407 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 

 
Oscillation Criteria for Solutions of Neutral Differential Equations of the 

Second-Order Emden-Fowler Type with Forcing Term 

 

Jihan Saad1*   and Hussain Ali Mohamad2   
1,2Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq.  

*Corresponding Author.  
 

 Received:22 May 2023                        Accepted:14 August 2023                   Published:20 January 2025  

 doi.org/10.30526/38.1.3515 

 
Abstract   

     In this paper, the oscillation property and the asymptotic behavior of solutions of neutral second-

order differential Equations of the Emden-Fowler type were studied under the influence of the 

coefficients of forces. It has been shown through this research that the coefficients of forces in 

addition to the Emden-Fowler type have a major role on the oscillation of solutions of neutral 

Equations. As well as its effect on the convergence and divergence of nonoscillatory solutions. For 

this purpose, some conditions are obtained to ensure that all solutions of the neutral Equations 

Emden-Fowler type oscillating or nonoscillating go to ∞, as t → ∞. Some of these conditions are 

the development of conditions similar to them in some of the well-known results included in the 

references, for example, condition (8) in this research with condition (4) in (9). The obtained results 

included some illustrative examples showing that the resulting conditions are easy to apply and 

guarantee oscillation. 

Keywords: Oscillation Criteria, Asymptotic Behavior, Emden-Fowler Type, Neutral Second Order 

with Forcing Term.   

 

1. Introduction 

      This paper aims to obtain sufficient conditions to ensure that every solution of the neutral force 

Equation of the second-order type Emden Fowler oscillates. Consider the Equation: 

(𝝃(𝒕)(𝝎′(𝒕))
𝜸

)
′

+ ∑ 𝒒𝒊(𝒕)𝒙𝜸(𝜹𝒊(𝒕))
𝒏

𝒊=𝟏
𝐬𝐠𝐧(𝒙)

= ∑ 𝒓𝒋(𝒕)
𝒌

𝒋=𝟏
.                                                                                                                   (𝟏) 

𝜔(𝑡) = 𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡)).                                                                                                                         (2) 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://doi.org/10.30526/38.1.3501
https://orcid.org/0009-0000-2264-7057
mailto:jihan.saad2103m@csw.uobaghdad.edu.iq
https://orcid.org/0000-0002-4684-786x
mailto:hussainam_math@csw.uobaghdad.edu.iq


IHJPAS. 2024, 38 (1) 
 

408 
 

 The number  𝛾  is a quotient of odd positive integers. 𝜏, 𝛿𝑖 ∈ ∁( [𝑡0, ∞)𝕋 , 𝑅), 𝑖 = 1,2, … , 𝑛,
lim
t→∞

𝜏(𝑡) = ∞ , lim
t→∞

𝛿𝑖(𝑡) = ∞ ,   sgn(𝑥) = ±1  if   𝑥 ≷ 0, 𝑠𝑔𝑛(0) = 0,  𝑝 ∈ ∁([𝑡0, ∞) , 𝑅+)  and 

𝑞𝑖, 𝑟𝑗 ∈ ∁( [𝑡0, ∞) , 𝑅), 𝑖 = 1,2, … , 𝑛, 𝑗 = 1,2, … , 𝑘. 

During this research, the following assumptions will be used as needed:            

  (M1) lim sup
𝑡→∞

∫ (
1

𝜉(𝑠)
)

1
𝛾

𝑑𝑠 = ∞;
𝑡

𝑡0

  

  (M2)  0 < 𝑝(𝑡) ≤ 𝑎,   𝜉(𝑡) > 0  

             𝑄𝑖(𝑡) = min
𝑡≥𝑡0

{ 𝑞𝑖(𝑡),   𝑞𝑖(𝜏(𝑡)), 𝑄(𝑡) = min{𝑄𝑖(𝑡), 𝑖 = 1,2, … , 𝑛} ,   and 

             𝐺𝑖(𝑡) = max
𝑡≥𝑡0

{ 𝑟𝑖(𝑡), 𝑟𝑖(𝜏(𝑡)),   𝑖 = 1,2, … , 𝑘} , 𝐺(𝑡) = max{𝐺𝑖(𝑡),  𝑖 = 1,2, … , 𝑘}.  

  (M3)  ∫ |𝐺(𝑡)|𝑑𝑡
∞

𝑇

< ∞,   𝑇 ≥ 𝑡0. 

  (M4) ∫ |𝑄(𝑡)|𝑑𝑡
∞

𝑇

= ∞, 𝑇 ≥ 𝑡0 . 

  The Emden-Fowler Equation has emerged in recent decades as a focus of interest for many 

researchers, specifically research in oscillation and the asymptotic behavior of the solutions of these 

Equations, Emden-Fowler Equation has been classified as unconventional Equations and its 

importance has emerged for its use in many applications, and for this reason many researches have 

appeared that produce a lot of conditions to ensure that each solution of these Equations oscillates, 

or that their non-oscillating solutions are convergent. Ahmed et al. (1) studied the second-order 

neutral dynamic linear Equation and established some conditions for the oscillation of every solution 

of this Equation. Yingzhu et al. (2) obtained oscillation conditions of each solution of second-order 

neutral nonlinear differential Equations. The obtained results in (3) are based on comparison 

theorems which enable to address the problem of second order Equation oscillation to first order 

Equation oscillation. Mehta et al. (4) investigated the Emden-Fowler Equation of the 

form 
𝑑

𝑑𝑡
(𝑡𝛼 𝑑𝑤

𝑑𝑡
) = 𝑡𝜎𝑤𝛼, where 𝑤(𝑡) = 𝑥(𝑡) ± 𝑞(𝑡)𝑥(𝜏(𝑡)), and established some conditions for 

all solutions to oscillate. Mohamad et al. (5, 6) discussed the oscillation property of third order neutral 

half-linear Equations and established some conditions to insure the oscillation of every solution of 

these Equations. Moaaz et al. (7) and Xu et al. (8) obtained oscillation conditions of each solution of 

second order neutral Emden-Fowler type  (𝑎(𝑡)[(𝑥(𝑡) − 𝑝(𝑡)𝑥(𝜏(𝑡)))′]𝛾)′ + 𝑞(𝑡)𝑥𝛾(𝜎(𝑡)) = 0,

t ≥ 𝑡0. Thandapani et al. (9) studied asymptotic properties of the third order quasi-linear neutral 

functional differential Equation (𝑎(𝑡)[(𝑥(𝑡) − 𝑝(𝑡)𝑥(𝜏(𝑡)))′′]𝛾)′ + 𝑞(𝑡)𝑥𝛾(𝜎(𝑡)) = 0, by using the 

Riccati transformation, and establishing some conditions which ensure that every solution of that 

Equation is either oscillatory or converges to zero. Hassan et al. (10) he dealt with new standards for 

the oscillation of half-linear differential Equations developed of the second order, where he concluded 

that the results obtained work to expand and develop modern standards for the same Equations that 

have been developed by many authors. Tripathy et al.  (11) find the necessary and sufficient 

conditions for volatility one of the impulsive neutral differential system solutions of the second order 

under certain conditions that ensure the occurrence of oscillation     . See Mehta et al. (12), and Vidhyaa 

et al. (13)  they studied the differential Equations and obtained the oscillation criteria for all the 



IHJPAS. 2024, 38 (1) 
 

409 
 

solutions of the neutral differential Equations of the second degree, half-linear 

(𝑘(𝑡)((ℎ(𝑡)𝑧′ (𝑡))′)𝜀)′ +  𝑘(𝑡)𝑥𝜀(𝑡)  =  0, 𝑡 ≥ 𝑡0,  where 𝑧(𝑡) =  𝑥(𝑡) +  𝑝(𝑡)𝑥(𝜏(𝑡)).  Dassios et 

al. (14) he studied the delayed and neutral differential Equations, where he focused on the stability 

of the important joins because these joins contain delays in each of the state variables and their time 

derivatives, where the proposed approach consists of model transformation that builds an equivalent 

set of algebraic differential Equations. Li et al. (15) and Marappan et al. (16) the oscillatory behavior 

of solutions of mixed nonlinear neutral differential Equations of the Emden-Fowler type was studied 

by applying the integral conditions and the integral average method. Baty (17) he studied second-

order Lane-Emden-Fowler differential Equations, third-order Emden-Fowler Equations, and fourth-

order Lane-Emden-Fowler Equations. He presented numerical methods using neural networks based 

on physics with the aim of solving higher-order differential Equations. Naeif et al. (18) he studied 

the oscillation and asymptotic behavior of a half-linear three-dimensional neutral system of second 

order and gave sufficient conditions to ensure oscillation or not.  See (19-22) they studied the optimal 

decomposition method for solving third-order nonlinear Emden-Fowler differential Equations, to 

avoid the singularity at x=0, by transforming the Emden-Fowler Equation into an integral Volterra 

Equation. 

Our paper was based on article (9), where a more general Equation with a forcing term is used 

and condition (2) in (9) has been developed into a more general case. A solution 𝒙(𝒕) is said to be 

oscillatory if it has arbitrarily large zeros on (𝒕𝟎, ∞), otherwise it is said to be nonoscillatory that is 

either eventually positive or eventually negative (6). 

  

2. Main Results 

In this section some results .established for oscillation for every solution of Equation (1).  In the 

beginning, it is shown that every non-oscillatory solution is achieves the following cases. 

Lemma 2.1: Assume that 𝑞𝑖(𝑡) ≥ 0, ∑ 𝑟𝑗(𝑡)𝑘
𝑗=1 ≤ 0, and (M1) holds. Let  𝑥(𝑡)  be a non- oscillatory 

solution of Equation (1). Then  𝜔′(𝑡) > 0,  and either lim
𝑡→∞

𝑥(𝑡) = ∞, or lim
𝑡→∞

  𝜉(𝑡)(𝜔′(𝑡))
𝛾

= 0.   

Proof: Assume that 𝑥(𝑡) be eventually positive solution of Equation (1). From (1) it follows that 
(𝜉(𝑡)(𝜔′(𝑡))𝛾)′ ≤ 0, that is  𝜉(𝑡)(𝜔′(𝑡))𝛾  is non-increasing for 𝑡 ≥ 𝑡0.  we claim that 

 𝜉(𝑡)(𝜔′(𝑡))𝛾 is eventually positive, otherwise if  𝜉(𝑡)(𝜔′(𝑡))𝛾 is eventually negative then there is 

 𝜇 < 0 and  𝑡1 ≥ 𝑡0 such that  𝜉(𝑡)(𝜔′(𝑡))𝛾 ≤ 𝜇 < 0, 𝑡 ≥ 𝑡1 , so it follows 

𝜔′(𝑡) ≤ (
𝜇

𝜉(𝑡)
)

1
𝛾

,   𝑡 ≥ 𝑡1 .                                                                                                                         (3) 

Integrating (3) from 𝑡1 to 𝑡 we get 

𝜔(𝑡) − 𝜔(𝑡1) ≤ 𝜇
1
𝛾 ∫ (

1

𝜉(𝑠)
)

1
𝛾𝑡

𝑡1

𝑑𝑠.                                                                                                         (4) 

Letting  𝑡 → ∞,  then from inequality (4) yields lim
𝑡→∞

𝜔(𝑡) = −∞, a contradiction. Hence our claim 

verified and   𝜉(𝑡)(𝜔′(𝑡))𝛾 > 0, that is  𝜔′(𝑡) > 0, 𝑡 ≥ 𝑡1 ≥ 𝑡0, then lim
𝑡→∞

  𝜉(𝑡)(𝜔′(𝑡))𝛾 = 𝑙 ≥ 0,  

thus  𝜉(𝑡)(𝜔′(𝑡))𝛾 ≥ 𝑙, 𝑡 ≥ 𝑡1 or 

𝜔′(𝑡) ≥ (
𝑙

𝜉(𝑡)
)

1
𝛾

,    𝑡 ≥ 𝑡1 .                                                                                                                        (5) 



IHJPAS. 2024, 38 (1) 
 

410 
 

Integrating (5) from 𝑡1 to 𝑡, it follows  𝜔(𝑡) − 𝜔(𝑡1) ≥ 𝑙
1

𝛾 ∫ (
1

𝜉(𝑠)
)

1

𝛾𝑡

𝑡1
𝑑𝑠.  

If  𝑙 > 0  then  lim
𝑡→∞

𝜔(𝑡) = ∞,  implies that  lim
𝑡→∞

𝑥(𝑡) = ∞. If  𝑙 = 0  then  lim
𝑡→∞

  𝜉(𝑡)(𝜔′(𝑡))𝛾 = 0.      

Lemma 2.2: Assume that  𝑞𝑖(𝑡) ≤ 0, ∑ 𝑟𝑗(𝑡)𝑘
𝑗=1 ≥ 0,  and (M1)  holds. Let 𝑥(𝑡) be a non-

oscillatory solution of Equation (1). Then the following statements hold: 

(a) 𝜔′(𝑡) > 0, and  lim
𝑡→∞

𝑥(𝑡) = ∞.  

(b) 𝜔′(𝑡) < 0, and  lim
𝑡→∞

𝜉(𝑡)(𝜔′(𝑡))𝛾 = 0. 

Proof: Assume that 𝑥(𝑡) be eventually positive solution of Equation (1). From Equation (1) we get 

(𝜉(𝑡)(𝜔′(𝑡))𝛾)′ ≥ 0 that is 𝜉(𝑡)(𝜔′(𝑡))𝛾 is non-decreasing, we have two cases to consider:  

1.   𝜉(𝑡)(𝜔′(𝑡))𝛾 > 0;   

2.    𝜉(𝑡)(𝜔′(𝑡))𝛾 < 0, 𝑡 ≥ 𝑡1 ≥ 𝑡0.    

Case 1:  𝜉(𝑡)(𝜔′(𝑡))𝛾 > 0  that is  𝜔′(𝑡) > 0, 𝑡 ≥ 𝑡1  then there exist  𝜇 > 0  such that  

𝜉(𝑡)(𝜔′(𝑡))𝛾 ≥ 𝜇, 𝑡 ≥ 𝑡2 ≥ 𝑡1 

𝜔′(𝑡) ≥ (
𝜇

𝜉(𝑡)
)

1
𝛾

,   𝑡 ≥ 𝑡2.                                                                                                                          (6) 

By integrating (6) from 𝑡2 to 𝑡 we get   𝜔(𝑡) − 𝜔(𝑡2) ≥ 𝜇
1

𝛾  ∫ (
1

𝜉(𝑠)
)

1

𝛾𝑡

𝑡2
𝑑𝑠.  

 As 𝑡 → ∞ it follows  lim
𝑡→∞

𝜔(𝑡) = ∞, which implies that lim
𝑡→∞

𝑥(𝑡) = ∞.  

Case 2:  𝜉(𝑡)(𝜔′(𝑡))𝛾 < 0  that is  𝜔′(𝑡) < 0, 𝑡 ≥ 𝑡1 and lim
𝑡→∞

 𝜉(𝑡)(𝜔′(𝑡))𝛾  = 𝑙 ≤ 0. we claim that 

 𝑙 = 0 otherwise 𝑙 < 0 thus 

𝜔′(𝑡) ≤ (
𝑙

𝜉(𝑡)
)

1
𝛾

,   𝑡 ≥ 𝑡2 ≥ 𝑡1 .                                                                                                                 (7)  

By integrating Equation (7) from 𝑡2 to t we get  𝜔(𝑡) − 𝜔(𝑡2) ≤ 𝑙
1

𝛾 ∫ (
1

𝜉(𝑠)
)

1

𝛾𝑡

𝑡2
𝑑𝑠.  

As  𝑡 → ∞ it follows that lim
𝑡→∞

𝜔(𝑡) = −∞,  a contradiction. Then  𝑙 = 0.  

 

Theorem 2.1. Assume that 𝑞𝑖(𝑡) ≥ 0, ∑ 𝑟𝑗(𝑡)𝑘
𝑗=1 ≤ 0,  (𝑀1) − (𝑀4) hold, and for any continuous 

functions 𝑢(𝑡), 𝑣(𝑡), 𝑢𝑣 > 0,  there exists  𝜆 > 0, such that 

𝑢𝛾(𝑡) + 𝑣𝛾(𝑡) ≥ 𝜆(𝑢(𝑡) + 𝑣(𝑡))
𝛾

.                                                                                                            (8) 

Then every solution of Equation (1) oscillates. 

Proof. Assume that Equation (1) has a non-oscillatory solution 𝑥(𝑡). For lack of prolongation and 

repetition, it can be assumed that  𝑥(𝑡) > 0, 𝑥(𝜏(𝑡)) > 0, 𝑥(𝛿𝑖(𝑡)) > 0, 𝑖 = 1,2, … , 𝑛,  for   𝑡 ≥ 𝑡0 . 

Let 𝑢(𝑡) = 𝑥(𝑡), 𝑣(𝑡) = 𝑝(𝑡) 𝑥(𝜏(𝑡)), 𝜂(𝑡) = max {𝑥(𝑡), 𝑥(𝜏(𝑡))}, then 

(𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡)))𝛾 ≤ (𝑥(𝑡) + 𝑎 𝑥(𝜏(𝑡)))𝛾 ≤ (𝜂(𝑡) + 𝑎 𝜂(𝑡))𝛾 = 𝜂𝛾(𝑡)(1 + 𝑎)𝛾. 

Since  𝜂(𝑡) = max {𝑥(𝑡), 𝑥(𝜏(𝑡))}  so there exists  휀 > 0, such that 

𝑥𝛾(𝑡) + 𝑝𝛾(𝑡)𝑥𝛾(𝜏(𝑡)) ≥ 휀𝜂𝛾(𝑡) =
𝜀(1+𝑎)𝛾

(1+𝑎)𝛾 𝜂𝛾(𝑡) =
𝜀

(1+𝑎)𝛾 (𝜂(𝑡) + 𝑎 𝜂(𝑡))𝛾 ≥   
𝜀

(1+𝑎)𝛾 (𝑥(𝑡) +

𝑝(𝑡)𝑥(𝜏(𝑡)))𝛾, 

Choose    𝜆 =
𝜀

(1+𝒶)𝛾 > 0. Hence, 



IHJPAS. 2024, 38 (1) 
 

411 
 

𝑥𝛾(𝑡) + 𝑝𝛾(𝑡)𝑥𝛾(𝜏(𝑡)) ≥ 𝜆(𝑥(𝑡) + 𝑝(𝑡)𝑥(𝜏(𝑡)))𝛾

= 𝜆𝜔𝛾(𝑡).                                                                                                                               (9) 

From Equation (1) it follows that: 

(𝜉(𝑡)(𝜔′(𝑡))𝛾)′ + ∑ 𝑞𝑖(𝑡)(𝑥(𝛿𝑖(𝑡))
𝛾𝑛

𝑖=1
+ 𝑎𝛾[𝜉(𝜏(𝑡))(𝜔′(𝜏(𝑡)))𝛾]′

+ 𝑎𝛾 ∑ 𝑞𝑖(𝜏(𝑡))(𝑥 (𝛿𝑖(𝜏(𝑡)))
𝛾𝑛

𝑖=1
 = ∑ (𝑟𝑗(𝑡)

𝑘

𝑗=1
 + 𝑎𝛾𝑟𝑗(𝜏(𝑡))                        (10) 

(𝜉(𝑡)(𝜔′(𝑡))𝛾)′ + 𝑎𝛾[𝜉(𝜏(𝑡))(𝜔′(𝜏(𝑡)))𝛾]′ + 𝑄(𝑡) ∑ [𝑥𝛾(𝛿𝑖(𝑡)
𝑛

𝑖=1
 + 𝑎𝛾𝑥𝛾(𝛿𝑖(𝜏(𝑡))]

− ∑ (𝑟𝑗(𝑡)
𝑘

𝑗=1
 + 𝑎𝛾𝑟𝑗(𝜏(𝑡)) ≤ 0 

(𝜉(𝑡)(𝜔′(𝑡))𝛾)′ + 𝑎𝛾[𝜉(𝜏(𝑡))(𝜔′(𝜏(𝑡)))𝛾]′  

+𝑄(𝑡) ∑ [𝑥𝛾(𝛿𝑖(𝑡)) + 𝑝𝛾(𝛿𝑖(𝑡))𝑥𝛾(𝛿𝑖(𝜏(𝑡))] 
𝑛

𝑖=1
− 𝐺(𝑡) ∑ (1 +

𝑘

𝑗=1
 𝑎𝛾) ≤ 0 

by using(9) the last inequality yields: 

(𝜉(𝑡)(𝜔′(𝑡))𝛾)′ + 𝑎𝛾[𝜉(𝜏(𝑡))(𝜔′(𝜏(𝑡)))𝛾]′ + 𝜆𝑄(𝑡) ∑ 𝜔𝛾(𝛿𝑖(𝑡))
𝑛

𝑖=1
− 𝑘𝐺(𝑡)(1 + 𝑎𝛾) ≤ 0 

Let 𝛿(𝑡) = min
𝑡≥𝑡1

{𝛿𝑖(𝑡), 𝑖 = 1,2 … , 𝑛 }, by lemma 2.1, 𝜔(𝑡)  is positive and increasing so there exist a 

constant  𝑏 > 0, and 𝑡2 ≥ 𝑡1  such that  𝜔(𝑡) ≥ 𝑏, 𝑡 ≥ 𝑡2. Hence the last inequality leads to:  

(𝜉(𝑡)(𝜔′(𝑡))𝛾)′ + 𝑎𝛾[𝜉(𝜏(𝑡))(𝜔′(𝜏(𝑡)))𝛾]′ + 𝑛𝜆𝑄(𝑡)𝜔𝛾(𝛿(𝑡)) − 𝑘𝐺(𝑡)(1 + 𝑎𝛾) ≤

0,                                                                                                                                                                         (11)  

𝑛𝜆𝑄(𝑡)𝑏𝛾 ≤ −(𝜉(𝑡)(𝜔′(𝑡))𝛾)′ − 𝑎𝛾[𝜉(𝜏(𝑡))(𝜔′(𝜏(𝑡)))𝛾
]

′

+ 𝑘𝐺(𝑡)(1 + 𝑎𝛾),                                                                                                                (12) 

Consequently, by integrating (12) from 𝑡2 to  𝑡  yields  

𝑛𝜆𝑏𝛾 ∫ 𝑄(𝑠)
𝑡

𝑡2

𝑑𝑠

≤ − ∫ (𝜉(𝑠)(𝜔′(𝑠))𝛾)′
𝑡

𝑡2

𝑑𝑠 − 𝑎𝛾 ∫ [𝜉(𝜏(𝑠))(𝜔′(𝑠))𝛾]′
𝑡

𝑡2

𝑑𝑠 + 𝑘(1 + 𝑎𝛾) ∫ 𝐺(𝑠)𝑑𝑠
𝑡

𝑡2

 

𝑛𝜆𝑏𝛾 ∫ 𝑄(𝑠)
𝑡

𝑡2

𝑑𝑠

≤ 𝜉(𝑡2)(𝜔′(𝑡2))𝛾 + 𝑎𝛾𝜉(𝜏(𝑡2))[𝜔′(𝜏(𝑡2))]𝛾
+ 𝑘(1 + 𝑎𝛾) ∫ 𝐺(𝑠)

𝑡

𝑡2

𝑑𝑠,                   (13) 

 Hence by (M3) it follows from (13)  ∫ 𝑄(𝑠)
∞

𝑡2
𝑑𝑠 < ∞,  contradicts (𝑀4).       

Theorem 2.2. Assume that 𝑞𝑖(𝑡) ≤ 0, 𝑖 = 1,2, … , 𝑛, ∑ 𝑟𝑖(𝑡)𝑘
𝑗=1 ≥ 0, 𝜉′(𝑡) > 0 on  [𝑡0, ∞) . Let 

(M1) − (M4) hold, and for any continuous functions𝑢(𝑡), 𝑣(𝑡), 𝑢𝑣 > 0, there exists 𝜆 > 0, such that 

(8) holds, in addition to the condition 

lim sup
𝑡→∞

∫ [
1

𝜉(𝑠)
∫ ∑ |𝑞𝑖(𝑣)|[1 − 𝑝(𝛿𝑖(𝑣))]𝛾

𝑛

𝑖=1

𝛼(𝑠)

𝑠

𝑑𝑣]

1
𝛾

𝑑𝑠
𝛼(𝑡)

𝑡

> 1.                                                                                                                                        (14) 

Then every solution of Equation (1) oscillates.  



IHJPAS. 2024, 38 (1) 
 

412 
 

Proof. Assume that Equation (1) has eventually positive solution  𝑥(t), that is  𝑥(𝑡) > 0, 𝑥(𝜏(𝑡)) >

0, 𝑥(𝛿𝑖(𝑡)) > 0, 𝑖 = 1,2, … , 𝑛. From Equation (1) we get (𝜉(𝑡)(𝜔′(𝑡))𝛾)′ ≥ 0, based on Lemma 

2.2, there are two cases that need to be investigated: 

(a) 𝜔′(𝑡) > 0, and  lim
𝑡→∞

𝑥(𝑡) = ∞.  

(b) 𝜔′(𝑡) < 0, and  lim
𝑡→∞

𝜉(𝑡)(𝜔′(𝑡))𝛾 = 0. 

 Case (a) proceeding as in the proof of theorem 2.1, we conclude that (11) holds.  

Letting   

𝑧(𝑡)
= 𝜉(𝑡)(𝜔′(𝑡))𝛾.                                                                                                                                             (15) 

Then 𝑧(𝑡) is positive and non-decreasing, hence (11) becomes: 

𝑧′(𝑡)+𝑎𝛾[𝑧(𝜏(𝑡))]
′

+ 𝑛𝜆𝑄(𝑡)𝜔𝛾(𝛿(𝑡)) < 𝑘𝐺(𝑡)(1 + 𝑎𝛾), for  𝑡

≥ 𝑡2                                                                                                                                         (16) 

𝜔(𝑡)  is positive and increasing so there exist a constant  𝑏 > 0, and 𝑡3 ≥ 𝑡2  such that  𝜔(𝑡) ≥ 𝑏,
𝑡 ≥ 𝑡3. Therefor (16) reduce to 

𝑧′(𝑡)+𝑎𝛾[𝑧(𝜏(𝑡))]
′

+ 𝑛𝜆𝑄(𝑡)𝑏𝛾 < 𝑘𝐺(𝑡)(1 + 𝑎𝛾), for  𝑡

≥ 𝑡3                                                                                                                                         (17) 

Integration (17) from 𝑡3 to 𝑡, where 𝑡 is sufficiently large 𝑡3, leads to 

∫ 𝑧′(𝑠)𝑑𝑠
𝑡

𝑡3

+𝑎𝛾 ∫ (𝑧(𝜏(𝑠)))′𝑑𝑠
𝑡

𝑡3

+ 𝑛𝜆𝑏𝛾 ∫ 𝑄(𝑠)𝑑𝑠
𝑡

𝑡3

< 𝑘(1 + 𝑎𝛾) ∫ 𝐺(𝑠)𝑑𝑠
𝑡

𝑡3

 

Since 𝑧(𝑡) is non-decreasing, then the last inequality becomes: 

𝑧(𝑡) − 𝑧(𝑡3) + 𝑎𝛾𝑧(𝜏(𝑡)) − 𝑎𝛾𝑧(𝜏(𝑡3)) + 𝑛𝜆𝑏𝛾 ∫ 𝑄(𝑠)
𝑡

𝑡3

𝑑𝑠 < 𝑘 (1 + 𝑎𝛾) ∫ 𝐺(𝑠)𝑑𝑠
𝑡

𝑡3

, 

There fore 

−𝑧(𝑡3)𝛾 − 𝑎𝛾𝑧(𝑡3) + 𝑛𝜆𝑏𝛾 ∫ 𝑄(𝑠)
𝑡

𝑡3

𝑑𝑠 < 𝑘 (1 + 𝑎𝛾) ∫ 𝐺(𝑠)𝑑𝑠,
𝑡

𝑡3

                                                (18) 

As  𝑡 → ∞  a contradiction will be got in (18). 

Case (b) in this case 

 𝜔(𝑡) > 0, 𝜔′(𝑡) < 0, lim
𝑡→∞

𝜉(𝑡)(𝜔′(𝑡))𝛾 = 0, (𝜉(𝑡)(𝜔′(𝑡))
𝛾

)′ ≥ 0 

Since  𝜉′(𝑡) > 0  and  𝜔(𝑡) is positive decreasing, so it can be conclude that  𝜔′′(𝑡) ≥ 0, for  𝑡 ≥ 𝑡2,  

 𝜔(𝑡) > 𝑥(𝑡),  𝑥(𝑡) = 𝜔(𝑡) − 𝑝(𝑡)𝑥(𝜏(𝑡)), 𝑥(𝛿𝑖(𝑡)) = 𝜔(𝛿𝑖(𝑡)) − 𝑝(𝛿𝑖(𝑡))𝑥(𝜏(𝛿𝑖(𝑡))) 

So Equation (1) become 

(𝜉(𝑡)(𝜔′(𝑡))
𝛾

)
′

+ ∑ 𝑞𝑖(𝑡)[𝜔(𝛿𝑖(𝑡)) − 𝑝(𝛿𝑖(𝑡))𝑥(𝜏(𝛿𝑖(𝑡)))]𝛾
𝑛

𝑖=1

= ∑ 𝑟𝑗(𝑡)
𝑘

𝑗=1
.                                                                                                                     (19) 

By integrating (19) from 𝑡 to 𝛼(𝑡), where 𝛼(𝑡) > 𝑡 and 𝜏 (𝛿𝑗 (𝛼(𝛼(𝑡)))) < 𝑡,  

𝛿𝑗(𝑡) = min{𝛿𝑖(𝑡) , 𝑖 = 1,2, … , 𝑛}, We get −𝜉(𝑡)(𝜔′(𝑡))
𝛾

≥ − ∫ ∑ 𝑞𝑖(𝑠)[𝜔(𝛿𝑖(𝑡)) −𝑛
𝑖=1

𝛼(𝑡)

𝑡

𝑝(𝛿𝑖(𝑡))𝜔(𝜏(𝛿𝑖(𝑡)))]𝛾 𝑑𝑠, 



IHJPAS. 2024, 38 (1) 
 

413 
 

−𝜉(𝑡)(𝜔′(𝑡))
𝛾

≥ − ∫ ∑ 𝑞𝑖(𝑠)𝜔𝛾(𝜏(𝛿𝑖(𝑡)))[1 − 𝑝(𝛿𝑖(𝑡))]𝛾
𝑛

𝑖=1

𝛼(𝑡)

𝑡

𝑑𝑠, 

𝜔′(𝑡) ≤ 𝜔 (𝜏 (𝛿𝑗(𝛼(𝑡)))) [
1

𝜉(𝑡)
∫ ∑ 𝑞𝑖(𝑠)[1 − 𝑝(𝛿𝑖(𝑡))]𝛾

𝑛

𝑖=1

𝛼(𝑡)

𝑡

𝑑𝑠]

1
𝛾

.                                  (20) 

Where 𝜏(𝛿𝑗(𝑡)) = min{𝜏(𝛿𝑖(𝑡)) , 𝑖 = 1,2, … , 𝑛],  integrating (20) from 𝑡 to 𝛼(𝑡) we get 

𝜔(𝛼(𝑡)) − 𝜔(𝑡) ≤ 𝜔(𝜏(𝛿𝑗(𝛼(𝛼(𝑡))))) ∫ [
1

𝜉(𝑠)
∫ ∑ 𝑞𝑖(𝑣)[1 − 𝑝(𝛿𝑖(𝑣))]𝛾

𝑛

𝑖=1

𝛼(𝑠)

𝑠

𝑑𝑣]

1
𝛾

𝑑𝑠
𝛼(𝑡)

𝑡

, 

1 ≥
𝜔(𝑡)

𝜔 (𝜏 (𝛿𝑗 (𝛼(𝛼(𝑡)))))

≥ − ∫ [
1

𝜉(𝑠)
∫ ∑ 𝑞𝑖(𝑣)[1 − 𝑝(𝛿𝑖(𝑣))]𝛾

𝑛

𝑖=1

𝛼(𝑠)

𝑠

𝑑𝑣]

1
𝛾

𝑑𝑠
𝛼(𝑡)

𝑡

. 

The last inequality contradicts the condition (14), thus case not valid also, hence every solution of 

Equation (1) oscillates. The proof is complete. 

 

 3.  Examples 

      In this section, two examples are given to illustrate the fulfillment of all necessary and sufficient 

conditions for the results presented in the previous section. 

 

Example 3.1. Consider the following Emden-Fowler Equation: 

[𝑥(𝑡) +
1

2
𝑥(𝑡 − 𝜋 )]

′′

= −
1

2
𝑥(𝑡 − 2𝜋) −

1

4
,   𝑡

≥ 0.                                                                                                                                         (21) 

Where 𝜉(𝑡) = 1 , 𝑝(𝑡) =
1

2
, 𝜏(𝑡) = 𝑡 − 𝜋 , 𝑞1(𝑡) = 𝑄(𝑡) =

1

2
, 𝑟1(𝑡) = −

1

2
,  and   𝛿(𝑡) = 𝑡 − 2𝜋,

𝑖 = 1,2, … , 𝑛, 𝛾 = 1. In reality  M1 − M3 are hold for every 𝑡 ≥ 𝑡0 = 0. And 

∫ 𝑄(𝑠)𝑑𝑠
∞

𝑡0

= ∫ 𝑑𝑠
∞

0

= ∞. 

Then (M4) is holds for every  𝑡 ≥
1

2
. Recall that (8) hold for  λ = 1. Hence all the conditions of 

Theorem 2.1 satisfy that according to Theorem 2.1, each solution of Equation (1) oscillates for 

example 𝑥(𝑡) = sin 𝑡 −
1

2
  such as this oscillation solution. See Figure (1) 

 
 

Figure 1.   𝑥(𝑡) = sin 𝑡 −
1

2
   

 



IHJPAS. 2024, 38 (1) 
 

414 
 

Example 3. 2.  Consider the following Emden-Fowler Equation: 

[𝑥(𝑡) + 2𝑒−𝜋𝑥(𝑡 − 𝜋 )]′′ − 4𝑒−
3𝜋
2 𝑥 (𝑡 −

3𝜋

2
) = 𝑒−𝑡,   𝑡

≥ 0.                                                                                                                                         (22) 

Where  𝜉(𝑡) = 1 , 𝑝(𝑡) = 2𝑒−𝜋, 𝜏(𝑡) = 𝑡 − 𝜋 , 𝑟(𝑡) = 𝑒−𝑡, 𝑞1(𝑡) = 𝑄(𝑡) = −4𝑒−
3𝜋

2 ,  and  𝛿(𝑡) =

𝑡 −
3𝜋

2
, 𝛾 = 1. In reality  M1 − M3 are hold for every 𝑡 ≥ 𝑡0 = 0.  And 

∫ |𝑄(𝑠)|𝑑𝑠
∞

𝑡0

= ∫ 4𝑒−
3𝜋
2 𝑑𝑠

∞

0

= ∞. 

Then (M4) is holds for every 𝑡 ≥ 0. Recall that (8) hold for  λ = 1. Hence all the conditions of 

Theorem 2.2 satisfy that according to Theorem 2.2, each solution of Equation (1) oscillates, for 

example  𝑥(𝑡) = 𝑒−𝑡(sin 𝑡 − 1)   is such an oscillatory solution. See Figure (2) 

 
 

Figure 𝟐. 𝜑(𝑡) = 𝑒−𝑡(sin 𝑡 − 1) 

 

4.  Conclusion    

       In this paper, we have studied the oscillation property of the solutions of neutral second-order 

differential Equations of the Emden-Fowler type. Some of the extracted conditions are the 

development of conditions known in the references, which ensure that either each solution of this 

Equation oscillates, or each nonoscillatory solution convergence to zero or tends to infinity as 𝑡 →
∞. Some examples are presented to clarify the results obtained. 

 

Acknowledgments  

I would like to extend my thinks to my teacher and supervisor, Dr. Hussain Ali Mohamad, for 

providing guidance and feedback throughout this project.  

 

Conflict of Interest 

     The authors declare that there are no competing interests regarding the publication of this paper. 

 

Funding 

     This work is not supported by any the Foundation. 

 

Ethical Clearance 

     Ethics of scientific research were carried out in accordance with international conditions. 



IHJPAS. 2024, 38 (1) 
 

415 
 

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