Nonlinear Neutral Delay Differential Equations Research Articles

Contractivity properties of a class of linear multistep methods for nonlinear neutral delay differential equations

In this paper, we show that under identical conditions which guarantee the contractivity of the theoretical solutions of general nonlinear NDDEs, the numerical solutions obtained by a class of linear multistep methods are also contractive.

New oscillatory criteria for higher-order nonlinear neutral delay differential equation

In this paper, we consider the higher-order nonlinear neutral delay differential equation (1.1) ( x ( t ) − p ( t ) x ( t − τ ) ) ( n ) + f ( t , x ( σ ( t ) ) ) = 0 where n is an odd number and n ⩾ 3 , τ > 0 , p , σ ∈ C ( [ t 0 , ∞ ) , R + ) , σ ( t ) ⩽ t , lim t → ∞ σ ( t ) = ∞ , f ∈ C ( [ t 0 , ∞ ) × R , R ) , f ( t , u ) is nondecreasing in u .

Existence and iterative approximations of nonoscillatory solutions of higher order nonlinear neutral delay differential equations

This paper studies a few existence results of nonoscillatory solutions for the following higher order nonlinear neutral delay differential equation: dndtnx(t)+cx(t-τ)+(-1)n-i+1didtih(t,x(h1(t)),x(h2(t)),…,x(hk(t)))+(-1)n+1f(t,x(f1(t)),x(f2(t)),…,x(fk(t)))=g(t),t⩾t0, where n is a positive integer, 0⩽i⩽n-1, c∈R⧹{-1}, τ>0, h∈Ci([t0,+∞)×Rk,R), f∈C([t0,+∞)×Rk,R), g∈C([t0,+∞),R), hl∈Ci([t0,+∞),R) and fl∈C([t0,+∞),R) with limt→+∞hl(t)=limt→+∞fl(t)=+∞,l=1,…,k constructs 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. In addition sufficient conditions for the existence of infinitely many nonoscillatory solutions for the above equation are given. Three nontrivial examples are given to illustrate the advantages of our results.

Necessary Conditions for the Solutions of Second Order Non-linear Neutral Delay Difference Equations to Be Oscillatory or Tend to Zero

We find necessary conditions for every solution of the neutral delay difference equationΔ(rnΔ(yn−pnyn−m))+qnG(yn−k)=fnto oscillate or to tend to zero asn→∞, whereΔis the forward difference operatorΔxn=xn+1−xn, andpn, qn, rnare sequences of real numbers withqn≥0, rn>0. Different ranges of{pn}, includingpn=±1, are considered in this paper. We do not assume thatGis Lipschitzian nor nondecreasing withxG(x)>0forx≠0. In this way, the results of this paper improve, generalize, and extend recent results. Also, we provide illustrative examples for our results.

Nonlinear stability of one-leg methods for delay differential equations of neutral type

This paper is devoted to investigations into numerical stability properties of one-leg methods for nonlinear neutral delay differential equations. At first, a series of new stability concepts, such as GS- stability, GAS- stability and Weak GS- stability, are introduced. Then it is proved that a strongly A - stable one-leg method with linear interpolation is GAS- stable, and that an A - stable one-leg method with linear interpolation is GS- stable and Weakly GS- stable. Some numerical experiments are given in the last section of this paper which confirm our results.

Bounded oscillation for second-order nonlinear neutral difference equations in critical and non-critical states

The sufficient and necessary conditions for the bounded oscillation of the second order nonlinear neutral delay difference equation Δ 2 [ x ( n ) - px ( n - τ ) ] = ∑ i = 1 m q i x ( n - σ i ) + f ( n , x ( n - η 1 ( n ) ) , … , x ( n - η l ( n ) ) ) are obtained in critical and non-critical cases.

Stability analysis of general linear methods for nonlinear neutral delay differential equations

The present paper is concerned with the numerical solution to initial value problems of neutral delay differential equations (NDDEs). The sufficient conditions for the stability and asymptotic stability of the general linear methods for a class R α, β of nonlinear systems of NDDEs are derived. Numerical test is given that confirm the theoretical results in the end.

Oscillation results for higher order nonlinear neutral delay difference equations

In this work, we shall consider higher order nonlinear neutral delay difference equation of the type Δ m ( y n + p n y n − l ) + q n y n − k α = 0 , where { p n } , { q n } are sequences of nonnegative real numbers, k and l are positive integers and α ∈ ( 0 , 1 ) is a ratio of odd positive integers. We obtain sufficient conditions for the oscillations of all solutions of this equation.

Oscillation of nonlinear neutral delay differential Equations

In this paper, we study the oscillatory behavior of first order nonlinear neutral delay differential equations. Several new sufficient conditions which ensure that all solutions are oscillatory are given. The obtained results extend and improve several known results in the literature. Some examples are considered to illustrate the main results.

Linearized oscillation of odd order nonlinear neutral delay differential equations (I)

In this paper, we proved that the odd order nonlinear neutral delay differential equation [ x ( t ) − p ( t ) g ( x ( t − τ ) ) ] ( n ) + q ( t ) h ( x ( t − σ ) ) = 0 has the same oscillatory character as its linearized equation [ x ( t ) − p 0 x ( t − τ ) ] ( n ) + q 0 x ( t − σ ) = 0 under some rather relaxed conditions on g ( u ) and h ( u ) , where p 0 = lim t → ∞ p ( t ) , q 0 = lim t → ∞ q ( t ) .

Bounded oscillation for second order non-linear neutral delay differential equations in critical and non-critical cases

Sufficient and necessary conditions for the bounded oscillation of the second order non-linear neutral delay differential equation [ x ( t ) - px ( t - τ ) ] ″ = ∑ i = 1 m q i x ( t - σ i ) + f ( t , x ( t - η 1 ( t ) ) , … , x ( t - η n ( t ) ) ) are obtained in critical and non-critical cases.

Classification and existence of nonoscillatory solutions of higher order nonlinear neutral difference equations

In this paper, we consider the higher order nonlinear neutral delay difference equation of the form $$\Delta ^r (x_n + px_{n - \tau } ) + f(n,x_{n - \sigma _1 (n)} ,x_{n - \sigma _2 (n)} ,...,x_{n - \sigma _m (n)} ) = 0.$$ We give an integrated classification of nonoscillatory solutions of the above equation according to their asymptotic behaviours. Necessary and sufficient conditions for the existence of nonoscillatory solutions with designated asymptotic properties are also established.

Oscillation for second-order nonlinear neutral delay difference equations

Using the Riccati transformation techniques, we will extend some oscillation criteria of [Appl. Math. Comput. 146 (2003) 791] and [Appl. Math. Comput. 142 (2003) 99] to the second-order nonlinear neutral delay difference equation Δ(a n( Δ(x n+p nx n−τ)) γ)+f(n,x n−σ)=0, n=0,1,2,… in the case when 0< γ<1, which answers a question posed by Saker [Appl. Math. Comput. 142 (2003) 99]. Two examples are considered to illustrate our main results.

First Order Nonlinear Neutral Delay Differential Equations

The author obtain results on the asymptotic behavior of the nonoscillatory solutions of first order nonlinear neutral differential equations. Keywords. Neutral differential equations, Oscillatory and Nonoscillatory solutions.

Non-Oscillatory Solutions of Higher Order Nonlinear Neutral Delay Difference Equation

Consider the forced higher order nonlinear neutral delay difference equation L_m( x(n) + cx(n - τ) ) + F ( n, x(σ(n) ) ) = g(n)\quad (n ≥ n_0). We obtain a global result (with respect to c ), which consists of some sufficient conditions for the existence of a non-oscillatory solution of the above equation. Our results improve and extend a number of existing results.

Existence and nonexistence of eventually positive solutions for nonlinear neutral differential equations

In this paper, the necessary and sufficient conditions of the existence and nonexistence of eventually positive solutions for a class of nonlinear neutral delay differential equations are obtained by Riccati-type transformation. By applying the necessary and sufficient conditions, the impulsive neutral differential equations are also investigated.

Interval criteria for oscillation of second-order nonlinear neutral differential equations

Some new oscillation criteria are established in this paper for the second-order nonlinear neutral delay differential equation [r(t)(x(t) + p(t)x(t−τ))′]′ + q(t)f(x(t−)δ)) = 0, where τ and δ are nonnegative constants, r, p, q ϵ C([ t 0, ∞), R), and f ϵ C( R, R). These results are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of [ t 0, ∞), rather than on the whole half-line. Our results are sharper than some previous results about neutral and delay differential equations. In particular, several examples that dwell upon the sharp conditions of our results are also included.

Linearized oscillations of first-order nonlinear neutral delay difference equations

In this paper, we establish an equivalence of oscillation for the first-order nonlinear neutral delay difference equation Δ(x n−px n−r)+ ∑ i=1 m q if i(x n−σ i )=0 and its corresponding linear equation Δ(x n−px n−r)+ ∑ i=1 m q ix n−σ i =0 under weaker conditions on f i .

Oscillation and stability of nonlinear neutral impulsive delay differential equations

In this paper, oscillation and stability of nonlinear neutral impulsive delay differential equation are studied. The main result of this paper is that oscillation and stability of nonlinear impulsive neutral delay differential equations are equivalent to oscillation and stability of corresponding nonimpulsive neutral delay differential equations. At last, two examples are given to illustrate the importance of this study.

Oscillation of second order nonlinear neutral delay difference equations

In this paper, we obtain sufficient conditions for the oscillation of all solutions of the second order nonlinear delay difference equation Δ[a n( Δ(x n+p nx g(n))) α]+f(n,x σ(n))=0, where Δ x n = x n+1 − x n is the forward difference operator, { p n } is a sequence of nonnegative real numbers and α⩾1 is a quotient of positive odd integers. Our results extend and improve the known results in the literature. An example that dwell up the advantage of our results is included.