MONOTONE ITERATIVE TECHNIQUE FOR NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

The objective of this paper is to develop monotone techniques for obtaining extremal solutions of initial value problem for nonlinear neutral delay differential equations.

DELAY-DEPENDENT TREATMENT OF LINEAR MULTISTEP METHODS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

This paper deals with a delay-dependent treatment of linear multistep methods for neutral delay differential equations y'(t) = ay(t) + by(t - τ) + cy'(t - τ), t > 0, y(t) = g(t), -τ≤ t ≤ 0, a,b andc ∈ R. The necessary condition for linear multistep methods to be Nτ(0)-stable is given. It is shown that the trapezoidal rule is Nτ(0)-compatible. Figures of stability region for some linear multistep methods are depicted.

OSCILLATION RESULTS FOR A SECOND ORDER NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Some new oscillation criteria are established for a second order neutral delay differential equations. These results improve oscillation results of Y.V. Rogo-vchenko for the retarded delay differential equations. The relevance of our theorems is illustrated with two carefully selected examples.

Numerical Solution for Neutral Delay Differential Equation of Constant or Proportional Type using Hybrid Block Method

Neutral Delay Differential Equation(NDDE)is a differential problem that has regularly existed in numerous occurrences and has represented a significant role in dealing with real-life phenomena,especially on their application in biological and physiological processes.A fifth-order two-point hybrid implicit multistep block method(2PIH5)has been formulated in this research for the numerical solution of Neutral Delay Differential Equation(NDDE).A Taylor series interpolation polynomial has been implemented in the formulation of the proposed 2PIH5.The order,consistency,and zero-stability for 2PIH5 have been illustrated.The analyses of convergence and stability test have been performed and discussed.The initial value problems for the first-order NDDE with constant or proportional delay have been solved using the proposed block method.Some numerical results for the proposed method have been presented to prove the adaptability and applicability of the proposed method in solving NDDE.The proposed method is proved to be comparable with the other existing methods.It is assumed to be reliable and efficient for solving the first-order NDDE with constant or proportional delay.

Existence of Nonoscillatory Solutions for a Second-Order Nonlinear Neutral Delay Differential Equation

A new second-order nonlinear neutral delay differential equation r（t） x（t） ＋ P（t）x（t-τ） ＋ cr（t） x（t）-x（t-τ） ＋ F t,x（t-σ1）,x（t-σ2）,...,x（t-σn） = G（t）,t ≥ t0,where τ 0,σ1,σ2,...,σn ≥ 0,P,r ∈ C（[t0,＋∞）,R）,F ∈ C（[t0,＋∞）×Rn,R）,G ∈ C（[t0,＋∞）,R） and c is a constant,is studied in this paper,and some sufficient conditions for existence of nonoscillatory solutions for this equation are established and expatiated through five theorems according to the range of value of function P（t）.Two examples are presented to illustrate that our works are proper generalizations of the other corresponding results.Furthermore,our results omit the restriction of Q1（t） dominating Q2（t）（See condition C in the text）.

NONLINEAR STABILITY OF NATURAL RUNGE-KUTTA METHODS FOR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Presents the stability analysis of theoretical solutions for a class of nonlinear neutral delay-differential equations (NDDE). Discussion on the numerical analogous results of the natural Runge-Kutta (NRK) methods for the same class of nonlinear NDDE; Review of the related concepts and results on RK methods; Information on the asymptotic stability and global stability of the induced NRK method.

EXISTENCE AND MANN ITERATIVE APPROXIMATIONS OF NONOSCILLATORY SOLUTIONS TO SECOND-ORDER NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In this paper, we establish a few existence results of nonoscillatory solutions to second-order nonlinear neutral delay differential equations, construct several Manntype iterative approximation schemes for these nonoscillatory solutions, and give some error estimates between the approximate solutions and the nonoscillatory solutions. And finally we give an example to illustrate our results.

LINEARIZED OSCILLATIONS FOR ODD-ORDERNEUTRAL DELAY DIFFERENTIAL EQUATIONSWITH ASYMPTOTIC PERIODIC COEFFICIENTS

In this paper, we prove that, under appropriate hypotheses, the odd-order nonlinear neutral delay differential equation has the same oscillatory character as the associated linear equation with periodic coefficients Our method is based on the establishment of a new comparison theorem for the oscillation of Eq.(E). Hence we prove that the parameter set such that Eq.(E) has a nonoscillatory solution is closed in certain metric space, and avoid the difficulty to set the necessary and sufficient condition for the oscillation of Eq.(E).

A NEW OSCILLATION CRITERION FOR FIRST ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION

In this paper, a new sufficient condition for the oscillation of all solutions of first order neutral delay differential equations is obtained. Secondly, the result can also be extended to a general neutral differential equation, and many known results in the literatures are improved.

CONDITIONS FOR OSCILLATION OF NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In this paper,we study one type of odd-order neutral delay differential equation. Suffcient conditions for the oscillation of every solution to this type of differential equation are given.

OSCILLATION THEOREMS FOR DELAY AND NEUTRAL DELAY DIFFERENTIAL EQUATIONS

Oscillation criteria for the delay differential equationwhere p, T are non-negative real-valued continuous functions are investigated in the case when the numberssatisfy 0 < k < 1/e and 1/e < L < 1. The present result improves almost all results of the literature concerning it. Furthermore, it is established that all solutions of the odd-order neutral delay differential equationwhere , are oscillatory ifThis result generalizes a theorem of Gopalsamy et. al (Czech. Math. J., 42 (1992),313-323) and also extends a very well-known result of Ladas (Applicable Analysis, 9(1979),93-98).

Existence of Periodic Solutions of Nonlinear Neutral Differential Equation with Deviating Argument

Existence criteria is established for the periodic solution of the nonlinear neutral delay differential equation x′(t)=f(t,x(t),x(t-τ 1(t)),x′(t-τ 2(t)))+p(t) by means of an abstract continuous theorem of k-set contractive operator and some analysis technique.

Fixed Points and Asymptotic Properties of Neutral Stochastic Delay Differential Equations

This paper discusses a linear neutral stochastic differential equation with variable delays. By using fixed point theory, the necessary and sufficient conditions are given to ensure that the trivial solution to such an equation is pth moment asymptotically stable. These conditions do not require the boundedness of delays, nor derivation of delays. An example was also given for illustration.

STABILITIES OF (A,B,C) AND NPDIRK METHODS FOR SYSTEMS OF NEUTRAL DELAY-DIFFERENTIAL EQUATIONS WITH MULTIPLE DELAYS

Consider the following neutral delay-differential equations with multiple delays (NMDDE)where γ> 0, L, Mj and Nj are constant complex- value d×d matrices. A sufficient condition for the asymptotic stability of NMDDE system (0.1) is given. The stability of Butcher's (A,B,C)-method for systems of NMDDE are studied. In addition, we present a parallel diagonally-implicit iteration RK (PDIRK) methods(NPDIRK) for systems of NMDDE, which is easier to be implemented than fully implicit RK methods. We also investigate the stability of a special class of NPDIRK methods by analyzing their stability behaviors of the solutions of (0.1).

Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration(PDA)feedback,which can be used to understand human balance in quiet standing.The closed-loop system is described by a neutral delay differential equation(NDDE).The optimal feedback gains(OFGs)that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4.Such a property is called multiplicity-induced dominancy of time-delay systems,and has been discussed intensively by many authors for retarded delay differential equations(RDDEs).This paper shows that multiplicity-induced dominancy can be achieved in NDDEs.In addition,the OFGs are delay-dependent,and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays.Thus,the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains.The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.

MEAN SQUARE STABILITY AND DISSIPATIVITY OF SPLIT- STEP THETA METHOD FOR NONLINEAR NEUTRAL STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH POISSON JUMPS

In this paper, a split-step 0 （SST） method is introduced and used to solve the non- linear neutral stochastic differential delay equations with Poisson jumps （NSDDEwPJ）. The mean square asymptotic stability of the SST method for nonlinear neutral stochastic differential equations with Poisson jumps is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the SST method with ∈ E （0, 2 -√2） is asymptotically mean square stable for all positive step sizes, and the SST method with ∈ E （2 -√2, 1） is asymptotically mean square stable for some step sizes. It is also proved in this paper that the SST method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.

CONVERGENCE RATE OF THE TRUNCATED EULER-MARUYAMA METHOD FOR NEUTRAL STOCHA STIC DIFFERENTIAL DELAY EQUATIONS WITH MARKOVIAN SWITCHING

The key aim of this paper is to show the strong convergence of the truncated Euler-Maruyama method for neutral stochastic differential delay equations(NSDDEs)with Markovian switching(MS)without the linear growth condition.We present the truncated Euler-Maruyama method of NSDDEs-MS and consider its moment boundedness under the local Lipschitz condition plus Khasminskii-type condition.We also study its strong convergence rates at time T and over a finite interval[0,T].Some numerical examples are given to illustrate the theoretical results.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION

The boundedness of the every solution and the asymptotic behavior of all solutions of the nonlinear neutral delay differential equation [x（t） - P（t）x（t - r）]＇ ＋ Q1（t）f（x（t -σ1）） - Q2（t）f（x（t - σ2）） = 0, t 〉 t0 are investigated, where τ,σ1, σ2 ∈ （0, ∞）, P∈C（[t0, ∞）, R）, and Q1, Q2 ∈C（[t0,∞）,R）, f ∈ C（R, R）. The sufficient conditions obtained improve the existing results in the literatures.

OSCILLATION CRITERIA FOR SECOND ORDER QUASILINEAR NEUTRAL EQUATION

We present some new oscillation criteria for second order neutral delay differential equations with distributed deviating argument using the Riccati technique and some inequalities. The results obtained improve and extend some known results.