cdots \hfill \\ & & -z\left({\varphi }_m\left(n\right)\right)\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_m\left(r_1\right),{\varphi }_m\left(n\right))c_m\left(r_1\right)\ge 0,r\in \{\mathcal{l},2,\dots ,m\}.\hfill \end{array}$$

Taking the sum over $r=\mathcal{l},\dots ,m$, we obtain

$$\begin{array}{ccc}\hfill & & -(m-\mathcal{l}+1)z(n-1)-z\left({\varphi }_{\mathcal{l}}\left(n\right)\right)\left(\sum _{r=\mathcal{l}}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_{\mathcal{l}}\left(r_1\right),{\varphi }_{\mathcal{l}}\left(n\right))c_{\mathcal{l}}\left(r_1\right)-1\right)\hfill \\ & & -z\left({\varphi }_{\mathcal{l}+1}\left(n\right)\right)\left(\sum _{r=\mathcal{l}}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_{\mathcal{l}+1}\left(r_1\right),{\varphi }_{\mathcal{l}+1}\left(n\right))c_{\mathcal{l}+1}\left(r_1\right)-1\right)\hfill \\ & & -\cdots -z\left({\varphi }_m\left(n\right)\right)\left(\sum _{r=\mathcal{l}}^m\sum _{r_1=n}^{{\varphi }_r\left(n\right)}{C}_k^{-1}({\theta }_m\left(r_1\right),{\varphi }_m\left(n\right))c_m\left(r_1\right)-1\right)\ge 0.\hfill \end{array}$$

(16)

Combining this with the positivity of $z\left(n\right)$ and condition (15), we obtain

$$0>-(m-\mathcal{l}+1)z(n-1)\ge 0,$$

which yields a contradiction and completes the proof of the theorem. □

Remark 1.

It is worth noting that one of the significant consequences of Theorem 1 is that none of the conditions

$$\sum _{r=1}^m\sum _{r_1=n}^{{\theta }_{\max }\left(n\right)}{c}_r\left(r_1\right)<A<+\infty ,\mathit{where}A>0,{\theta }_{\max }\left(n\right)=\underset{1\le r\le m}{\max }{\theta }_r\left(n\right),$$

or

$$\sum _{r=n+1}^{{\theta }_k\left(n\right)}{c}_l\left(r\right)<A<+\infty ,\mathit{where}A>0,k\ne l,k,l=1,2,\dots ,m,$$

where each ${\theta }_r\left(n\right)$ denotes a non-decreasing sequence of positive integers, is a necessary condition for the non-oscillation of Equation (1).

3. Numerical Results and Simulations

In this section, we provide a comparative analysis between our results and those reported in previous studies, as summarized in Table 1. It is clear that all existing works establish only sufficient conditions for the oscillation of Equation (1). In contrast, our results not only yield new and sharper oscillation criteria but also reveal that certain formulas are not sufficient to ensure the oscillatory behavior of all solutions of Equation (1).

We also present two numerical examples to illustrate the validity and sharpness of the obtained results, as well as the new concept of solutions termed generalized zeros. The first example applies one of the main results (Theorem 4) to demonstrate that a class of difference equations of the form (1) is oscillatory, whereas previously known criteria fail to detect this behavior. Solution simulations and graphical representations further illustrate the oscillatory nature and highlight regions where the earlier conditions are ineffective. The second example addresses a qualitative property that, to the best of the authors’ knowledge, has not been previously studied for difference equations with advanced arguments, namely the distance between generalized zeros, and the simulations indicate that this distance decreases as the sequence of coefficients increases, highlighting the relevance of this property for a deeper understanding of the solution dynamics.

Table 1. Comparison of existing results with the present study.

Reference	Type of Equation	Argument Structure	Main Contribution
Braverman & Karpuz (2011) [8]	Delay differential and difference equations	Single non-monotone delay argument	New oscillation criterion; Monotone-delay results do not extend to non-monotone delay.
Chatzarakis and Pašić [11]	Difference equations with delay and advanced arguments	Several non-monotone arguments	Improved sufficient conditions using summation inequalities.
Present study	Difference equations with advanced arguments	Several non-monotone advanced arguments	Sharp and generalized oscillation conditions; Monotone-single advanced arguments results can not be extend to several monotone advanced arguments.

Table 1. Comparison of existing results with the present study.

Reference	Type of Equation	Argument Structure	Main Contribution
Braverman & Karpuz (2011) [8]	Delay differential and difference equations	Single non-monotone delay argument	New oscillation criterion; Monotone-delay results do not extend to non-monotone delay.
Chatzarakis and Pašić [11]	Difference equations with delay and advanced arguments	Several non-monotone arguments	Improved sufficient conditions using summation inequalities.
Present study	Difference equations with advanced arguments	Several non-monotone advanced arguments	Sharp and generalized oscillation conditions; Monotone-single advanced arguments results can not be extend to several monotone advanced arguments.

Example 1.

Consider the first-order linear difference equation with several advanced arguments

$$\nabla z\left(n\right)-c_1\left(n\right)z\left({\theta }_1\left(n\right)\right)-c_2\left(n\right)z\left({\theta }_2\left(n\right)\right)-c_3\left(n\right)z\left({\theta }_3\left(n\right)\right)=0,n\in \mathbb{N},$$

(17)

where

$${\theta }_1\left(n\right)=\left\{\begin{array}{cc}n+1,\hfill & \mathit{if}n=2k,\hfill \\ n+3,\hfill & \mathit{if}n=2k+1,\hfill \end{array}\right.k\in \mathbb{N},$$

and

$${\theta }_2\left(n\right)=n+⌊\frac{1}{2\delta }⌋,{\theta }_3\left(n\right)=n+⌊\frac{1}{\delta }⌋,$$

where $0<\delta <\frac{1}{3}$, and $⌊\frac{1}{\delta }⌋$ denotes the greatest integer less than or equal to $\frac{1}{\delta }$.

Furthermore, $c_1\left(n\right)={\delta }^2$, and

$$c_2\left(n\right)=c_3\left(n\right)=\left\{\begin{array}{cc}a,\hfill & ifn={\alpha }_k,{\alpha }_k+1,{\alpha }_k+2,\dots ,{\alpha }_{k+⌊{\displaystyle \frac{1}{\delta }}⌋},\hfill \\ 0,\hfill & \mathit{otherwise},\hfill \end{array}\right.k\in \mathbb{N},$$

where $a=\frac{2}{3}(\delta +ϵ)$, $ϵ>0$, and the sequence $\left\{{\alpha }_k\right\}$ is an unbounded sequence of positive integers that satisfies

$${\alpha }_{k+1}>{\alpha }_{k+⌊{\displaystyle \frac{1}{\delta }}⌋+1}+⌊{\displaystyle \frac{1}{\delta }}⌋for all k\in \mathbb{N}.$$

We will show below that all the hypotheses of Theorem 4 are satisfied for Equation (17), and hence it is oscillatory.

It follows directly that condition (14) holds, and

$${\varphi }_2\left(n\right)=\underset{r_1\ge n}{\min }{\theta }_2\left(r_1\right)={\theta }_2\left(n\right),{\varphi }_3\left(n\right)=\underset{r_1\ge n}{\min }{\theta }_3\left(r_1\right)={\theta }_3\left(n\right).$$

Consequently, for $j^{*}=2,3$, we obtain

$$\begin{array}{cc}\hfill \sum _{r=2}^3\sum _{r_1={\alpha }_k}^{{\varphi }_r\left({\alpha }_k\right)}{c}_{j^{*}}\left(r_1\right)& =\sum _{r_1={\alpha }_k}^{{\varphi }_2\left({\alpha }_k\right)}{c}_{j^{*}}\left(r_1\right)+\sum _{r_1={\alpha }_k}^{{\varphi }_3\left({\alpha }_k\right)}{c}_{j^{*}}\left(r_1\right)\hfill \\ \hfill & \ge a\left({\displaystyle \frac{1}{2\delta }}+{\displaystyle \frac{1}{\delta }}\right)={\displaystyle \frac{2}{3}}(\delta +ϵ)·{\displaystyle \frac{3}{2\delta }}=1+{\displaystyle \frac{ϵ}{\delta }}\mathit{for}\mathit{all}k\in \mathbb{N}.\hfill \end{array}$$

Therefore, condition (15) is satisfied with $\mathcal{l}=2$, and hence Equation (17) is oscillatory.

However, as we will demonstrate below, all known iterative and non-iterative oscillation criteria fail to prove the oscillation of Equation (17). For example,

$$\begin{array}{cc}\hfill \sum _{r=1}^3\sum _{r_1={\alpha }_{k+⌊{\displaystyle \frac{1}{\delta }}⌋+1}}^{{\theta }_r\left({\alpha }_{k+⌊{\displaystyle \frac{1}{\delta }}⌋+1}\right)}{c}_r\left(r_1\right)& \le \sum _{r=1}^3\sum _{r_1={\alpha }_{k+⌊{\displaystyle \frac{1}{\delta }}⌋+1}}^{{\theta }_{\min }\left({\alpha }_{k+⌊{\displaystyle \frac{1}{\delta }}⌋+1}\right)}{c}_r\left(r_1\right)\hfill \\ \hfill & \le \sum _{r=1}^3\sum _{r_1={\alpha }_{k+⌊{\displaystyle \frac{1}{\delta }}⌋+1}}^{{\alpha }_{k+⌊{\displaystyle \frac{1}{\delta }}⌋+1}+⌊{\displaystyle \frac{1}{\delta }}⌋}{c}_r\left(r_1\right)=0\mathit{for}\mathit{all}k\in \mathbb{N},\hfill \end{array}$$

where ${\theta }_{\min }\left(n\right)={\min }_{1\le r\le m}{\theta }_r\left(n\right)={\theta }_3\left(n\right)$. Therefore,

$$\underset{n\to \infty }{lim\; inf}\sum _{r=1}^3\sum _{r_1=n+1}^{{\theta }_r\left(n\right)}{c}_r\left(r_1\right)\le \underset{n\to \infty }{lim\; inf}\sum _{r=1}^3\sum _{r_1=n+1}^{{\theta }_{\min }\left(n\right)}{c}_r\left(r_1\right)=0.$$

Consequently, ([10], Theorem 3.2) cannot be applied to Equation (17). Furthermore, we observe that

$$\theta \left(n\right)=\underset{1\le r\le m}{\min }{\theta }_r\left(n\right)={\theta }_1\left(n\right)\le n+3,$$

and

$$\varphi \left(n\right)=\underset{1\le r\le n}{\min }{\varphi }_r\left(n\right)\le \theta \left(n\right)\le n+3.$$

Moreover, define

$$\begin{array}{cc}\hfill G\left(n\right)& =G_0\left(n\right)=\sum _{r=1}^3{c}_r\left(n\right),\hfill \\ \hfill G_k\left(n\right)& =G\left(n\right)\left[1+\sum _{r=n+1}^{\theta \left(n\right)}G\left(r\right)exp\left(\sum _{r_1=r+1}^{\theta \left(r\right)}G\left(r_1\right)\prod _{r_2=r_1+1}^{\theta \left(r_1\right)}{\displaystyle \frac{1}{1-G_{k-1}\left(r_2\right)}}\right)\right],k=1,2,\dots .\hfill \end{array}$$

Therefore,

$$G\left(n\right)=G_0\left(n\right)=\sum _{r=1}^3{c}_r\left(n\right)\le {\delta }^2+\frac{4}{3}(\delta +ϵ)=W(\delta ,ϵ),$$

and

$$\begin{array}{cc}\hfill G_1\left(n\right)& =G\left(n\right)\left[1+\sum _{r=n+1}^{\theta \left(n\right)}G\left(r\right)exp\left(\sum _{r_1=r+1}^{\theta \left(r\right)}G\left(r_1\right)\prod _{r_2=r_1+1}^{\theta \left(r_1\right)}{\displaystyle \frac{1}{1-G_0\left(r_2\right)}}\right)\right]\hfill \\ \hfill & \le W(\delta ,ϵ)\left[1+3W(\delta ,ϵ)exp\left(3W(\delta ,ϵ){\left({\displaystyle \frac{1}{1-W(\delta ,ϵ)}}\right)}^3\right)\right]<R_1(\delta ,ϵ)<+\infty .\hfill \end{array}$$

Similarly,

$$G_k\left(n\right)\le R_k(\delta ,ϵ)<+\infty ,k=1,2,\dots .$$

Consequently,

$$\sum _{r=n}^{\theta \left(n\right)}G\left(r\right)\prod _{r_1=\varphi \left(n\right)+1}^{\varphi \left(r\right)}{\displaystyle \frac{1}{1-G_k\left(r_1\right)}}\le 4W(\delta ,ϵ){\left({\displaystyle \frac{1}{1-R_k(\delta ,ϵ)}}\right)}^3<1$$

for sufficiently small δ and ϵ. Hence, ([1], Theorem 2.7) is not satisfied.

Likewise, it can be shown that the remaining oscillation conditions are not satisfied for Equation (17).

This example demonstrates that condition (15) yields sharper results than existing ones. The numerical results confirm the oscillatory behavior of Equation (17). As seen in Figure 1, the solution $z\left(n\right)$ oscillates for $\delta =\frac{1}{10}$ and $\delta =\frac{1}{15}$ with $ϵ={10}^{-4}$, while smaller $\delta$ leads to more sign changes, motivating further study of the distance between generalized zeros (see Example 2).

In addition, we illustrate, through several graphical representations, a comparison between condition (15) and ([1], Theorem 2.7). Specifically, we define the function

$$F_k\left(\delta \right)\ge \sum _{r=n}^{\theta \left(n\right)}G\left(r\right)\prod _{r_1=\varphi \left(n\right)+1}^{\varphi \left(r\right)}{\displaystyle \frac{1}{1-G_k\left(r_1\right)}}-1,$$

where $ϵ={10}^{-4}$. We plot the relationship between $F_k\left(\delta \right)$ and the parameter $\delta$. As illustrated in Figure 2 and

Table 2. Computed values of the iterative criteria functions $F_1\left(\delta \right)$, $F_2\left(\delta \right)$, and $F_3\left(\delta \right)$ for various parameter $\delta$.

$\mathit{\delta }$	${\mathit{F}}_1\left(\mathit{\delta }\right)$	${\mathit{F}}_2\left(\mathit{\delta }\right)$	${\mathit{F}}_3\left(\mathit{\delta }\right)$
0.067	−0.429338	−0.286404	−0.157764
0.100	0.449848	0.594833	0.754316
0.020	−0.881088	−0.692979	−0.523681
0.040	−0.729753	−0.556778	−0.401100
0.060	−0.520528	−0.368476	−0.231628
0.080	−0.191859	−0.072673	0.034594

, the oscillation condition of ([1], Theorem 2.7), for $k=1,2,$ and 3, is not satisfied on the intervals

$$[0,0.08776083032],[0,0.08584232790],[0,0.08545505006],$$

respectively. However, as shown above, Equation (17) is oscillatory for all $0<\delta <\frac{1}{3}$ and $ϵ>0$.

Example 2.

Consider the difference equation with advanced argument

$$\nabla z\left(n\right)-\mu z(n+1)=0,n\in \mathbb{N}.$$

(18)

This equation represents a particular case of Equation (4) with $h\left(n\right)=n+1$ and $c\left(n\right)=\mu$. In this example, we illustrate the concept of generalized zeros for solutions of advanced difference equations and their relation to both the sequences of coefficients and advanced arguments. Here, a generalized zero is a positive integer at which the solution is zero or has a different sign than at the preceding integer.

Numerical simulations of Equation (18) reveal a clear dependence between the parameter μ and the distance between consecutive generalized zeros of the solutions. The numerical results show that the maximum interval length T, over which the solution remains positive (or negative), decreases as μ increases. Specifically, for $\mu =\frac{1}{3},\frac{1}{2},\frac{3}{4},and 1$, the corresponding lengths are $T\simeq 5,4,3,$ and 3, respectively (see Figure 3). These findings highlight the importance of examining the spacing between successive generalized zeros in advanced-type difference equations, as such an analysis provides deeper insight into the qualitative oscillatory dynamics of these equations.

It is worth noting that a systematic and independent study of the distance between generalized zeros for advanced difference equations is still needed to establish sharp and verifiable criteria for measuring this property. The distance between zeros has been widely studied in the context of delay differential equations; therefore, several analytical ideas developed in the continuous setting may be adapted and extended to the discrete framework. However, studying this property for advanced difference equations presents significant difficulties, due to the fundamentally different nature of their behavior in comparison with delay differential equations. In particular, the analysis of a lower bound for the distance between generalized zeros of solutions is not a straightforward extension of the corresponding results for differential equations, since it depends on the properties of the initial function, which, to the best of the authors’ knowledge, have not yet been defined for advanced difference equations.

4. Conclusions

In this work, we obtained new oscillation criteria for first-order difference equations with several, not necessarily monotone, advanced arguments. Our results substantially strengthen and extend the existing results in the literature. Theorem 1 shows that there are fundamental differences between equations with a single advanced argument and those with several advanced arguments. Moreover, the analytical approach presented in Theorem 2 is new and sufficiently flexible to be applied to further qualitative investigations, including the study of the distribution of generalized zero and related properties of difference equations with generalized delays or advanced arguments.

5. Conjectures

Motivated by the analytical results and numerical observations presented in this work, we conclude with the following conjectures, which may serve as a basis for future investigations.

Conjecture 1.

Assume that the deviating arguments ${\theta }_r\left(n\right)$, $r=1,2,\dots ,m$, are non-decreasing. Then, Equation (1) is oscillatory if the sharp condition

$$\underset{n\to \infty }{\mathrm{lim\; sup}}\sum _{r=1}^m\sum _{r_1=n+1}^{{\theta }_r\left(n\right)}{c}_r\left(r_1\right)>1,$$

is satisfied.

Conjecture 2.

The analytical methods and techniques proposed for delay differential equations in the study of upper and lower bounds of the distance between zeros can be suitably adapted and generalized to difference equations with advanced arguments. In particular, such extensions may lead to effective criteria for estimating both lower and upper bounds for the distance between generalized zeros of solutions.

We believe that the resolution of these conjectures would significantly enhance the current understanding of oscillatory behavior and zero distribution for advanced-type difference equations.