Nonlinear Neutral Delay Differential Equations Research Articles

Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

Some sufficient conditions are established for the oscillation of second-order neutral differential equation , , where . The results complement and improve those of Grammatikopoulos et al. Ladas, A. Meimaridou, Oscillation of second-order neutral delay differential equations, Rat. Mat. 1 (1985), Grace and Lalli (1987), Ruan (1993), H. J. Li (1996), H. J. Li (1997), Xu and Xia (2008).

Remarks on the paper [Appl. Math. Comput. 207 (2009) 388–396

In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations ( r ( t ) ψ ( x ( t ) ) | Z ′ ( t ) | α - 1 Z ′ ( t ) ) ′ + q ( t ) f ( x ( σ ( t ) ) ) = 0 , t ⩾ t 0 > 0 , where Z ( t ) = x ( t ) + p ( t ) x ( t - τ ) and α > 0 , 0 ⩽ p ( t ) < 1 . On the other hand, some new oscillation criteria are established for the second-order nonlinear neutral delay differential equations [ r ( t ) [ x ( t ) + p ( t ) x ( τ ( t ) ) ] ′ ] ′ + q ( t ) f ( x ( σ ( t ) ) ) = 0 , t ⩾ t 0 > 0 , where ∫ t 0 ∞ d t r ( t ) < ∞ , 0 ⩽ p ( t ) ⩽ p 0 < + ∞ . The results obtained here complement and correct some known results in [1]. Some examples are given to illustrate the main results.

Stability of linear multistep methods for nonlinear neutral delay differential equations in Banach space

This paper is devoted to investigating the nonlinear stability properties of linear multistep methods for the solution to neutral delay differential equations in Banach space. Two approaches to numerically treating the “neutral term” are considered, which allow us to prove several results on numerical stability of linear multistep methods. These results provide some criteria for choosing the step size such that the numerical method is stable. Some examples of application and a numerical experiment, which further confirms the main results, are given.

Solvability and iterative algorithms for a higher order nonlinear neutral delay differential equation

The paper is concerned with the higher order nonlinear neutral delay differential equation r ( t ) ( x ( t ) + p ( t ) x ( t - τ ) ) ( m ) ( n - m ) + ( - 1 ) n - m + 1 f t , x σ 1 ( t ) , … , x ( σ l ( t ) ) = g ( t ) , t ⩾ t 0 . Using the Banach fixed-point theorem, we establish five existence results of uncountably many bounded nonoscillatory solutions for the above equation, construct five Mann iterative sequences with mixed errors for approximating these nonoscillatory solutions and discuss five error estimates between the approximate solutions and these nonoscillatory solutions. To dwell the importance and applications of our results, five nontrivial examples are constructed.

Necessary and sufficient conditions for oscillation of higher order nonlinear neutral delay differential equations

We first show that the n order superlinear neutral delay differential equation with unstable type $$ [x(t) - px(t - \tau )]^{(n)} = q(t)|x(t - \sigma )|^{\beta - 1} x(t - \sigma ),t \geqslant t_0 $$ always has an unbounded and nonoscillatory solution, where n > 1 is an even integer, σ > 0, τ > 0, β > 1, 0 < p < 1 and q ∈ C([t0, ∞), [0, ∞)). Then a necessary and sufficient condition is obtained for all bounded solution of the above equation to be oscillatory.

Comment on the paper: “Forced oscillation of certain neutral hyperbolic equations with continuous distributed deviating arguments” [Math. Comput. Modeling 49 (2009) 1211–1220

In recent years, many results on oscillation criteria of second-order nonlinear neutral delay differential equations with distributed deviating argument have been obtained, but some of these criteria are incorrect due to a mistake in a crucial auxiliary result. We give two counterexamples.

New oscillation criteria for second-order nonlinear neutral delay differential equations

In this paper, several new oscillation criteria for the second-order nonlinear neutral delay differential equation [ r ( t ) | ( x ( t ) + p ( t ) x [ σ ( t ) ] ) ′ | m − 1 ( x ( t ) + p ( t ) x [ σ ( t ) ] ) ′ ] ′ + q ( t ) f ( x [ τ ( t ) ] ) = 0 , t ≥ t 0 are established. These oscillation criteria extend and improve some known results. An interesting example illustrating the importance of our results is also provided.

Existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations

In this paper, the Banach fixed-point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference equation Δ a ( n ) Δ x ( n ) + b ( n ) x ( n - τ ) + Δ h ( n , x ( h 1 ( n ) ) , x ( h 2 ( n ) ) , … , x ( h k ( n ) ) ) + f ( n , x ( f 1 ( n ) ) , x ( f 2 ( n ) ) , … , x ( f k ( n ) ) ) = c ( n ) , n ⩾ n 0 , where τ , k ∈ N , n 0 ∈ N 0 , a , b , c : N n 0 → R with a ( n ) > 0 for n ∈ N n 0 , h , f : N n 0 × R k → R and h l , f l : N n 0 → Z with lim n → ∞ h l ( n ) = lim n → ∞ f l ( n ) = + ∞ , l ∈ { 1 , 2 , … , k } . A few Mann type iterative approximation schemes with errors are suggested, and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed. Seven nontrivial examples are given to illustrate the advantages of our results.

Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments.

Global solvability for a second order nonlinear neutral delay difference equation

This paper studies the global existence of solutions of the second order nonlinear neutral delay difference equation Δ ( a n Δ ( x n + b x n − τ ) ) + f ( n , x n − d 1 n , x n − d 2 n , … , x n − d k n ) = c n , n ≥ n 0 with respect to all b ∈ R . A few results on global existence of uncountably many bounded nonoscillatory solutions are established for the above difference equation. Several nontrivial examples which dwell upon the importance of the results obtained in this paper are also included.

Oscillation of nonlinear neutral delay differential equations of second-order with positive and negative coefficients

Some oscillation criteria for the following second-order neutral differential equation [x(t)\pm r(t) f( x(t-\gamma))]''+p(t) g(x(t-\alpha)) -q(t) g(x(t-\beta )) = s(t) where t\geq t_0, \gamma,\alpha,\beta \in R^+ with \alpha \geq \beta, r \in C^2([t_0,\infty ), R^+) , p,q\in C([t_0,\infty ),R^+) and f,g\in C(R,R), s\in C([ t_0,\infty),R) have been obtained. Our results are not restricted with boundedness of solutions.

Stability of continuous Runge–Kutta-type methods for nonlinear neutral delay-differential equations

This paper provides results on the correct simulation, when using continuous Runge–Kutta methods, of certain stability properties of nonlinear neutral delay-differential equations (NDDEs) y ′ ( t ) = f ( t , y ( t ) , y ( t - τ ( t ) ) , y ′ ( t - τ ( t ) ) ) ( t ⩾ t 0 ) . In particular, it is shown that certain continuous Runge–Kutta methods based upon the backward Euler method or the 2-stage Lobatto IIIC method, combined with linear interpolation, are GRN -stable and asymptotically stable for NDDEs.

A note on oscillation criteria of second order nonlinear neutral delay differential equations

In recent years, many results on oscillation criteria of second order nonlinear neutral delay differential equations have been obtained, but some of these criteria are incorrect due to a mistake in a crucial auxiliary result. This note examines several of these results, provides some counterexamples and points out what causes the problems.

Oscillation and Asymptotic Behavior for nth-order Nonlinear Neutral Delay Dynamic Equations on Time Scales

In this paper, we derive some sufficient conditions for the oscillation and asymptotic behavior of the nth-order nonlinear neutral delay dynamic equations $$\begin{array}{rcl}&&\left\{a(t)\Psi(x(t))\left[|(x(t)+p(t)x(\tau(t)))^{\Delta ^{n-1}}|^{\alpha-1}(x(t)+p(t)x(\tau(t)))^{\Delta^{n-1}}\right]^{\gamma}\right\}^{\Delta}\\[12pt]&&{}\quad +\lambda F(t,x(\delta(t)))=0,\end{array}$$ on time scales, where α>0 is a constant, γ>0 is a quotient of odd positive integers and λ=±1. Our results in this paper not only extend and improve some known results but also present a valuable unified approach for the investigation of oscillation and asymptotic behavior of nth-order nonlinear neutral delay differential equations and nth-order nonlinear neutral delay difference equations. Examples are provided to show the importance of our main results.

Infinitely many nonoscillatory solutions for second order nonlinear neutral delay differential equations

In this paper we consider the second order nonlinear neutral delay differential equation [ a ( t ) ( x ( t ) + b ( t ) x ( t − τ ) ) ′ ] ′ + [ h ( t , x ( h 1 ( t ) ) , x ( h 2 ( t ) ) , … , x ( h k ( t ) ) ) ] ′ + f ( t , x ( f 1 ( t ) ) , x ( f 2 ( t ) ) , … , x ( f k ( t ) ) ) = g ( t ) , t ≥ t 0 , where τ > 0 , a , b , g ∈ C ( [ t 0 , + ∞ ) , R ) with a ( t ) > 0 for t ≥ t 0 , h ∈ C 1 ( [ t 0 , + ∞ ) × R k , R ) , f ∈ C ( [ t 0 , + ∞ ) × R k , R ) , h l ∈ C 1 ( [ t 0 , + ∞ ) , R ) and f l ∈ C ( [ t 0 , + ∞ ) , R ) with lim t → + ∞ h l ( t ) = lim t → + ∞ f l ( t ) = + ∞ , l = 1 , … , k . Under suitable conditions, by making use of the Banach fixed point theorem, we show the existence of infinitely many nonoscillatory solutions, which are uncountable, for the above equation, suggest several Mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions. Five nontrivial examples are given to illustrate the advantages of our results.

Dissipativity of θ-methods for nonlinear delay differential equations of neutral type

This paper is concerned with the analytic and numerical dissipativity of nonlinear neutral delay differential equations (NDDEs) of the form y ′ ( t ) = F ( y ( t ) , G ( y ( t − τ ) , y ′ ( t − τ ) ) ) . The basic idea is to reformulate the original problem eliminating the dependence on the derivative of the solution in the past values. A dissipativity criteria for nonlinear NDDEs is given. Dissipativity properties of one-leg θ-methods and linear θ-methods for the underlying systems are investigated. It is shown that, for 1 2 < θ ⩽ 1 , both one-leg θ-method and linear θ-method are dissipative. Numerical examples illustrate the theoretical results.

Convergence of waveform relaxation methods for neutral delay differential equations

This paper gives sufficient conditions for existence and uniqueness of solutions and for the convergence of waveform relaxation methods for nonlinear neutral delay differential equations (NDDEs). By investigations into the continuous time iteration and the discrete time iteration processes that are obtained from algebraically stable and diagonally stable Runge–Kutta methods, it is shown that the method is convergent under the assumptions that the splitting function satisfies the one-sided Lipschitz condition with respect to some arguments and the Lipschitz condition with respect to the others.

Dissipativity of [formula omitted]-methods for a class of nonlinear neutral delay differential equations

This paper is concerned with the dissipativity of theoretical solution and numerical solution of a class of nonlinear neutral delay differential equations. At first, a sufficient condition for the dissipativity of theoretical solution of the mentioned problem above is given, then the dissipativity results are obtained for one-leg and linear θ -methods when they are applied to these problems.

On the oscillation of second order nonlinear neutral delay difference equations

In this paper sufficient conditions are obtained for oscillation of all solutions of a class of nonlinear neutral delay difference equations of the form

Oscillation and nonoscillation of nonlinear neutral delay difference equations

In this paper, the authors establish some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral delay difference equation$$ \Delta^2 (x_n-p_nx_{n-k}) + q_nf(x_{n-\ell}) = 0, ~~n \ge n_0 $$where $ \{p_n\} $ and $ \{q_n\} $ are non-negative sequences with $ 0$