Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 1 (2024) 

 

 

89 
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Oscillatory and Property Conditions for Third Order Difference 

Equation with Negative Middle-Term 

 

S. Kaleeswari 1, S. Rangasri 2 
1,2Department of Mathematics, Nallamuthu Gounder Mahalingam College, Pollachi, Tamilnadu, India. 

 e-mail: kaleesdesika@gmail.com, e-mail: rangasrisuresh97@gmail.com.  

 

Article History: 

Received: 18-09-2023 

Revised: 25-10-2023 

Accepted: 19-11-2023 

Abstract:  

This paper presents a detailed analysis of the oscillation and Property A conditions for 

third-order advanced difference equations with negative middle terms. The authors 

employ the generalized Riccati transformation technique and summation by parts to 

establish sufficient criteria for the solutions to be oscillatory or have Property A. The 

main results and their proofs are presented clearly and concisely. The difference equations 

have been a topic of interest in various scientific fields such as economics, physics, and 

mathematical biology. The results presented in this paper can be used to inform the 

development of more accurate models for real-world problems in various scientific fields. 

Additionally, this paper explores the economic theory of business cycles and its 

mathematical foundations, specifically Samuelson's model, which outlines five potential 

trajectories or trends that economic activity can follow based on different combinations 

of the marginal propensity to consume multiplier parameter(α) and the accelerator 

coefficient (β). 

Keywords: Advanced equation, property A, oscillation criteria, Riccati transformation, 

semi-canonical. 

1. Introduction 

In this article, we will discuss the economic theory of business cycles and its mathematical foundations. 

Specifically, we will explore Samuelson's model, which outlines five potential trajectories or trends 

that economic activity can follow. These paths are determined by different combinations of the 

marginal propensity to consume (α) and the accelerator coefficient (β). Each combination results in a 

unique pattern of economic movement. We will examine the model in detail and highlight its key 

features, including the dynamic of economic cycles and their relationship to changes in consumption 

and investment. 

The article will also delve into the mathematical foundations of the model, including second-order 

difference equations in the context of business cycles. We will explore the oscillatory behavior and 

Property A conditions for third-order difference equations, which have applications in economics, 

physics, and mathematical biology. The paper will highlight the connection between these equations 

and Samuelson's model, demonstrating how they can be used to gain insights into the behavior of 

economic cycles. 

Finally, we will present practical examples to demonstrate the main results, providing to apply in real-

world scenarios. This chapter aims to provide readers with a comprehensive overview of Samuelson's 



Communications on Applied Nonlinear Analysis 

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model and its mathematical foundations, highlighting its relevance and potential implications for 

various fields. 

Third-order advanced difference equation 

𝛥 (𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑))
𝜅
))− 𝑝(𝜑)(𝛥𝑦(𝜑+ 1))

𝜅 + 𝑞(𝜑)𝑓(𝑦(𝜎(𝜑))) = 0,𝜑 ≥ 𝜑0     (1) 

where {𝑎(𝜑)}, {𝑏(𝜑)}, {𝑝(𝜑)} and {𝑞(𝜑)} are the positive real sequences for 𝜑 ∈ 𝑁. 

Terms considered: 

H1) 𝜅 is a quotient of odd positive integers; 

H2) 𝜎(𝜑) is an increasing sequence such that 𝜎(𝜑) ≥ 𝜑+ 1 for all 𝜑 ≥ 𝜑0; 

H3) ∑𝑠=𝜑0
∞  

1

𝑎(𝑠)
< ∞,∑𝑠=𝜑0

∞  
1

𝑏(𝑠)
= ∞; 

H4) function 𝑓: 𝑅 → 𝑅 is continuous such that 𝑢𝑓(𝑢) > 0 for 𝑢 ≠ 0 and 𝑓(𝑢𝑣) ≥ 𝑓(𝑢)𝑓(𝑣) :  

A solution of (1) is a real sequence {𝑦(𝜑)} defined for all 𝜑 ≥ 𝜑0.  A nontrivial solution {𝑦(𝜑)} of (1) 

is said to be oscillatory if it is neither positive nor negative, otherwise it is non-oscillatory.  By Property 

A of (1) it is meant that every solution 𝑦(𝜑) of (1) is strictly increasing  

𝑦(𝜑)) > 0, 𝑑(𝜑)𝛥(𝑦(𝜑))𝜅 < 0, 𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅)) > 0,  

𝛥 (𝑐(𝜑)𝛥 (𝑑(𝜑)(𝛥(𝑦(𝜑))
𝜅
))) < 0 

Recent research has explored the oscillation criteria of difference equations, primarily focused on [1]-

[4] , [15]. However, third-order equations are also essential, as they have applications in various fields, 

including economics, physics, and mathematical biology ([7],[10], [12]). For instance, Samuelson's 

business cycles theory, a central economic theory related to the business cycle, can be modelled by 

linear second and third-order difference equations. This capability allows advanced arguments to 

depict events dependent on another action. Recent studies have investigated the oscillatory criteria and 

Property A conditions for difference equations, and their findings are available in the literature 

([6],[8],[9],[11]). This paper's results and recent research can inform the development of more accurate 

models for real-world problems. 

In [13], the authors used advanced arguments to demonstrate the oscillatory and Property B conditions 

for difference equation,  

𝛥 (𝑎𝑛(𝛥(𝑏𝑛𝛥𝑥𝑛))
𝛼
)− 𝑝𝑛𝑓(𝑥𝜎(𝑛)) = 0,𝑛 ≥ 𝑛0 

where 𝑎𝑛, 𝑏𝑛, and 𝑝𝑛 are the positive real sequences. 

In [14], the oscillatory behavior of difference equation was investigated 

𝛥 (𝑐(𝑙)𝛥(𝑑(𝑙)(𝛥𝑥(𝑙))
𝛼
))− 𝑎(𝑙)(𝛥𝑥(𝑙 + 1))

𝛼 − 𝑏(𝑙)𝑔(𝑥(𝜎(𝑙))) = 0, 𝑙 ≥ 𝑙0 

Proceeding the manner, the aim of this paper is to commence sufficient condition to ensure that 

solution of (1) are oscillatory and also it is closely relevant to the auxiliary second order equation 



Communications on Applied Nonlinear Analysis 

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Vol 31 No. 1 (2024) 

 

 

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𝛥(𝑎(𝜑)𝛥(𝑧(𝜑))) −
𝑝(𝜑)

𝑏(𝜑+1)
𝑥(𝜑 + 1)                                                    (2) 

We also derive a solution for equation (1) that possesses Property A, and its corresponding second-

order equation aligns with Samuelson’s business cycle model. 

2. Main Results 

Let us define: 

𝑐(𝜑) = 𝑎(𝜑)𝑧(𝜑)𝑧(𝜑 + 1), 𝑑(𝜑) =
𝑏(𝜑)

𝑧(𝜑)
, 𝑄(𝜑) = 𝑞(𝜑)𝑧(𝜑 + 1), 

  𝐶(𝜑) = ∑
1

𝑐(𝑠)

𝜑−1
𝑠=𝜑1

   , 𝐷(𝜑) = ∑
1

𝑑
1
𝜅(𝑠)

𝜑−1
𝑠=𝜑1

   , 𝐸(𝜑) = ∑
1

𝑑
1
𝜅(𝑠)

𝜑−1
𝑠=𝜑1

  (∑
1

𝑐(𝑠)

𝜑−1
𝑠=𝜑1

  )

1

𝜅
. 

Theorem 2.1. Assume that 

∑
1

𝑑
1
𝜅(𝑠)

∞
𝑠=𝜑0   = ∞,                                                                                       (3) 

and 

𝐿𝑦 (𝜑) = 𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑))𝜅)) − 𝑝(𝜑)(𝛥𝑦(𝜑 + 1))𝜅 

Then 𝐿𝑦 (𝜑) can be written as 

𝐿𝑦 (𝜑) =
1

𝑧(𝜑 + 1)
𝛥 (𝑎(𝜑)𝑧(𝜑)𝑧(𝜑 + 1)𝛥 (

𝑏(𝜑)

𝑧(𝜑)
(𝛥𝑦(𝜑))𝜅)). 

Proof. 

1

𝑧(𝜑+1)
 𝛥(𝑎(𝜑)𝑧(𝜑)𝑧(𝜑 + 1)𝛥 (

𝑏(𝜑)

𝑧(𝜑)
(𝛥𝑦(𝜑))𝜅))   

=  
1

𝑧(𝜑+1)
[𝛥(𝑎(𝜑)(𝛥(𝑏(𝜑)(𝛥𝑦(𝜑))𝜅𝑧(𝜑 + 1)) − 𝑏(𝜑 + 1)(𝛥𝑦(𝜑 + 1))𝜅𝛥𝑧(𝜑)))]   

=  
1

𝑧(𝜑+1)
[𝑧(𝜑 + 1)𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑 + 1))𝜅)) −

𝑏(𝜑+1)

𝑧(𝜑+1)
(𝛥𝑦(𝜑 + 1))𝜅𝛥(𝑎(𝜑𝛥𝑧(𝜑)))]   

=  𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑 + 1))𝜅)) − 𝑝(𝜑)(𝛥𝑦(𝜑 + 1))𝜅.  

Hence the proof. 

Corollary 2.2. Assume that the solution {𝑧(𝜑)} of (2) is positive. Then the semi-canonical difference 

equation (1) is in equivalent canonical form 

𝛥 (𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))
𝜅
))+𝑄(𝜑)𝑓(𝑦(𝜎(𝜑))) = 0.                                      (4) 

Lemma 2.3. (see [2]) If 

𝑎(𝜑) ≥ 1,∑
𝑝(𝜑)

𝑏(𝜑+1)
∞
𝜑=𝜑0   < ∞, 𝑙𝑖𝑚 𝑠𝑢𝑝𝜑→∞  𝑛 ∑

𝑝(𝑠)

𝑏(𝑠+1)
∞
𝑠=𝜑   <

1

4
,                          (5) 

then the solution of equation (2) is positive. 



Communications on Applied Nonlinear Analysis 

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Lemma 2.4. (see [2]) Assume that (5) holds. Then ∃ a non-oscillatory solution to the equation (2) 

fulfilling 

∑
1

𝑎(𝜑)𝑧(𝜑)𝑧(𝜑+1)
∞
𝜑=𝑛0   = ∞.                                                                                 (6) 

Remark: 

From (3) 

∑
𝑧(𝑠)

𝑏(𝑠)
∞
𝑠=𝜑0   = ∞,                                                                                                (7) 

where {𝑦(𝜑)} is a positive solution. 

In the continuation, we examine the positive solution of (4) without losing generality. So, we 

introduce the following classes: 

𝑦(𝜑) ∈ 𝑁0: 𝑦(𝜑)) > 0, 𝑑(𝜑)𝛥(𝑦(𝜑))
𝜅 < 0, 𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅)) > 0, 

𝛥 (𝑐(𝜑)𝛥 (𝑑(𝜑)(𝛥(𝑦(𝜑))
𝜅
))) < 0 

𝑦(𝜑) ∈ 𝑁1: 𝑦(𝜑)) > 0, 𝑑(𝜑)𝛥(𝑦(𝜑))
𝜅 > 0, 𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅)) > 0, 

𝛥 (𝑐(𝜑)𝛥 (𝑑(𝜑)(𝛥(𝑦(𝜑))
𝜅
))) < 0. 

for all 𝜑 ≥ 𝜑2 ≥ 𝜑1. 

Lemma 2.5. Suppose that {𝑦(𝜑)} is a positive solution of (4) which satisfies Case 𝑁1 and 

(∑
1

𝑐(𝑠)

𝜑−1
𝑠=𝜑3

  ∑  𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞
𝑡=𝑠  )

1

𝜅 = ∞.                                                      (8) 

Then 

(i) {
𝑦(𝜑)

𝐷(𝜑)
} is increasing ∀ 𝜑 ≥ 𝑁, 

(ii) {
𝑦(𝜑)

𝐸(𝜑)
} is decreasing ∀ 𝜑 ≥ 𝑁, 

(iii) {(
𝑑(𝜑)(𝛥𝑦(𝜑))𝜅

𝐶(𝜑)
)} is decreasing ∀ 𝜑 ≥ 𝑁. 

Proof. Let {𝑦(𝜑)} be a positive solution to (4) that fulfills {𝑦(𝜑)} ∈ 𝑁1 for all 𝜑 ∈ 𝑁. Since 

𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅) is decreasing, we get 

𝑑(𝜑)(𝛥𝑦(𝜑))𝜅   = 𝑑(𝜑1)(𝛥𝑦(𝜑1))
𝜅
+ ∑

𝑐(𝑖)𝛥(𝑑(𝑖)(𝛥𝑦(𝑖))𝜅)

𝑐(𝑖)

𝜑−1
𝑖=𝜑1

     ≥ 𝐶(𝜑)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅). 

This gives 

𝛥 (
𝑑(𝜑)(𝛥𝑦(𝜑))𝜅

𝐶(𝜑)
) =

𝐶(𝜑)𝑐(𝜑)𝛥(𝑑(𝜑(𝛥𝑦(𝜑))
𝜅
))−𝑑(𝜑)(𝛥𝑦(𝜑))

𝜅

𝑐(𝜑)𝐶(𝜑)𝐶(𝜑+1)
≤ 0. 

So (
𝑑(𝜑)(𝛥𝑦(𝜑))𝜅

𝐶(𝜑)
) is decreasing and further, this fact yields 



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𝑦(𝜑)   = 𝑦(𝜑1) + (∑
𝐶
1
𝜅(𝑠)𝑑

1
𝜅(𝑠)(𝛥𝑦(𝑠))

𝐶
1
𝜅(𝑠)𝑑(𝑠)

𝜑−1
𝑠=𝜑1

  )    ≥ 𝐸(𝜑) (
𝑑
1
𝜅(𝜑)(𝛥𝑦(𝜑))

𝐶
1
𝜅(𝜑)

).                      (9) 

Hence 

𝛥 (
𝑦(𝜑)

𝐸(𝜑)
) =

𝐸(𝜑)𝑑
1
𝜅(𝜑)𝛥𝑦(𝜑)−𝑦(𝜑)𝐶

1
𝜅(𝜑)

𝑑
1
𝜅(𝜑)𝐸(𝜑)𝐸(𝜑+1)

≤ 0, 

which gives 
𝑦(𝜑)

𝐸(𝜑)
 is decreasing. 

Next, since {𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑)} is increasing for all 𝜑 ≥ 𝑁, it is apparent ∀ 𝜑 ≥ 𝜑2 ≥ 𝜑1 

𝑦(𝜑)  = 𝑦(𝜑2) + ∑
𝑑
1
𝜅(𝑠)

𝑑
1
𝜅(𝑠)
𝛥𝑦(𝑠)

𝜑−1
𝑠=𝜑2

     ≤ 𝑦(𝜑2) + 𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑)∑
1

𝑑
1
𝜅(𝑠)

𝜑−1
𝑠=𝜑2

       

= 𝑦(𝜑2) − 𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑)∑
1

𝑑
1
𝜅(𝑠)

𝜑1−1
𝑠=𝜑2

  + 𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑)∑
1

𝑑
1
𝜅(𝑠)

𝜑−1
𝑠=𝜑2

   .                     (10) 

We claim that 𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑) → ∞ as 𝜑 → ∞. Suppose 𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑) → 2𝑤 as 𝜑 → ∞, we have 

𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑) ≥ 𝑤. This gives 𝑦(𝜑) ≥ 𝑤𝐷(𝜑), summing (4) from 𝜑 to ∞, we have 

𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))
𝜅
) ≥

1

𝑐(𝜑)
𝑓(𝑤)∑ 𝑄(𝑠)𝑓 (𝐷(𝜎(𝑠)))∞

𝑠=𝜑  . 

Again summing from 𝜑3 to 𝜑− 1, we obtain 

𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑) ≥ 𝑓
1

𝜅(𝑤) (∑
1

𝑐(𝑠)

𝜑−1
𝑠=𝜑3

  ∑ 𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞
𝑡=𝑠   )

1

𝜅
 ,                          (11) 

for all 𝜑 ≥ 𝜑3. 

2𝑤 ≥ 𝑓
1

𝜅(𝑤) (∑
1

𝑎(𝑠)

𝜑−1
𝑠=𝜑3

  ∑ 𝑄(𝑡)𝑓(𝐵(𝜎(𝑡)))∞
𝑡=𝑠   )

1

𝜅
, 

which contradicts (8) so 𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑) → ∞ as 𝜑 →∞. Hence, (10) becomes 

𝑦(𝜑) ≤ 𝑑
1

𝜅(𝜑)𝛥𝑦(𝜑)𝐷(𝜑). 

Since 

𝛥 (
𝑦(𝜑)

𝐷(𝜑)
) =

𝑑
1
𝜅(𝜑)𝐷(𝜑)𝛥𝑦(𝜑)−𝑦(𝜑)

𝑑
1
𝜅(𝜑)𝐷(𝜑)𝐷(𝜑+1))

≥ 0, 

𝑦(𝜑)

𝐷(𝜑)
 is increasing. This concludes the proof. 

Theorem 2.6. Let (8) hold. Assume that (2) has a positive solution {𝑧(𝜑)} and 

𝑙𝑖𝑚 𝑠𝑢𝑝𝑢→∞  
𝑢𝜅

𝑓(𝑢)
= 𝐾 < ∞.                                                               (12) 

 



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If 

𝑙𝑖𝑚 𝑠𝑢𝑝𝜑→∞  

(

  
 

𝐸𝜅(𝜎(𝜑))

𝐶𝜅(𝜎(𝜑))
𝑓 (

1

𝐸(𝜎(𝜑))
)∑ 𝐶(𝑠 + 1)𝑄(𝑠)𝑓 (𝐸(𝜎(𝑠)))

𝜎(𝜑)−1
𝑠=𝜑1

+𝐸𝜅(𝜎(𝜑))𝑓 (
1

𝐸(𝜎(𝜑))
)∑ 𝑄(𝑡)𝑓 (𝐸(𝜎(𝑡)))

𝜑−1
𝑡=𝜎(𝜑)

+ 𝐸𝜅(𝜎(𝜑))𝑓 (
1

𝐷(𝜎(𝜑))
)∑  𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞

𝑡=𝜑 )

  
 
> 𝐾,                (13) 

then 𝑦(𝜑) ∉ 𝑁1 which implies (1) has Property 𝐴. 

Proof: 

Assume that {𝑦(𝜑)} is a positive solution of (1) and so a solution of (4) which belonging to either 𝑁0 

or 𝑁1 for all 𝜑 ≥ 𝑁. Now we take 𝑦(𝜑) ∈ 𝑁1, summation of (4) from 𝜑 to ∞ gives 

𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))
𝜅
) ≥

1

𝑐(𝜑)
∑ 𝑄(𝑠)𝑓 (𝑦(𝜎(𝑠)))∞
𝑠=𝜑  . 

Summing the aforementioned inequality from 𝜑1 to 𝜑− 1, we get 

𝑑(𝜑)(𝛥𝑦(𝜑))𝜅   ≥ ∑
1

𝑐(𝑠)

𝜑−1
𝑠=𝜑1

  ∑ 𝑄(𝑡)𝑓 (𝑦(𝜎(𝑡))) ∞
𝑡=𝑠   

= ∑ 𝐶(𝑠 + 1)𝑄(𝑠)𝑓 (𝑦(𝜎(𝑠)))
𝜑−1
𝑠=𝜑1

  + 𝐶(𝜑)∑ 𝑄(𝑡)𝑓(𝑦(𝜎(𝑡)))∞
𝑡=𝜑    . 

From (9), we obtain 

𝐶(𝜑)𝑦𝜅(𝜑)

𝐸𝜅(𝜑)
≥ ∑ 𝐶(𝑠 + 1)𝑄(𝑠)𝑓(𝑦(𝜎(𝑠)))

𝜑−1
𝑠=𝜑1

 + 𝐶(𝜑)∑ 𝑄(𝑡)𝑓(𝑦(𝜎(𝑡)))∞
𝑡=𝜑  . 

or 

𝐶(𝜎(𝜑))𝑦𝜅(𝜎(𝜑))

𝐸𝜅(𝜎(𝜑))
≥ ∑ 𝐶(𝑠 + 1)𝑄(𝑠)𝑓 (𝑦(𝜎(𝑠)))

𝜎(𝜑)−1
𝑠=𝜑1

  + 𝐶(𝜎(𝜑)) ∑ 𝑄(𝑡)𝑓 (𝑦(𝜎(𝑡)))
𝜑−1
𝑡=𝜎(𝜑)     

+𝐶(𝜎(𝜑))∑ 𝑄(𝑡)𝑓 (𝑦(𝜎(𝑡)))∞
𝑡=𝜑  .  

Using the monotonic properties (i)-(ii) of Lemma 2.5 and applying 𝑓, we get 

 
𝐶(𝜎(𝜑))𝑦𝜅(𝜎(𝜑))

𝐸𝜅(𝜎(𝜑))
≥ 𝑓 (

𝑦(𝜎(𝜑))

𝐸(𝜎(𝜑))
)∑ 𝐶(𝑠 + 1)𝑄(𝑠)𝑓 (𝐸(𝜎(𝑠)))𝜎(𝜑)−1

𝑠=𝜑1
    

+𝐶(𝜎(𝜑))𝑓 (
𝑦(𝜎(𝜑))

𝐸(𝜎(𝜑))
)∑ 𝑄(𝑡)𝑓 (𝐸(𝜎(𝑡)))

𝜑−1
𝑡=𝜎(𝜑)     

   +𝐶(𝜎(𝜑))𝑓 (
𝑦(𝜎(𝜑))

𝐷(𝜎(𝜑))
)∑ 𝑄(𝑡)𝑓 (𝐷(𝜎(𝑡)))∞

𝑡=𝜑 , 

or 

 
𝑦𝜅(𝜎(𝜑))

𝑓(𝑦(𝜎(𝜑)))
≥
𝐸𝜅(𝜎(𝜑))

𝐶(𝜎(𝜑))
𝑓 (

1

𝐸(𝜎(𝜑))
)∑ 𝐶(𝑠 + 1)𝑄(𝑠)𝑓 (𝐸(𝜎(𝑠)))𝜎(𝜑)−1

𝑠=𝜑1
    

+𝐸𝜅(𝜎(𝜑))𝑓 (
1

𝐸(𝜎(𝜑))
)∑ 𝑄(𝑡)𝑓 (𝐸(𝜎(𝑡)))

𝜑−1
𝑡=𝜎(𝜑)                

          +𝐸𝜅(𝜎(𝜑))𝑓 (
1

𝐷(𝜎(𝜑))
)∑  𝑄(𝑡)𝑓 (𝐷(𝜎(𝑡)))∞

𝑡=𝜑 .   



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Taking limsup as 𝜑 → ∞ on both side of above inequality, we get contradiction with (13). Hence 

𝑦(𝜑) ∉ 𝑁1, so 𝑦(𝜑) satisfies the case 𝑁0. Therefore (1) has Property A. 

Corollary 2.7. Let (8) hold, 𝑓(𝑢) = 𝑢𝜅 and (2) have a positive solution {𝑧(𝜑)}. If 

𝑙𝑖𝑚 𝑠𝑢𝑝𝜑→∞   

(

 
 

1

𝐶(𝜎(𝜑))
∑ 𝐶(𝑠 + 1)𝑄(𝑠)𝐸𝜅(𝜎(𝑠))
𝜎(𝜑)−1
𝑠=𝜑1

     

+∑ 𝑄(𝑡)𝑓 (𝐸(𝜎(𝑡)))
𝜑−1
𝑡=𝜎(𝜑)     

+
𝐸𝜅(𝜎(𝜑))

𝐷𝜅(𝜎(𝜑))
∑ 𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞
𝑡=𝜑    )

 
 
> 1.                                     (14) 

then 𝑦(𝜑) ∉ 𝑁1 which implies (1) has Property 𝐴 

Throughout the proof of the following theorem, we use 𝐻(𝜑, 𝑗): 𝜑, 𝑗 ∈ 𝑁, 𝜑 ≥ 𝑗 ≥ 0 to denote the 

double sequence satisfying 

𝐻(𝜑, 𝜑) = 0 𝑓𝑜𝑟 𝜑 ≥ 𝜑0 ; 

𝐻(𝜑, 𝑗) > 0 𝑓𝑜𝑟 𝜑 > 𝑗 ≥ 𝜑0; 

 𝛥2𝐻(𝜑, 𝑗) = 𝐻(𝜑, 𝑗 + 1) − 𝐻(𝜑, 𝑗) < 0 𝑓𝑜𝑟 𝜑 > 𝑗 ≥ 𝜑0 . 

Theorem 2.8.(H1)-(H3) hold and (2) has a positive solution {𝑧(𝜑)}. If 

𝑙𝑖𝑚 𝑠𝑢𝑝𝜑→∞  
1

𝐻(𝜑,𝑀)
∑  [𝐻(𝜑, 𝑗)

𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))

𝑦(𝜎(𝑗))𝜅
−
ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗))

4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗)
]

𝜑−1
𝑗=𝑀 = ∞,          (15) 

and 

∑ (
1

𝑑(𝑡)
∑

1

𝑐(𝑗)

∞
𝑗=𝑡   ∑ 𝑄(𝑖)𝑓(𝑦(𝜎(𝑖)))∞

𝑖=𝑗   )

1

𝜅∞
𝑡=𝜑   = ∞,                                       (16) 

then (1) is oscillatory. 

Proof. Let 𝑦(𝜑) be a nonoscillatory solution of (4). Without loss of generality, 𝑦(𝜑) is eventually 

positive such that 𝑦(𝜑) ∈ 𝑁0 or 𝑦(𝜑) ∈ 𝑁1. If 𝑦(𝜑) ∈ 𝑁1, then we have to define for 𝜌(𝜑) > 0, 

𝑤(𝜑) = 𝜌(𝜑)
𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅)

(𝑦(𝜎(𝜑)))𝜅
, 

then 𝑤(𝜑) > 0 for 𝜑 ≥ 𝜑1, and 

𝛥𝑤(𝜑)   = 𝛥(𝜌(𝜑)
𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))

𝜅
)

(𝑦(𝜎(𝜑)))
𝜅 )    

= 𝛥𝜌(𝜑)
𝑐(𝜑+ 1)𝛥(𝑑(𝜑 + 1)(𝛥𝑦(𝜑+ 1))

𝜅
)

(𝑦(𝜎(𝜑+ 1)))𝜅
+ 𝜌(𝜑)𝛥(

𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))
𝜅
)

(𝑥(𝜎(𝜑)))𝜅
)    

=
𝑤(𝜑+ 1)

𝜌(𝜑+ 1)
𝛥𝜌(𝜑) + 𝜌(𝜑)

𝛥 (𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))
𝜅
))

(𝑦(𝜎(𝜑)))𝜅

−
𝜌(𝜑)𝑤(𝜑+ 1)

(𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+ 1)
𝛥(𝑦(𝜎(𝜑)))

𝜅
.  

From (4), we have 



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𝛥𝑤(𝜑) =
𝑤(𝜑+1)

𝜌(𝜑+1)
𝛥𝜌(𝜑) −

𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑)))

𝑦(𝜎(𝜑))𝜅
−

𝜌(𝜑)𝑤(𝜑+1)

(𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+1)
𝛥(𝑦(𝜎(𝜑)))𝜅.               (17) 

Since 𝛥(𝑑(𝜑)𝛥(𝑦(𝜑))
𝜅
) is decreasing, so 𝑑(𝜑)𝛥(𝑦(𝜑))𝜅 = 𝑑(𝑗)𝛥(𝑦(𝑗))𝜅 + ∑𝑠=𝑗

𝜑−1
 𝛥(𝑑(𝑠)𝛥𝑦(𝑠)), 

𝜑 ≥ 𝑗 ≥ 𝜑2, which implies 

𝑑(𝜑)(𝛥𝑦(𝜑))𝜅 ≥ (𝜑 − 𝑗)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅). 

and so 

(𝛥𝑦(𝜎(𝜑)))𝜅 ≥
(𝜎(𝜑)−𝑗)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅)

𝑐(𝜑)𝑑(𝜎(𝜑))
.                                         (18) 

From (17) and (18), we carry 

𝛥𝑤(𝜑) ≤
𝑤(𝜑+1)

𝜌(𝜑+1)
𝛥𝜌(𝜑)−

𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑)))

𝑦(𝜎(𝜑))𝜅
−

𝜌(𝜑)𝑤(𝜑+1)

(𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+1)

(𝜎(𝜑)−𝑗)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))
𝜅
)

𝑐(𝜑)𝑑(𝜎(𝜑))
, 

 𝛥𝑤(𝜑) ≤ −
𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑)))

𝑦(𝜎(𝜑))𝜅
+

𝛥𝜌(𝜑)

𝜌(𝜑+1)
𝑤(𝜑+ 1) −

(𝜎(𝜑)−𝑗)

𝜌(𝜑+1)𝑐(𝜑)𝑑(𝜎(𝜑))
𝑤2(𝜑+ 1). 

Multiplying both side by 𝐻(𝜑, 𝑗) and then summing up from 𝑀 to 𝜑− 1 for all 𝜑 ≥𝑀, we obtain 

∑ 𝐻(𝜑, 𝑗)
𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))

𝑦(𝜎(𝑗))𝜅
𝜑−1
𝑗=𝑀     ≤ ∑ 𝐻(𝜑, 𝑗)

𝛥𝜌(𝑗)

𝜌(𝑗+1)
𝑤(𝑗 + 1)

𝜑−1
𝑗=𝑀     

− ∑ 𝐻(𝜑, 𝑗)
(𝜎(𝑗)−𝑗)

𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗))
𝑤2(𝑗 + 1)

𝜑−1
𝑗=𝑀   − ∑ 𝐻(𝜑, 𝑗)𝑤(𝑗)

𝜑−1
𝑗=𝑀  .  (19) 

Using summation by parts, we see that 

−∑  𝑤(𝑗)𝐻(𝜑, 𝑗)
𝜑−1
𝑗=𝑀 = 𝐻(𝜑,𝑀)𝑤(𝑀) + ∑ 𝑤(𝑗 + 1)𝛥2𝐻(𝜑, 𝑗)

𝜑−1
𝑗=𝑀  .                   (20) 

Substituting (20) in (19), we carry 

∑ 𝐻(𝜑, 𝑗)
𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))

𝑦(𝜎(𝑗))𝜅
𝜑−1
𝑗=𝑀    ≤ 𝐻(𝜑,𝑀)𝑤(𝑀) + ∑ ℎ(𝜑, 𝑗)𝑤(𝑗 + 1)

𝜑−1
𝑗=𝑀   

          −∑ 𝐻(𝜑, 𝑗)
(𝜎(𝑗)−𝑗)

𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗))
𝑤2(𝑗 + 1)

𝜑−1
𝑗=𝑀    ,    

where 

ℎ(𝜑, 𝑗) = 𝛥2𝐻(𝜑, 𝑗) + 𝐻(𝜑, 𝑗)
𝛥𝜌(𝑗)

𝜌(𝑗+1)
. 

 ∑ 𝐻(𝜑, 𝑗)
𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))

𝑦(𝜎(𝑗))𝜅
𝜑−1
𝑗=𝑀    ≤ 𝐻(𝜑,𝑀)𝑤(𝑀) + ∑

ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗))

4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗)

𝜑−1
𝑗=𝑀    , 

or 

∑ [𝐻(𝜑, 𝑗)
𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))

𝑦(𝜎(𝑗))𝜅
−
ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗))

4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗)
]

𝜑−1
𝑗=𝑀   ≤ 𝐻(𝜑,𝑀)𝑤(𝑀).          (21) 

Taking 𝑙𝑖𝑚𝑠𝑢𝑝 as 𝜑 →∞, we obtain 

𝑙𝑖𝑚 𝑠𝑢𝑝𝜑→∞  
1

𝐻(𝜑,𝑀)
∑ [𝐻(𝜑, 𝑗)

𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))

𝑦(𝜎(𝑗))𝜅
−
ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗))

4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗)
]

𝜑−1
𝑗=𝑀   ≤ 𝑤(𝑀). 

which contradicts (15). 



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Now we take 𝑦(𝜑) ∈ 𝑁0, summing (4) from 𝜑 to ∞, we have 

 𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))
𝜅
) ≥ ∑ 𝑄(𝑡)𝑓 (𝑦(𝜎(𝑡)))  ∞

𝑡=𝜑 , 

𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))
𝜅
)    ≥

1

𝑐(𝜑)
∑   𝑄(𝑡)𝑓(𝑦(𝜎(𝑡))) ∞
𝑡=𝜑 . 

Again summing from 𝜑 to ∞ 

(𝛥𝑦(𝜑))𝜅 ≤ −
1

𝑑(𝜑)
∑

1

𝑐(𝑡)
∞
𝑡=𝜑   ∑ 𝑄(𝑗)𝑓 (𝑦(𝜎(𝑗)))∞

𝑗=𝑡   , 

𝛥𝑦(𝜑) ≤ − (
1

𝑑(𝜑1)
∑

1

𝑐(𝑡)
∞
𝑡=𝜑   ∑ 𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))∞

𝑗=𝑡   )

1

𝜅
. 

Summing once again from 𝜑1 to ∞, we obtain 

𝑦(𝜑1) ≥ ∑
1

𝑑(𝑡)
(∑

1

𝑐(𝑗)

∞
𝑗=𝑡   ∑ 𝑄(𝑖)𝑓(𝑦(𝜎(𝑖)))∞

𝑖=𝑗   )

1

𝜅∞
𝑡=𝜑1 , 

which contradicts (16). Therefore, every solution of (1) is oscillatory. Hence the proof. 

3 Examples 

Example 3.1. Look into the equation 

𝛥(𝜑2𝛥(𝜑(𝛥𝑦(𝜑))))−
(2𝜑+1)(𝜑+1)

𝜑+2
(𝛥(𝜑 + 1)) +

𝜑+1

𝜑+2
𝑓(𝑦(3𝜑))) = 0,𝜑 ≥ 1.           (22) 

Here 𝑎(𝜑) = 𝜑2, 𝑏(𝜑) = 𝜑, 𝑝(𝜑) =
(2𝜑+1)(𝜑+1)

𝜑+2
, 𝑞(𝜑) =

𝜑+1

𝜑+2
, 𝜅 = 1,    

𝜎(𝜑) = 3𝜑, 𝑓(𝑦) = 𝑦𝜅 , 𝑐(𝜑) = 𝜑2(𝜑 + 1)(𝜑 + 2), 𝑑(𝜑) =
𝜑

𝜑+1
,    

𝑄(𝜑) = 𝜑 + 1, 𝐶(𝜑) ≈
1

𝜑2(𝜑+1)(𝜑+2)
, 𝐷(𝜑) ≈

𝜑+1

𝜑
, 𝐸(𝜑) ≈

1

𝜑3(𝜑+2)
.   

Now 

∑
1

𝑎(𝑠)
∞
𝑠=1   = ∑

1

𝑠2
∞
𝑠−1   < ∞,   ∑

1

𝑏
1
𝜅(𝑠)

∞
𝑠=1   = ∑  

1

𝑠
∞
𝑠=1 = ∞, 

and the auxiliary second order equation (4) becomes, 

𝛥(𝜑2𝛥𝑧(𝜑))−
(2𝜑+1)(𝜑+1)

(𝜑+2)
𝑧(𝜑+ 1) = 0.                                                   (23) 

It has a nonoscillatory solution 𝑧(𝜑) = 𝜑 + 1. Moreover, the positive sequence 𝜌(𝜑) = 𝑙 which 

implies that 𝛥𝜌(𝜑) = 1 and 

𝐻(𝜑, 𝜑) = 0, 𝜑 ≥ 𝜑0 , 

𝐻(𝜑, 𝑗) = 𝜑 − 𝑗 > 0, 𝜑 > 𝑗 ≥ 𝜑0, 

 𝛥2[𝐻(𝜑, 𝑗)] = 𝐻(𝜑, 𝑗 + 1)−𝐻(𝜑, 𝑗) = −1 < 0,𝜑 > 𝑗 ≥ 𝜑0, 

ℎ(𝜑, 𝑗) =
1

𝜑+1
[(𝜑 − 𝑗) − (𝜑 + 1)]. 

From the above values, (15) becomes 



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𝑙𝑖𝑚 𝑠𝑢𝑝𝜑→∞  
1

𝐻(𝜑,𝑀)
∑ [𝑗(𝜑 − 𝑗)(𝑗 + 1) −

3𝑗3(2(𝑗+1)+(𝜑−𝑗)

8𝑗(3𝑗+1)
]

𝜑−1
𝑗=𝑀   = ∞, 

and (16) becomes 

∑ (
𝑡+1

𝑡
∑

1

𝑗2(𝑗+2)(𝑗+1)

∞
𝑗=𝑡   ∑   (𝑖 + 1)∞

𝑖=𝑗 )∞
𝑡=𝜑   = ∞. 

Therefore, conditions of theorem 2.8 are satisfied, so every solution of (22) is oscillatory. 

Example 3.2. Look over the equation 

𝛥(𝜑2𝛥(𝜑(𝛥𝑦(𝜑))))−
(2𝜑+1)(𝜑+1)

𝜑+2
(𝛥(𝜑 + 1)) +

8𝜑2

𝜑+1
𝑓(𝑦(3𝜑))) = 0,𝜑 ≥ 1               (24) 

Here 𝑎(𝜑) = 𝜑2, 𝑏(𝜑) = 𝜑, 𝑝(𝜑) =
(2𝜑+1)(𝜑+1)

𝜑+2
, 𝑞(𝜑) =

8𝜑2

𝜑+1
, 𝜅 = 1,  

𝜎(𝜑) = 𝜑 + 1, 𝑓(𝑦) = 𝑦𝜅 , 𝑐(𝜑) = 𝜑2(𝜑 + 1)(𝜑 + 2), 𝑑(𝜑) =
𝜑

𝜑+1
,  

𝑄(𝜑) = 𝜑 + 1, 𝐶(𝜑) ≈
1

𝜑2(𝜑+1)(𝜑+2)
, 𝐷(𝜑) ≈

𝜑+1

𝜑
, 𝐸(𝜑) ≈

1

𝜑3(𝜑+2)
.   

 Now 

∑
1

𝑎(𝑠)
∞
𝑠=1   = ∑

1

𝑠2
∞
𝑠−1   < ∞,   ∑

1

𝑏
1
𝛼(𝑠)

∞
𝑠=1   = ∑

1

𝑠
∞
𝑠=1   = ∞, 

and equation (4) becomes, 

𝛥(𝜑2𝛥𝑧(𝜑))−
(2𝜑+1)(𝜑+1)

(𝜑+2)
𝑧(𝜑+ 1) = 0.                                                         (25) 

It has a nonoscillatory solution 𝑧(𝜑) = 𝜑 + 1. Moreover, (8) becomes 

∑
1

𝜑2(𝜑+1)(𝜑+2)

𝜑−1
𝑠=𝜑3

 ∑ 8𝑡(𝑡 + 2)∞
𝑡=𝑠   = ∞, 

and from Corollary 2.7, (14) becomes 

𝑙𝑖𝑚 𝑠𝑢𝑝𝜑→∞   [(𝜑 + 1)
2(𝜑 + 2)(𝜑 + 3)∑ (𝜑 + 1)2(𝜑 + 2)(𝜑 + 3)

𝜑−1
𝑠=𝜑1

  + ∑ (
1

(𝑡+1)(𝑡+3)
) +

𝜑−1
𝑡=𝜑+1

(𝜑+1)3(𝜑+3)

𝜑+2
  ∑ 𝑡 + 2∞
𝑡=𝜑 ]    > 1. 

So every solution of (24) are oscillatory. 

4 Application 

 Samuelson is the originator of the business cycle model [12]. This model is built on a set of 

underlying assumptions. 

Yearly income 𝑦(𝜑) is equal to the sum of Capital Investment 𝐺(𝜑), Consumption 𝐶(𝜑) and Private 

Investment 𝐼(𝜑) 

𝑦(𝜑) = 𝐺(𝜑) + 𝐶(𝜑) + 𝐼(𝜑).                                                           (26) 

Capital Investment 𝐺(𝜑) remains constant. 

𝐺(𝜑) = 𝐺.                                                                                    (27) 



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Consumption 𝐶(𝜑) depends on the previous year income and on marginal tendency to consume, it is 

denoted as 𝛼. 

𝐶(𝜑) = 𝛼𝑦(𝜑 − 1),                                                                    (28) 

where the multiplier parameter 0 < 𝛼 < 1. 

Private Investment I(r) depends on consumption changes and on the accelerator factor 𝛽, where  

𝛽 > 0  

𝐼(𝜑) = 𝛽(𝐶(𝜑) − 𝐶(𝜑 − 1)) = 𝛼𝛽(𝑦(𝜑 − 1) − 𝑦(𝜑 − 2)).                         (29) 

Substitute (27), (28) and (29), the yearly income 𝑦(𝜑) can be determined as a second-order 

difference equation 

𝑦(𝜑) = 𝐺 + 𝛼𝑦(𝜑 − 1) + 𝛼𝛽𝑦(𝜑 − 1) − 𝛼𝛽𝑦(𝜑 − 2).                                     (30) 

The model developed by Paul Samuelson is a cornerstone in the field of economics. It 

delineates five possible paths or trends that economic activities might follow. These trajectories are 

shaped by the interplay of two fundamental parameters: the marginal propensity to consume (α) and 

the accelerator coefficient (β). 

The marginal propensity to consume (α) quantifies the proportion of additional income 

consumers spend rather than save. Conversely, the accelerator coefficient (β) is a metric that gauges 

the sensitivity of investment expenditures to shifts in GDP. 

Each unique pairing of α and β yields a specific pattern of economic activity, which can range 

from stable growth to cyclical oscillations. This spectrum of outcomes underscores the intricate 

relationship between consumption, investment, and aggregate economic activity. 

Samuelson’s model offers a holistic blueprint for deciphering the mechanics of economic 

cycles. It equips economists with the tools to scrutinize and forecast the impact of alterations in 

consumer behavior and investment on the course of the economy. This model is particularly 

instrumental in policy formulation, as it aids in devising strategies to navigate economic cycles 

effectively. 

In essence, Samuelson’s model is a potent instrument furnishing invaluable insights into 

economic cycle dynamics. It is indispensable in economic prognostication and policy design, 

contributing significantly to our understanding of economic phenomena. This model’s versatility and 

predictive power make it an essential tool in the economist’s toolkit. Its ability to capture the nuanced 

interdependencies within an economy underscores its enduring relevance in economic analysis and 

policy planning. 

Table 1: Five Different Path Cycle 

Case Values Behavior of the Cycles 

1 𝛼 = 0.5, 𝛽 = 0 Cycle less path 

2 𝛼 = 0.5, 𝛽 = 1 Damped Fluctuations 

3 𝛼 = 0.5, 𝛽 = 2 Fluctuation of constant Amplitude 

4 𝛼 = 0.5, 𝛽 = 3 Explosive Cycles 

5 𝛼 = 0.5, 𝛽 = 4 Cycle less Explosive Path 



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Samuelson’s model, a cornerstone in economic theory, illustrates the behavior of economic 

cycles based on varying values of the marginal propensity to consume (α) and the accelerator 

coefficient (β). 

1. When α=0.5 and β=0, the economy follows a cycle-less path, indicating stability without 

fluctuations. This suggests a balanced economy where changes in income do not significantly 

affect consumption or investment patterns. 

2. With α=0.5 and β=1, the economy experiences damped fluctuations, suggesting cycles gradually 

diminishing over time. This indicates an economy that, while initially reactive to changes in 

income, eventually stabilizes. 

3. For α=0.5 and β=2, the economy undergoes fluctuations of constant amplitude, indicating 

regular cyclical patterns. This represents an economy with consistent cycles, reflecting a direct, 

proportional relationship between income changes and consumption or investment. 

4. When α=0.5 and β=3, the economy exhibits explosive cycles, suggesting rapid and potentially 

unstable growth. This scenario might occur when an increase in income significantly boosts 

consumption and investment, leading to accelerated economic growth. 

5. Lastly, with α=0.5 and β=4, the economy follows a cycle-less explosive path, indicating steady, 

rapid growth without cyclical fluctuations. This could represent an economy where income 

changes lead to increased consumption and investment but without the cyclical patterns seen in 

other scenarios. 

These patterns provide valuable insights into how changes in consumption and investment 

behaviors, driven by shifts in income, can impact the cyclical behavior of an economy. Understanding 

these dynamics is crucial for economists and policymakers to effectively manage economic stability 

and growth. Thus, Samuelson’s model is a powerful tool in economic analysis and policy planning. 

Based on these assumptions, the nonoscillatory second-order difference equation (25) is 

𝑦(𝜑) = 𝐺 + (1 +
(2𝜑−3)(𝜑−1)

𝜑
) 𝑦(𝜑 − 1) +

(𝜑−2)2

(𝜑−1)2
𝑦(𝜑 − 1) −

(𝜑−2)2

(𝜑−1)2
𝑦(𝜑 − 2).             (31) 

It is a corresponding equation of (30). Here 𝑦(𝜑) denotes the current year income of the business 

and 

𝛼 = (1 +
(2𝜑−3)(𝜑−1)

𝜑
) =  𝑇ℎ𝑒𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟 , 

and 

𝛽 = (
𝜑(𝜑−2)2

[𝜑+(2𝜑−3)(𝜑−1)](𝜑−1)2
) =  𝑇ℎ𝑒𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑜𝑟 , 

Illustration: Let's consider an illustration to understand this equation better. Suppose a steel company 

owner invests an autonomous investment of Rs. 2 crore. The previous two years' income is unknown, 

and we refer to the Path Cycle table to determine the values of autonomous investment, current 

consumption, induced investment, and yearly income for the next five years and from the Path cycle 

table we take 𝛼 = 0.5, 𝛽 = 1 



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Table 2: Business Income 

Time 

Period (𝜑) 

Autonomous 

Investment 𝐺(𝜑) 

Current 

Consumption 𝐶(𝜑) 

Induced Investment 

𝐼(𝜑) 

Yearly Income 

𝑌(𝜑) 

1 2,00,00,000 0 0 2,00,00,000 

2 2,00,00,000 1,00,00,000 1,00,00,000 4,00,00,000 

3 2,00,00,000 2,00,00,000 1,00,00,000 5,00,00,000 

4 2,00,00,000 2,50,00,000 50,00,000 5,00,00,000 

5 2,00,00,000 2,50,00,000 0 4,50,00,000 

The data represents a business’s income over five years. The autonomous investment, G(φ), 

remains constant at 2,00,00,000. In the first year, no consumption or induced investment leads to a 

yearly income, Y(φ), of 2,00,00,000. In the second year, consumption, C(φ), increases to 1,00,00,000 

and induced investment, I(φ), to 1,00,00,000, doubling the yearly income to 4,00,00,000. In the third 

year, consumption doubles, and the yearly income increases to 5,00,00,000. In the fourth year, 

consumption increases further, but induced investment decreases, keeping the yearly income steady. 

In the fifth year, induced investment drops to zero, reducing the yearly income to 4,50,00,000. This 

suggests a need for strategies to maintain or increase induced investment. 

The nonoscillatory second-order difference equation (25) can be written as a corresponding 

equation. Here, the notation " φ " denotes the current year's income of a business, " φ -1" represents 

the income of the previous year, and " φ -2" represents the income of the year before that. 

As per the table, in the first year, there is no autonomous investment or consumption, and 

hence, the yearly income is Rs. 2 crore. In the second year, the autonomous investment is still Rs. 2 

crore, but there is a consumption of Rs. 3 crore, resulting in an induced investment of Rs. 1.5 crore. 

This brings the yearly income to Rs. 4.5 crore. Similarly, in the third year, with a consumption of Rs. 

4 crore, the induced investment is Rs. 4 crore, leading to a yearly income of Rs. 10 crore. 

By using the second-order difference equation, we can determine the yearly income for the 

remaining two years based on the values of consumption and induced investment. This equation helps 

to provide a better understanding of the dynamics of economic cycles and how changes in consumption 

and investment can impact them. 

 
Figure 1 : 



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This graph indicates the five-year income of the business. it will be changed based on the 

Consumption. 

5 Conclusion 

The article shows that Property A solution is closely related to Samuelson’s business cycle 

model, which is a prominent economic theory of the fluctuations in output and employment. The 

authors illustrate the importance of their main results by giving some examples and comparing them 

with existing literature. They also suggest some possible directions for future research on this topic. 

The research paper discussed the oscillatory behavior and Property A condition for a positive middle 

term. This equation has applications in economics, physics, and mathematical biology. The article 

establishes new oscillatory behavior and Property A conditions for the equation, which can be 

considered an improvement to previous results. The authors use a generalized Riccati transformation 

technique and some auxiliary second-order difference equations. Additionally, the article derives a 

solution for the equation that possesses Property A and Oscillatory. 

The article also highlights the connection between their Property A solution and Samuelson’s 

business cycle model. This model is a well-known economic theory that explains the fluctuations in 

output and employment. By drawing on this connection, the authors illustrate the importance of their 

main results by giving some examples and comparing them with existing literature. 

The article also suggests possible directions for future research on this topic, highlighting the 

potential for further exploration and applications of the results in various fields. The application of 

Samuelson's business cycle model in a second-order difference equation (2) is presented, and the yearly 

income is calculated based on the model. This demonstrates the practical relevance of the research and 

its potential implications for real-world scenarios. 

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