Order Neutral Differential Equations Research Articles

Existence results for positive periodic solutions to first order neutral differential equations with distributed deviating arguments

We take into account the first order nonlinear neutral differential equation with distributed deviating arguments.
 Using Krasnoselskii's fixed point theorem, we give some new criteria for the existence of positive periodic solutions to this equation. The theorems we have established are illustrated by an example.

New oscillation and non-oscillation criteria for third order neutral differential equations with distributed deviating arguments

New oscillation and non-oscillation criteria for third order neutral differential equations with distributed deviating arguments

Existence of Nonoscillatory Solutions of Higher Order Nonlinear Neutral Differential Equations

In this paper, an n-th order neutral nonlinear differential equation is studied. By using the Banach contraction principle, some sufficient conditions are established for the existence of nonoscillatory solutions of nonlinear n-th order neutral differential equation. An example is included to illustrate the results obtained.

PERIODIC SOLUTIONS OF A CLASS OF SECOND ORDER NEUTRAL DIFFERENTIAL EQUATIONS WITH MULTIPLE DIFFERENT DELAYS

The present work mainly probes into the existence and uniqueness of periodic solutions for a class of second-order neutral differential equations with multiple delays. Our approach is based on using Banach and Krasnoselskii's fixed point theorems as well as the Green's function method. Besides, two examples are exhibited to validate the effectiveness of our findings which complement and extend some relevant ones in the literature.

Multiplicity–Induced–Dominancy property for second–order neutral differential equations with application in oscillation damping

This paper addresses the exponential stability of linear time-delay systems of neutral type. In general, it is quite a challenge to establish conditions on the parameters of the system in order to guarantee such a stability. Recent works emphasized the link between maximal multiplicity and dominant roots. Indeed, conditions for a given multiple root to be necessarily dominant are investigated, this property is known as Multiplicity-Induced-Dominancy (MID). The aim of this paper is to explore the effect of multiple roots with admissible multiplicities exhibiting, under appropriate conditions, the validity of the MID property for second-order neutral time-delay differential equations with a single delay. The ensuing control methodology is summarized in a five-steps algorithm that can be exploited in the design of higher-order systems. The main ingredient of the proposed method is the dominancy proof for multiple spectral values based on frequency bounds established via integral equations. As an illustration, the stabilization of the classical oscillator benefits from the obtained results.

Oscillation of Second Order Nonlinear Neutral Differential Equations

The study of the oscillatory behavior of solutions to second order nonlinear differential equations is motivated by their numerous applications in the natural sciences and engineering. In the presented research, some new oscillation criteria for a class of damped second order neutral differential equations with noncanonical operators are established. The results extend and improve on those reported in the literature. Moreover, some examples are provided to show the significance of the results.

Oscillation criteria of second order differential equations with positive and negative coefficients

In this paper we obtain oscillation criteria for solutions of homogeneous and nonhomogeneous cases of second order neutral differential equations with positive and negative coefficients. Our results improve and extend the results of [Oscillation criteria for a class of second order neutral delay differential equations, Appl. Math. Comput. \textbf{210}, 303--312, 2009].

Unbounded oscillation criteria for fourth order neutral differential equations of non-canonical type

Abstract In this work, sufficient conditions are established for oscillation of all unbounded solutions of a class of fourth order neutral differential equations of the form ( r ( t ) ( y ( t ) + p ( t ) y ( t − τ ) ) ″ ) ″ + q ( t ) G ( y ( t − α ) ) − h ( t ) H ( y ( t − σ ) ) = 0 $$\begin{array}{}\displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\alpha))-h(t)H(y(t-\sigma)) = 0\end{array} $$ under the non-canonical type assumption ∫ 0 ∞ t r ( t ) d t < ∞ $$\begin{array}{}\displaystyle \int\limits_{0}^{\infty}\frac{t}{r(t)}{\rm d} t \lt \infty\end{array} $$ for various ranges of p(t) with ∣p(t)∣ < ∞.

On the behaviour of solutions to a kind of third order neutral stochastic differential equation with delay

This article demonstrates the behaviour of solutions to a kind of nonlinear third order neutral stochastic differential equations. Setting x^{prime }(t)=y(t), y^{prime }(t) =z(t) the third order differential equation is ablated to a system of first order differential equations together with its equivalent quadratic function to derive a suitable downright Lyapunov functional. This functional is utilised to obtain criteria which guarantee stochastic stability of the trivial solution and stochastic boundedness of the nontrivial solutions of the discussed equations. Furthermore, special cases are provided to verify the effectiveness and reliability of our hypotheses. The results of this paper complement the existing decisions on system of nonlinear neutral stochastic differential equations with delay and extend many results on third order neutral and stochastic differential equations with and without delay in the literature.

On stability and square integrability of solutions to some third order neutral differential equations

On stability and square integrability of solutions to some third order neutral differential equations

Oscillation Theorems for Third Order Neutral Delay Difference Equation with Negative Coefficient in the Neutral Term

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On the behaviour of solutions to a kind of third order nonlinear neutral differential equation with delay

This paper presents a novel class of third order nonlinear nonautonomous neutral differential equation with delay. The third order neutral differential equation is cut down to a system of first order, a suitable complete Lyapunov-Krasovskii's functional is constructed and used, to obtain standard conditions on the nonlinear functions to ensure stability and uniform asymptotic stability of the trivial solutions, the existence of a unique periodic solution, uniform boundedness and uniform ultimate boundedness of solutions when the forcing term is nonzero. The obtained results are new and include many prominent results on neutral and nonneutral delay differential equations in literature. Finally, the practicability and reliability of the theoretical results are demonstrated.

Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator

Oscillatory behavior of higher order neutral differential equation with multiple functional delays under derivative operator

Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations

AbstractIn this article, sufficient conditions are obtained so that every solution of the neutral difference equationΔm(yn−pnL(yn−s))+qnG(yn−k)=0,$$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})=0, \end{equation*}$$or every unbounded solution ofΔm(yn−pnL(yn−s))+qnG(yn−k)−unH(yα(n))=0,n≥n0,$$\begin{equation*}\Delta^{m}\big(y_n-p_n L(y_{n-s})\big) + q_nG(y_{n-k})-u_nH(y_{\alpha(n)})=0,\quad n\geq n_0, \end{equation*}$$oscillates, wherem=2 is any integer, Δ is the forward difference operator given by Δyn=yn+1−yn; Δmyn= Δ(Δm−1yn) and other parameters have their usual meaning. The non linear functionL∈C(ℝ, ℝ) inside the operator Δmincludes the caseL(x) =x. Different types of super linear and sub linear conditions are imposed onGto prevent the solution approaching zero or ±∞. Further, all the three possible cases,pn≥ 0,pn≤ 0 andpnchanging sign, are considered. The results of this paper generalize and extend some known results.

On Oscillatory and Asymptotic Behavior of Higher Order Neutral Differential Equations with Impulsive Conditions

On Oscillatory and Asymptotic Behavior of Higher Order Neutral Differential Equations with Impulsive Conditions

Approximate Controllability of Second Order Neutral Stochastic Integro Differential Equations with Impulses Driven By Fractional Brownian Motion

Approximate Controllability of Second Order Neutral Stochastic Integro Differential Equations with Impulses Driven By Fractional Brownian Motion

Solutions of linear uncertain fractional order neutral differential equations

Uncertain fractional order neutral differential equation is an important model to describe the evolution process of uncertain dynamical system. This paper devotes to studying linear uncertain fractional order neutral differential equations. After providing the analytic solutions for linear uncertain fractional order neutral differential equations by the Mittag-Leffler function, the author investigates the inverse uncertainty distribution of the solution to linear uncertain fractional order neutral differential equation by the α-path and has a discussion of the dependence of the solution on initial function based on the generalized Gronwall inequality.

Oscillations in second order linear neutral impulsive differential equations with constant delays

Sufficient conditions are established for the oscillation of all solutions of a class of second order neutral differential equations with constant delay arguments and constant impulsive jumps for the cases where the coefficient p is a constant and when p is a variable. Examples are provided for clarity.

Oscillation results for a certain class of fourth-order nonlinear delay differential equations

In this work, we study the oscillation of the fourth order neutral differential equations with delay argument. By means of generalized Riccati transformation technique, we obtain new oscillation criteria for oscillation of this equation. An example is given to clarify the main results in this paper.

Nonlinear Second Order Impulsive Difference Equations and their Oscillation Properties

Abstract In this work, we state and discuss the sufficient conditions for oscillation and nonoscillation of a class of nonlinear second order neutral impulsive difference equations with fixed moments of impulsive effect for various ranges of the neutral coefficient p(n).