scholarly journals Qualitative Behavior of Solutions of Second Order Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 777 ◽  
Author(s):  
Clemente Cesarano ◽  
Omar Bazighifan
Keyword(s):  
Differential Equations ◽  
Second Order ◽  
Oscillation Criterion ◽  
Qualitative Behavior ◽  
Second Order Differential Equations ◽  
Future Developments ◽  
Correct Direction ◽  
Riccati Substitution ◽  
Delay Differential ◽  
The Right

In this work, we study the oscillation of second-order delay differential equations, by employing a refinement of the generalized Riccati substitution. We establish a new oscillation criterion. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. We illustrate the results with some examples.

scholarly journals New Conditions for the Oscillation of Second-Order Differential Equations with Sublinear Neutral Terms

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1159
Author(s):  
Shyam Sundar Santra ◽  
Omar Bazighifan ◽  
Mihai Postolache
Keyword(s):  
Fluid Dynamics ◽  
Neural Networks ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Neutral Differential Equations ◽  
Second Order Differential Equations ◽  
Delay Differential

In continuous applications in electrodynamics, neural networks, quantum mechanics, electromagnetism, and the field of time symmetric, fluid dynamics, neutral differential equations appear when modeling many problems and phenomena. Therefore, it is interesting to study the qualitative behavior of solutions of such equations. In this study, we obtained some new sufficient conditions for oscillations to the solutions of a second-order delay differential equations with sub-linear neutral terms. The results obtained improve and complement the relevant results in the literature. Finally, we show an example to validate the main results, and an open problem is included.

Almost-conservative second-order differential equations

1975 ◽  
Vol 77 (1) ◽  
pp. 159-169 ◽  
Author(s):  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Differential Equations ◽  
Rate Of Convergence ◽  
Second Order ◽  
Good Function ◽  
Second Order Differential Equations ◽  
Right Hand ◽  
Image Position ◽  
The Right

During the last thirty years an immense amount of research has been done on differential equations of the formwhere ε > 0 is small. It is usually assumed that the perturbing term on the right-hand side is a ‘good’ function of its arguments and that its dependence on t is purely trigonometric; this means that there is an expansion of the formwhere the ωn are constants, and that one may impose any conditions on the rate of convergence of the series which turn out to be convenient. Without loss of generality we can assumeand for convenience we shall sometimes write ω0 = 0. Often f is assumed to be periodic in t, in which case it is implicit that the period is independent of x and ẋ (We can also allow f to depend on ε, provided it does so in a sensible manner.)

scholarly journals Asymptotic and Oscillatory Behavior of Solutions of a Class of Higher Order Differential Equation

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1434 ◽  
Author(s):  
Elmetwally M. Elabbasy ◽  
Clemente Cesarano ◽  
Omar Bazighifan ◽  
Osama Moaaz
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Higher Order ◽  
Oscillatory Behavior ◽  
Future Developments ◽  
Correct Direction ◽  
Delay Differential ◽  
The Right

The objective of this paper is to study asymptotic behavior of a class of higher-order delay differential equations with a p-Laplacian like operator. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and show us the correct direction for future developments. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and comparison principles. This new theorem complements and improves a number of results reported in the literature. Some examples are provided to illustrate the main results.

scholarly journals More Effective Conditions for Oscillatory Properties of Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 278 ◽  
Author(s):  
Taher A. Nofal ◽  
Omar Bazighifan ◽  
Khaled Mohamed Khedher ◽  
Mihai Postolache
Keyword(s):  
Differential Equations ◽  
Middle Term ◽  
Riccati Substitution ◽  
Delay Differential ◽  
The Right ◽  
Nonlinear Delay Differential Equation ◽  
Effective Conditions ◽  
Oscillation Of Solutions ◽  
Nonlinear Delay ◽  
Comparison Technique

In this work, we present several oscillation criteria for higher-order nonlinear delay differential equation with middle term. Our approach is based on the use of Riccati substitution, the integral averaging technique and the comparison technique. The symmetry contributes to deciding the right way to study oscillation of solutions of this equations. Our results unify and improve some known results for differential equations with middle term. Some illustrative examples are provided.

scholarly journals New Oscillation Results for Second-Order Neutral Differential Equations with Deviating Arguments

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1937
Author(s):  
Yakun Wang ◽  
Fanwei Meng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Second Order ◽  
Deviating Arguments ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
First Order ◽  
Delay Differential ◽  
The Right

In this paper, we focus on the second-order neutral differential equations with deviating arguments which are under the canonical condition. New oscillation criteria are established, which are based on a first-order delay differential equation and generalized Riccati transformations. The idea of symmetry is a useful tool, not only guiding us in the right way to study this function but also simplifies our proof. Our results are generalizations of some previous results and we provide an example to illustrate the main results.

scholarly journals Behavior of Non-Oscillatory Solutions of Fourth-Order Neutral Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 477 ◽  
Author(s):  
Osama Moaaz ◽  
Rami Ahmad El-Nabulsi ◽  
Omar Bazighifan
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fourth Order ◽  
Riccati Transformation ◽  
Oscillatory Solutions ◽  
Neutral Differential Equations ◽  
Future Developments ◽  
Generalized Riccati Transformation ◽  
Correct Direction ◽  
The Right

In this paper, we deal with the asymptotics and oscillation of the solutions of fourth-order neutral differential equations of the form r t z ‴ t α ′ + q t x α g t = 0 , where z t : = x t + p t x δ t . By using a generalized Riccati transformation, we study asymptotic behavior and derive some new oscillation criteria. Our results extend and improve some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. An example is given to illustrate the importance of our results.

scholarly journals On the qualitative behavior of the solutions to second-order neutral delay differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shyam Sundar Santra ◽  
Hammad Alotaibi ◽  
Omar Bazighifan
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Qualitative Behavior ◽  
Multiple Delays ◽  
Neutral Delay ◽  
Open Problems ◽  
Neutral Delay Differential Equations ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractDifferential equations of second order appear in numerous applications such as fluid dynamics, electromagnetism, quantum mechanics, neural networks and the field of time symmetric electrodynamics. The aim of this work is to establish necessary and sufficient conditions for the oscillation of the solutions to a second-order neutral differential equation. First, we have taken a single delay and later the results are generalized for multiple delays. Some examples are given and open problems are presented.

scholarly journals Oscillatory Behavior of Fourth-Order Differential Equations with Neutral Delay

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 371 ◽  
Author(s):  
Osama Moaaz ◽  
Rami Ahmad El-Nabulsi ◽  
Omar Bazighifan
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Neutral Delay ◽  
Neutral Differential Equations ◽  
Future Developments ◽  
Correct Direction ◽  
The Right

In this paper, new sufficient conditions for oscillation of fourth-order neutral differential equations are established. One objective of our paper is to further improve and complement some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. An example is given to illustrate the importance of our results.

scholarly journals New Oscillation Theorems for Second-Order Differential Equations with Canonical and Non-Canonical Operator via Riccati Transformation

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1111
Author(s):  
Shyam Sundar Santra ◽  
Abhay Kumar Sethi ◽  
Osama Moaaz ◽  
Khaled Mohamed Khedher ◽  
Shao-Wen Yao
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Riccati Transformation ◽  
Neutral Delay ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
Neutral Delay Differential Equations ◽  
Delay Differential ◽  
Oscillation Theorems

In this work, we prove some new oscillation theorems for second-order neutral delay differential equations of the form (a(ξ)((v(ξ)+b(ξ)v(ϑ(ξ)))′))′+c(ξ)G1(v(κ(ξ)))+d(ξ)G2(v(ς(ξ)))=0 under canonical and non-canonical operators, that is, ∫ξ0∞dξa(ξ)=∞ and ∫ξ0∞dξa(ξ)<∞. We use the Riccati transformation to prove our main results. Furthermore, some examples are provided to show the effectiveness and feasibility of the main results.

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