Third-order Neutral Differential Equations Research Articles

Existence and bounds for Kneser-type solutions to noncanonical third-order neutral differential equations

This article focuses on the existence and asymptotic behavior of Kneser-type solutions to third-order noncanonical differential equations with a delay or advanced argument in the neutral term $$ \Big(r_2(t)\big(r_1(t)z'(t)\big)'\Big)'+g(t)x(t)=0, $$ where \( z(t)=x(t)+p(t)x(\tau(t))\). This equation is transformed into a canonical equation, which reduces the number of classes of positive solutions from 4 to 2. This is done without assuming extra conditions, and greatly simplifies the process of obtaining conditions for the existence of Kneser-type solutions. Also we obtain lower and upper bounds for these solutions, and obtain their rate of convergence to zero. Two examples are provided to illustrate our main results, one with a delay neutral term, and one with an advanced neutral term. For more information see https://ejde.math.txstate.edu/Volumes/2024/55/abstr.html

Investigating Oscillatory Behavior in Third-Order Neutral Differential Equations with Canonical Operators

In this study, we aim to set new criteria regarding the asymptotic behavior of the neutral differential equation of the third order. These criteria are designed to ensure that this equation is oscillatory using comparisons with first-order differential equations and Riccati substitution. The results we obtained improve some of the results found in the literature. Some examples are provided to illustrate the applicability of our results and compare them with results found in some previous studies.

Uniform stability, boundedness and square integrability for non-autonomous third-order neutral differential equations with delay

Uniform stability, boundedness and square integrability for non-autonomous third-order neutral differential equations with delay

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.

Positive periodic solutions of third-order neutral differential equations with delayed derivative terms

By applying fixed point index theory on cones, the existence results of positive 2π-periodic solutions are obtained for the third-order neutral differential equation with delayed derivative terms in nonlinearity(x(t)−cx(t−δ))‴+a(t)x(t)=f(t,x(t),x(t−τ0),x′(t−τ1),x″(t−τ2)),t∈R under the condition that the nonlinear term f satisfies superlinear or sublinear growth, where constants δ>0,|c|<1, a:R→(0,+∞) and f:R×[0,+∞)2×R2→[0,+∞) are continuous functions which are 2π periodic in t and constants τ0,τ1,τ2>0.

On the Asymptotic Behavior of Class of Third-Order Neutral Differential Equations with Symmetrical and Advanced Argument

In this paper, we aimed to study some asymptotic properties of a class of third-order neutral differential equations with advanced argument in canonical form. We provide new and simplified oscillation criteria that improve and complement a number of existing results. We also show some examples to illustrate the importance of our results.

Existence and uniqueness criterion of a periodic solution for a third-order neutral differential equation with multiple delay

In this paper, we study the existence and uniqueness of a periodic solution for a third-order neutral delay differential equation (NDDE) by applying Mawhin’s continuation theorem of coincidence degree and analysis techniques. An illustrative example is given as an application to support our results. To confirm the accuracy of our results, we also present a plot of the behavior of the periodic solution.

Oscillation and Asymptotic Behavior of the Third-Order Neutral Differential Equation with Damping and Distributed Deviating Arguments

Oscillation and Asymptotic Behavior of the Third-Order Neutral Differential Equation with Damping and Distributed Deviating Arguments

Third-Order Neutral Differential Equation with a Middle Term and Several Delays: Asymptotic Behavior of Solutions

This study aims to investigate the asymptotic behavior of a class of third-order delay differential equations. Here, we consider an equation with a middle term and several delays. We obtain an iterative relationship between the positive solution of the studied equation and the corresponding function. Using this new relationship, we derive new criteria that ensure that all non-oscillatory solutions converge to zero. The new findings are an extension and expansion of relevant findings in the literature. We apply our results to a special case of the equation under study to clarify the importance of the new criteria.

Third-Order Neutral Differential Equations with Damping and Distributed Delay: New Asymptotic Properties of Solutions

In this paper, we are interested in studying the oscillation of differential equations with a damping term and distributed delay. We establish new criteria that guarantee the oscillation of the third-order differential equation in terms of oscillation of the second-order linear differential equation without a damping term. By using the Riccati transformation technique and the principle of comparison, we obtain new results on the oscillation for the studied equation. The results show significant improvement and extend the previous works. Symmetry contributes to determining the correct methods for solving neutral differential equations. Some examples are provided to show the significance of our results.

Oscillation of third-order neutral differential equations with oscillatory operator

A third-order damped neutral sublinear differential equation for which its differential operator is oscillatory is studied. Sufficient conditions are given under which every solution is either oscillatory or the derivative of its neutral term is oscillatory (or it tends to zero).

ASYMPTOTIC BEHAVIOR OF KNESER’S SOLUTION FOR SEMI-CANONICAL THIRD-ORDER HALF-LINEAR ADVANCED NEUTRAL DIFFERENTIAL EQUATIONS

In this paper, we study the properties of positive solutions of the thirdorder neutral differential equations with advanced argument of the form (p(t)(q(t)(z′(t))α)′)′ + f(t)xα(τ (t)) = 0, where z(t) = x(t) + g(t)(x(σ(t))). First we obtain conditions for the existence of Kneser type solutions and then provide lower and upper estimate that yield the rate of convergence to zero of such solutions. An example is provided to illustrate the importance of the main results.

Oscillation criteria for third-order neutral differential equations with unbounded neutral coefficients and distributed deviating arguments

This paper focuses on the oscillation criteria for the third-order neutral differential equations with unbounded neutral coefficients and distributed deviating arguments. Using comparison principles, new sufficient conditions improve some known existing results substantially due to less constraints on the considered equation. At last, two examples are established to illustrate the given theorems.

NEW OSCILLATION CRITERIA FOR THIRD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH DISTRIBUTED DEVIATING ARGUMENTS

New oscillation criteria for a class of third-order neutral differential equations with distributed deviating arguments are established. Examples illustrating the results are provided and some suggestions for further research are indicated.

Oscillation criteria of third-order neutral differential equations with damping and distributed deviating arguments

Our objective in this paper is to study the oscillatory and asymptotic behavior of the solutions of third-order neutral differential equations with damping and distributed deviating arguments. New oscillation criteria are established, which are based on a refinement generalized Riccati transformation. An important tool for this investigation is the integral averaging technique. Moreover, we provide an example to illustrate the main results.

Symmetry and Its Role in Oscillation of Solutions of Third-Order Differential Equations

Symmetry plays an essential role in determining the correct methods for the oscillatory properties of solutions to differential equations. This paper examines some new oscillation criteria for unbounded solutions of third-order neutral differential equations of the form (r2(ς)((r1(ς)(z′(ς))β1)′)β2)′ + ∑i=1nqi(ς)xβ3(ϕi(ς))=0. New oscillation results are established by using generalized Riccati substitution, an integral average technique in the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems.

Qualitative Behavior of Unbounded Solutions of Neutral Differential Equations of Third-Order

New oscillatory properties for the oscillation of unbounded solutions to a class of third-order neutral differential equations with several deviating arguments are established. Several oscillation results are established by using generalized Riccati transformation and a integral average technique under the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems.

On the Qualitative Behavior of Third-Order Differential Equations with a Neutral Term

In this paper, we analyze the asymptotic behavior of solutions to a class of third-order neutral differential equations. Using different methods, we obtain some new results concerning the oscillation of this type of equation. Our new results complement related contributions to the subject. The symmetry plays a important and fundamental role in the study of oscillation of solutions to these equations. An example is presented in order to clarify the main results.

Oscillatory behavior of solutions of certain third-order neutral differential equation with continuously distributed delay

This paper concerns the oscillatory behavior of the solutions of the third-order neutral differential equation with continuously distributed delay. We prove many new sufficient conditions which ensure that every solution to be either oscillatory or converges to zero asymptotically. Examples dwelling upon the importance of our theorems are also included.

Philos-Type Oscillation Results for Third-Order Differential Equation with Mixed Neutral Terms

The motivation for this paper is to create new Philos-type oscillation criteria that are established for third-order mixed neutral differential equations with distributed deviating arguments. The key idea of our approach is to use the triple of the Riccati transformation techniques and the integral averaging technique. The established criteria improve, simplify and complement results that have been published recently in the literature. An example is also given to demonstrate the applicability of the obtained conditions.