Neutral Differential Equations Research Articles

New Oscillation Conditions Test for Neutral Differential Equations

In this work, we investigate new oscillation conditions of delay differential equations of second-order. We start by finding new relations and inequalities between the corresponding function and its solution, also with its derivatives. The new inequalities allow us to improve the asymptotic and oscillation conditions of the solutions of the this equation. By using an improved method and approach, we obtain new criteria that test the all solutions oscillation. The article provides some examples that highlight the significance of new conditions and properties.

Analysis of Oscillation in Neutral Differential Equations of Even Order with Specific Delay Properties

This research addresses a limitation in the current "Kamenev-Type" criteria used by mathematicians to study the behavior (oscillation) of solutions to a specific kind of equation (even order neutral differential equations). These equations describe scenarios where the rate of change depends on both the current state and a delayed version of it. By tackling this shortcoming, the paper introduces new and enhanced results for understanding oscillation in these equations. This advancement not only refines the "Kamenev-Type" criteria but also surpasses many other established methods for analyzing oscillation in this area of mathematics.

Second-Order Neutral Differential Equations with a Sublinear Neutral Term: Examining the Oscillatory Behavior

This article highlights the oscillatory properties of second-order Emden–Fowler delay differential equations featuring sublinear neutral terms and multiple delays, encompassing both canonical and noncanonical cases. Through the proofs of several theorems, we investigate criteria for the oscillation of all solutions to the equations under study. By employing the Riccati technique in various ways, we derive results that expand the scope of previous research and enhance the cognitive understanding of this mathematical domain. Additionally, we provide three illustrative examples to demonstrate the validity and applicability of our findings.

Oscillation Criteria For Solution Of The Fourth Order Nonlinear Neutral Differential Equations

In this research , we study the properties of oscillation for all neutral differential equations Its importance in solving many engineering, physical and chemical problems and giving better results in a timely manner within a specific period of time some basic necessary and sufficient condition are established for every solution of solutions of the fourth order nonlinear To become oscillatory , Examples are given insupport our results

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

Euler Method for a Class of Linear Impulsive Neutral Differential Equations

This paper presents a new numerical scheme for a class of linear impulsive neutral differential equations with constant coefficients based on the Euler method. We rigorously establish the first-order convergence of the proposed numerical approach. Additionally, the asymptotical stability of the exact solutions and numerical solutions of impulsive neutral differential equations are studied. To substantiate our findings, two illustrative examples are provided, demonstrating the theoretical conclusions of this paper.

Approximate controllability of damped second-order neutral differential equations with state-dependent delay

ABSTRACT This article discusses the approximate controllability of damped second-order neutral differential equations with state-dependent delay. The results are obtained using strongly continuous cosine family theories and fixed-point theories. Assuming approximate controllability of the corresponding equation, we obtain sufficient conditions for approximate controllability of the nonlinear second-order equations. An application are provided to demonstrate how the proposed result can be applied.

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.

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.

Comparison Theorems for Oscillation of Higher-Order Neutral Delay Differential Equations

In this work, we study the oscillatory features of a class of neutral differential equations with multiple delays. We present novel oscillation conditions for this equation by using a comparison method. We create conditions that exclude the positive solution of the examined equation. Using the feature of symmetry between non-oscillatory solutions (positive and negative), these conditions also exclude negative solutions without adding additional restrictions. In our study, we take into account the even and odd cases of the order of the equation. Furthermore, we study the asymptotic properties in three different cases of functional coefficients. Our results are a fundamental extension and generalization of previous relevant findings, and this fact has been demonstrated through comparisons.

Asymptotic behavior of neutral partial differential equation under the light of semi‐Fredholm perturbation

Through this work, we establish the periodicity of solutions for the following neutral differential equation defined as . Applying the theory of perturbation of semi‐Fredholm operators, we purpose an a priori estimate on the bounded operator to derive the periodicity of bounded solutions on without taking of consideration the compactness of the semigroup generated by the part of . Moreover, we introduce an application includes numerical simulations to depict the applicability of theoretical findings.

Group Classification of Second-Order Linear Neutral Differential Equations

In this paper, we shall extend the method of obtaining symmetries of ordinary differential equations to second-order non-homogeneous functional differential equations with variable coefficients. The existing research for delay differential equations defines a Lie-Bäcklund operator and uses the invariant manifold theorem to obtain the infinitesimal generators of the Lie group. However, we shall use a different approach that requires Taylor’s theorem for a function of several variables to obtain a Lie invariance condition and the determining equations for second-order functional differential equations. Certain standard results from the theory of ordinary differential equations have been employed to simplify the equation under study. The symmetry analysis of this equation was found to be non-trivial for arbitrary variable coefficients. In such cases, by selecting certain specific functions, arising in most practical models, we find the symmetries that are seen to be in terms of Bessel’s functions, Mathieu functions, etc. We then make a complete group classification of the second-order linear neutral differential equation, for which there is no existing literature.

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

Classifications of Solutions of Second Order Nonlinear Neutral Delay Difference Equations With Positive and Negative Coefficients

Classifications of Solutions of Second Order Nonlinear Neutral Delay Difference Equations With Positive and Negative Coefficients

Existence of Periodic Solutions for Totally Nonlinear Neutral Differential Equations With Variable Delay

Existence of Periodic Solutions for Totally Nonlinear Neutral Differential Equations With Variable Delay

Kamenev-Type Criteria for Testing the Asymptotic Behavior of Solutions of Third-Order Quasi-Linear Neutral Differential Equations

This paper aims to study the asymptotic properties of nonoscillatory solutions (eventually positive or negative) of a class of third-order canonical neutral differential equations. We use Riccati substitution to reduce the order of the considered equation, and then we use the Philos function class to obtain new criteria of the Kamenev type, which guarantees that all nonoscillatory solutions converge to zero. This approach is characterized by the possibility of applying its conditions to a wider area of equations. This is not the only aspect that distinguishes our results; we also use improved relationships between the solution and the corresponding function, which in turn is reflected in a direct improvement of the criteria. The findings in this article extend and generalize previous findings in the literature and also improve some of these findings.

On Explicit Abstract Neutral Differential Equations with State-Dependent Delay

On Explicit Abstract Neutral Differential Equations with State-Dependent Delay

Existence Results for Positive Periodic Solutions to First-Order Neutral Differential Equations

Existence Results for Positive Periodic Solutions to First-Order Neutral Differential Equations

Approximate controllability of fractional differential equations driven by Markovian switching and Lévy noise with infinite delay

Motivated by applications of nonlinear dynamical systems and based on some existing works, we introduce a general class of Atangana-Baleanu nonlinear fractional neutral differential equations driven by Markovian switching and Lévy noise with infinite delay. In this paper, we aim to derive new results on the existence of solutions as well as approximate controllability for the corresponding Atangana-Baleanu equations in Banach spaces. Under suitable conditions with the help of semigroup theory, fixed point theorem as well as stopping time argument, we prove the main results. As an application, we present two examples with numerical simulations to support our theory. The results obtained in this paper improve and extend some related conclusions on this topic.

An averaging result for fractional variable-order neutral differential equations with variable delays driven by Markovian switching and Lévy noise

In this paper, we derive new results on the averaging principle for a class of Caputo neutral stochastic system driven by Markovian switching and Lévy noise with variable delays and time-varying fractional order. Under a set of appropriate conditions, we showed that solutions of the averaged stochastic systems approach the solutions of the original stochastic systems in the sense of both convergences in mean square and convergence in probability. Finally, we attach two examples with numerical simulations to justify the validity of our theory.