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20 pages, 330 KB

Open AccessArticle

Third-Order Neutral Differential Equations with Non-Canonical Forms: Novel Oscillation Theorems

by Barakah Almarri, Belal Batiha, Omar Bazighifan and Fahd Masood

Axioms 2024, 13(11), 755; https://doi.org/10.3390/axioms13110755 - 31 Oct 2024

Cited by 8 | Viewed by 1216

This paper explores the asymptotic and oscillatory properties of a class of third-order neutral differential equations with multiple delays in a non-canonical form. The main objective is to simplify the non-canonical form by converting it to a canonical form, which reduces the complexity [...] Read more.

This paper explores the asymptotic and oscillatory properties of a class of third-order neutral differential equations with multiple delays in a non-canonical form. The main objective is to simplify the non-canonical form by converting it to a canonical form, which reduces the complexity of the possible cases of positive solutions and their derivatives from four cases in the non-canonical form to only two cases in the canonical form, which facilitates the process of inference and development of results. New criteria are provided that exclude the existence of positive solutions or Kneser-type solutions for this class of equations. New criteria that guarantee the oscillatory behavior of all solutions that satisfy the conditions imposed on the studied equation are also derived. This work makes a qualitative contribution to the development of previous studies in the field of neutral differential equations, as it provides new insights into the oscillatory behavior of neutral equations with multiple delays. To confirm the strength and effectiveness of the results, three examples are included that highlight the accuracy of the derived criteria and their practical applicability, which enhances the value of this research and expands the scope of its use in the field. Full article

(This article belongs to the Special Issue Difference, Functional, and Related Equations)

10 pages, 236 KB

Open AccessArticle

Third-Order Noncanonical Neutral Delay Differential Equations: Nonexistence of Kneser Solutions via Myshkis Type Criteria

by Gunasekaran Nithyakala, George E. Chatzarakis, Govindasamy Ayyappan and Ethiraju Thandapani

Mathematics 2024, 12(18), 2847; https://doi.org/10.3390/math12182847 - 13 Sep 2024

Cited by 3 | Viewed by 901

Abstract

The purpose of this paper is to add some new asymptotic and oscillatory results for third-order neutral delay differential equations with noncanonical operators. Without assuming any extra conditions, by using the canonical transform technique, the studied equation is changed to a canonical type [...] Read more.

The purpose of this paper is to add some new asymptotic and oscillatory results for third-order neutral delay differential equations with noncanonical operators. Without assuming any extra conditions, by using the canonical transform technique, the studied equation is changed to a canonical type equation, and this reduces the number of classes of nonoscillatory solutions into two instead of four. Then, we obtain Myshkis type sufficient conditions for the nonexistence of Kneser type solutions for the studied equation. Finally, employing these newly obtained criteria, we provide conditions for the oscillation of all solutions of the studied equation. Examples are presented to illustrate the importance and the significance of the main results. Full article

11 pages, 283 KB

Open AccessArticle

Investigation of the Oscillatory Properties of Solutions of Differential Equations Using Kneser-Type Criteria

by Yousef Alnafisah and Osama Moaaz

Axioms 2023, 12(9), 876; https://doi.org/10.3390/axioms12090876 - 13 Sep 2023

Cited by 1 | Viewed by 1391

Abstract

This study investigates the oscillatory properties of a fourth-order delay functional differential equation. This study’s methodology is built around two key tenets. First, we propose optimized relationships between the solution and its derivatives by making use of some improved monotonic features. By using [...] Read more.

This study investigates the oscillatory properties of a fourth-order delay functional differential equation. This study’s methodology is built around two key tenets. First, we propose optimized relationships between the solution and its derivatives by making use of some improved monotonic features. By using a comparison technique to connect the oscillation of the studied equation with some second-order equations, the second aspect takes advantage of the significant progress made in the study of the oscillation of second-order equations. Numerous applications of functional differential equations of the neutral type served as the inspiration for the study of a subclass of these equations. Full article

(This article belongs to the Special Issue Differential Equations and Asymptotic Analysis: Recent Advances and Applications)

15 pages, 827 KB

Open AccessArticle

New Conditions for Testing the Asymptotic Behavior of Solutions of Odd-Order Neutral Differential Equations with Multiple Delays

by Fahd Masood, Osama Moaaz, Sameh S. Askar and Ahmad Alshamrani

Axioms 2023, 12(7), 658; https://doi.org/10.3390/axioms12070658 - 2 Jul 2023

Cited by 5 | Viewed by 1393

Abstract

The purpose of this research is to investigate the asymptotic and oscillatory characteristics of odd-order neutral differential equation solutions with multiple delays. The relationship between the solution and its derivatives of different orders, as well as their related functions, must be understood in [...] Read more.

The purpose of this research is to investigate the asymptotic and oscillatory characteristics of odd-order neutral differential equation solutions with multiple delays. The relationship between the solution and its derivatives of different orders, as well as their related functions, must be understood in order to determine the oscillation terms of the studied equation. In order to contribute to this subject, we create new and significant relationships and inequalities. We use these relationships to create conditions in which positive and N-Kneser solutions of the considered equation are excluded. To obtain these terms, we employ the comparison method and the Riccati technique. Furthermore, we use the relationships obtained to create new criteria, so expanding the existing literature on the field. Finally, we provide an example from the general case to demonstrate the results’ significance. The findings given in this work provide light on the behavior of odd-order neutral differential equation solutions with multiple delays. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

11 pages, 282 KB

Open AccessArticle

Higher-Order Delay Differential Equation with Distributed Deviating Arguments: Improving Monotonic Properties of Kneser Solutions

by Shaimaa Elsaeed, Osama Moaaz, Ghada AlNemer and Elmetwally M. Elabbasy

Symmetry 2023, 15(2), 502; https://doi.org/10.3390/sym15020502 - 14 Feb 2023

Viewed by 1504

Abstract

This study aims to investigate the oscillatory behavior of the solutions of an even-order delay differential equation with distributed deviating arguments. We first study the monotonic properties of positive decreasing solutions or the so-called Kneser solutions. Then, by iterative deduction, we improve these [...] Read more.

This study aims to investigate the oscillatory behavior of the solutions of an even-order delay differential equation with distributed deviating arguments. We first study the monotonic properties of positive decreasing solutions or the so-called Kneser solutions. Then, by iterative deduction, we improve these properties, which enables us to apply them more than once. Finally, depending on the symmetry between the positive and negative solutions of the studied equation and by combining the new condition for the exclusion of Kneser solutions with some well-known results in the literature, we establish a new standard for the oscillation of the investigated equation. Full article

(This article belongs to the Special Issue The Qualitative Theory of Functional Differential Equations and their Applications)

14 pages, 321 KB

Open AccessArticle

Oscillation Results of Third-Order Differential Equations with Symmetrical Distributed Arguments

by Belgees Qaraad, Omar Bazighifan, Ali Hasan Ali, Areej A. Al-Moneef, Awatif Jahman Alqarni and Kamsing Nonlaopon

Symmetry 2022, 14(10), 2038; https://doi.org/10.3390/sym14102038 - 29 Sep 2022

Cited by 11 | Viewed by 1877

Abstract

This paper is concerned with the oscillation and asymptotic behavior of certain third-order nonlinear delay differential equations with distributed deviating arguments. By establishing sufficient conditions for the nonexistence of Kneser solutions and existing oscillation results for the studied equation, we obtain new criteria [...] Read more.

This paper is concerned with the oscillation and asymptotic behavior of certain third-order nonlinear delay differential equations with distributed deviating arguments. By establishing sufficient conditions for the nonexistence of Kneser solutions and existing oscillation results for the studied equation, we obtain new criteria which ensure that every solution oscillates by using the theory of comparison with first-order delay equations and the technique of Riccati transformation. Some examples are presented to illustrate the importance of main results. Full article

(This article belongs to the Special Issue Theories and Applications for Dynamical Systems, Symmetry Problems and Differential Equations)

11 pages, 289 KB

Open AccessArticle

Criteria for the Nonexistence of Kneser Solutions of DDEs and Their Applications in Oscillation Theory

by Osama Moaaz, Ioannis Dassios, Haifa Bin Jebreen and Ali Muhib

Appl. Sci. 2021, 11(1), 425; https://doi.org/10.3390/app11010425 - 4 Jan 2021

Cited by 9 | Viewed by 2335

Abstract

The objective of this study was to improve existing oscillation criteria for delay differential equations (DDEs) of the fourth order by establishing new criteria for the nonexistence of so-called Kneser solutions. The new criteria are characterized by taking into account the effect of [...] Read more.

The objective of this study was to improve existing oscillation criteria for delay differential equations (DDEs) of the fourth order by establishing new criteria for the nonexistence of so-called Kneser solutions. The new criteria are characterized by taking into account the effect of delay argument. All previous relevant results have neglected the effect of the delay argument, so our results substantially improve the well-known results reported in the literature. The effectiveness of our new criteria is illustrated via an example. Full article

(This article belongs to the Special Issue Methods in Dynamical Systems, Mathematics of Networks, and Optimization for Modelling in Engineering)

16 pages, 274 KB

Open AccessArticle

Oscillation Theory for Non-Linear Neutral Delay Differential Equations of Third Order

by Osama Moaaz, Ioannis Dassios, Waad Muhsin and Ali Muhib

Appl. Sci. 2020, 10(14), 4855; https://doi.org/10.3390/app10144855 - 15 Jul 2020

Cited by 27 | Viewed by 3010

Abstract

In this article, we study a class of non-linear neutral delay differential equations of third order. We first prove criteria for non-existence of non-Kneser solutions, and criteria for non-existence of Kneser solutions. We then use these results to provide criteria for the under [...] Read more.

In this article, we study a class of non-linear neutral delay differential equations of third order. We first prove criteria for non-existence of non-Kneser solutions, and criteria for non-existence of Kneser solutions. We then use these results to provide criteria for the under study differential equations to ensure that all its solutions are oscillatory. An example is given that illustrates our theory. Full article

15 pages, 271 KB

Open AccessArticle

Establishing New Criteria for Oscillation of Odd-Order Nonlinear Differential Equations

by Osama Moaaz, Jan Awrejcewicz and Ali Muhib

Mathematics 2020, 8(6), 937; https://doi.org/10.3390/math8060937 - 8 Jun 2020

Cited by 23 | Viewed by 2070

Abstract

By establishing new conditions for the non-existence of so-called Kneser solutions, we can generate sufficient conditions to ensure that all solutions of odd-order equations are oscillatory. Our results improve and expand the previous results in the literature. Full article

(This article belongs to the Special Issue Differential/Difference Equations: Mathematical Modeling, Oscillation and Applications)

12 pages, 267 KB

Open AccessArticle

New Results for Kneser Solutions of Third-Order Nonlinear Neutral Differential Equations

by Osama Moaaz, Belgees Qaraad, Rami Ahmad El-Nabulsi and Omar Bazighifan

Mathematics 2020, 8(5), 686; https://doi.org/10.3390/math8050686 - 1 May 2020

Cited by 18 | Viewed by 2127

Abstract

In this paper, we consider a certain class of third-order nonlinear delay differential equations

{(r {(w^{″})}^{α})}^{'} (v) + q (v) x^{β} (ς (v)) = 0,

for

v \geq v_{0},

where [...] Read more.

In this paper, we consider a certain class of third-order nonlinear delay differential equations

{(r {(w^{″})}^{α})}^{'} (v) + q (v) x^{β} (ς (v)) = 0,

for

v \geq v_{0},

where

w (v) = x (v) + p (v) x (ϑ (v))

. We obtain new criteria for oscillation of all solutions of this nonlinear equation. Our results complement and improve some previous results in the literature. An example is considered to illustrate our main results. Full article

(This article belongs to the Special Issue Differential/Difference Equations: Mathematical Modeling, Oscillation and Applications)

11 pages, 749 KB

Open AccessArticle

New Aspects for Non-Existence of Kneser Solutions of Neutral Differential Equations with Odd-Order

by Osama Moaaz, Dumitru Baleanu and Ali Muhib

Mathematics 2020, 8(4), 494; https://doi.org/10.3390/math8040494 - 2 Apr 2020

Cited by 29 | Viewed by 2263

Abstract

Some new oscillatory and asymptotic properties of solutions of neutral differential equations with odd-order are established. Through the new results, we give sufficient conditions for the oscillation of all solutions of the studied equations, and this is an improvement of the relevant results. [...] Read more.

Some new oscillatory and asymptotic properties of solutions of neutral differential equations with odd-order are established. Through the new results, we give sufficient conditions for the oscillation of all solutions of the studied equations, and this is an improvement of the relevant results. The efficiency of the obtained criteria is illustrated via example. Full article

(This article belongs to the Special Issue Differential/Difference Equations: Mathematical Modeling, Oscillation and Applications)

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