Oscillatory Behavior Of Solutions Research Articles

Neutral Emden–Fowler Differential Equation of Second Order: Oscillation Criteria of Coles Type

In this work, we study the asymptotic and oscillatory behavior of solutions to the second-order general neutral Emden–Fowler differential equation (avηxvz′v)′ + qvFxgv = 0, where v≥v0 and the corresponding function z = x + px∘h. Besides the importance of equations of the neutral type, studying the qualitative behavior of solutions to these equations is rich in analytical points and interesting issues. We begin by finding the monotonic features of positive solutions. The new properties contribute to obtaining new and improved relationships between x and z for use in studying oscillatory behavior. We present new conditions that exclude the existence of positive solutions to the examined equation, and then we establish oscillation criteria through the symmetry property between non-oscillatory solutions. We use the generalized Riccati substitution method, which enables us to apply the results to a larger area than the special cases of the considered equation. The new results essentially improve and extend previous results in the literature. We support this claim by applying the results to an example and comparing them with previous findings. Moreover, the reduction of our results to Euler’s differential equation introduces the well-known sharp oscillation criterion.

Oscillatory behavior of solutions of second-order non-linear differential equations with mixed non-linear neutral terms

This study primarily seeks to expand upon these developments by encompassing neutral differential equations of mixed type, incorporating both delay and advanced terms, particularly in the case of the canonical operator. The presented results are derived from the application of the comparison method, Riccati transformation, and integral averaging technique. These methodologies lead to substantial improvements and extensions of existing results found in the literature. Additionally, illustrative examples are provided to demonstrate the practical implications of the developed criteria.

On the Oscillatory Behavior of Solutions of Second-Order Non-Linear Differential Equations with Damping Term

In this paper, we discuss the oscillatory behavior of solutions of two general classes of nonlinear second-order differential equations. New criteria are obtained using Riccati transformations and the integral averaging techniques. The obtained results improve and generalize some recent criteria in the literature. Moreover, a traditional condition is relaxed. Three examples are given to justify the results.

Oscillation criteria for third-order semi-canonical differential equations with unbounded neutral coefficients

In this paper, we investigate the oscillatory behavior of solutions to a class of third-order differential equations of the form Lz(t) + f (t)yβ (σ(t)) = 0, where Lz(t) = (p(t)(q(t)zt(t))t)t is a semi-canonical operator and z(t) = y(t) + g(t)y(τ (t)). The main idea is to convert the semi-canonical operator into canonical form and then obtain some new sufficient conditions for the oscillation of all solutions. The obtained results essentially improve and complement to the known results. Examples are provided to illustrate the main results. Mathematics Subject Classification (2010): 34C10, 34K11, 34K40. Received 19 September 2021; Accepted 20 January 2022

On the oscillatory behavior of solutions of canonical and noncanonical even-order neutral differential equations with distributed deviating arguments

On the oscillatory behavior of solutions of canonical and noncanonical even-order neutral differential equations with distributed deviating arguments

New Conditions for Testing the Oscillation of Solutions of Second-Order Nonlinear Differential Equations with Damped Term

This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the studied equation without additional restrictions. Our approach improves the standard integral averaging technique to obtain simpler oscillation theorems for new classes of nonlinear differential equations. Two examples are presented to illustrate the importance of our findings.

Oscillatory behavior of solutions of third order semi-canonical dynamic equations on time scale

<p>This paper investigates the oscillatory behavior of nonlinear third-order dynamic equations on time scales. Our main approach is to transform the equation from its semi-canonical form into a more tractable canonical form. This transformation simplifies the analysis of oscillation behavior and allows us to derive new oscillation criteria. These criteria guarantee that all solutions to the equation oscillate. Our results extend and improve upon existing findings in the literature, particularly for the special cases where $ \mathbb{T} = \mathbb{R} $ and $ \mathbb{T} = \mathbb{Z} $. Additionally, we provide illustrative examples to demonstrate the practical application of the developed criteria.</p>

Oscillatory behavior of a fifth differential equation with unbounded neutral coefficients

"The authors study the oscillatory behavior of solutions to a class of fifth-order differential equations with unbounded neutral coefficients. The results are obtained by a comparison with first order delay differential equations whose oscillatory characters are known. Two examples illustrating the results are provided, one of which is applied to Euler type equations. Keywords: Oscillation, fifth-order, neutral differential equation."

On the monotonic properties and oscillatory behavior of solutions of neutral differential equations

Abstract In this work, we study new asymptotic properties of positive solutions of the even-order neutral differential equation with the noncanonical operator. The new properties are iterative, which means they can be used several times. We also use these properties to obtain new criteria for oscillation of the studied equation.

More Effective Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of a Class of Third-Order Functional Differential Equations

This paper delves into the investigation of quasi-linear neutral differential equations in the third-order canonical case. In this study, we refine the relationship between the solution and its corresponding function, leading to improved preliminary results. These enhanced results play a crucial role in excluding the existence of positive solutions to the investigated equation. By building upon the improved preliminary results, we introduce novel criteria that shed light on the nature of these solutions. These criteria help to distinguish whether the solutions exhibit oscillatory behavior or tend toward zero. Moreover, we present oscillation criteria for all solutions. To demonstrate the relevance of our results, we present an illustrative example. This example validates the theoretical framework we have developed and offers practical insights into the behavior of solutions for quasi-linear third-order neutral differential equations.

Oscillatory behavior of second-order nonlinear noncanonical neutral differential equations

Abstract This paper discusses the oscillatory behavior of solutions to a class of second-order nonlinear noncanonical neutral differential equations. Sufficient conditions for all solutions to be oscillatory are given. Examples are provided to illustrate all the main results obtained.

Oscillatory Properties Test for Even-Order Di§erential Equations of Neutral type

This paper presents a study on the oscillatory behavior of solutions to even-order neutral differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first-order delay differential equations, Riccati technique and integral averages technique. The results of this work contribute to a deeper understanding of even-order differential equations and their connections to various branches of mathematics and practical sciences. The findings emphasize the importance of continued research in this area.

More Effective Conditions for Testing the Oscillatory Behavior of Solutions to a Class of Fourth-Order Functional Differential Equations

This paper presents an investigation into the qualitative behavior of solutions for a specific class of fourth-order half-linear neutral differential equations. The main objective of this study is to improve the relationship between the solution and its corresponding function. By developing improved relationships, a novel criterion is proposed to determine the oscillatory behavior of the studied equation. The exclusion of positive solutions is achieved through a comparative approach in which the examined equation is compared to second-order equations. Additionally, the significance of the obtained results is demonstrated by applying them to various illustrative examples.

Oscillatory Behavior of Solutions of Third-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Oscillatory Behavior of Solutions of Third-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

New Oscillation Theorems for Second-Order Superlinear Neutral Differential Equations with Variable Damping Terms

In this paper, we investigate the oscillatory behavior of solutions of second-order differential equations with a superlinear neutral term and a damping term in noncanonical forms. We do not place any additional conditions on the damping term. Therefore, our results apply to new classes of equations that were not covered by the previous criteria. According to symmetry between the negative and positive solutions of the studied equation, we present some new criteria that ensure the oscillation of all solutions by using the Riccati transformation and comparison method. We provide an example to illustrate our main results.

Oscillatory Behaviour of Solutions of Delay Differential Equations of Fractional Order

Oscillatory Behaviour of Solutions of Delay Differential Equations of Fractional Order

On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations

ABSTRACT The authors examine the oscillatory behavior of solutions to a class of third order half-linear delay differential equations. The results are obtained by a comparison with first-order linear delay differential equations whose oscillatory characters are known. An example is provided to illustrate the results.

High-Order Nonlinear Functional Differential Equations: New Monotonic Properties and Their Applications

We provide streamlined criteria for evaluating the oscillatory behavior of solutions to a class of higher-order functional differential equations in the non-canonical case. We use a comparison approach with first-order equations that have standard oscillation criteria. Normally, in the non-canonical situation, the oscillation test requires three independent conditions, but we provide criteria with two-conditions without checking the additional conditions. Lastly, we give examples to highlight the significance of the findings.

Oscillatory Properties of Fourth-Order Advanced Differential Equations

This paper presents a study on the oscillatory behavior of solutions to fourth-order advanced differential equations involving p-Laplacian-like operator. We obtain oscillation criteria using techniques from first and second-order delay differential equations. The results of this work contribute to a deeper understanding of fourth-order differential equations and their connections to various branches of mathematics and practical sciences. The findings emphasize the importance of continued research in this area.

New Conditions for Testing the Asymptotic and Oscillatory Behavior of Solutions of Neutral Differential Equations of the Fourth Order

In this work, in the noncanonical case, we find new properties for a class of positive solutions of fourth-order differential equations. These properties allow us to obtain iterative criteria that exclude positive decreasing solutions, and we then establish sufficient conditions to guarantee that all solutions to the examined equation oscillate. The importance of applying the results to a special case of the investigated equation is demonstrated.