Second-order Neutral Differential Equations Research Articles

Homoclinic orbits for a singular second-order neutral differential equation

Using critical point theory and an approximation method, we study the existence of homoclinic orbits for a singular second-order neutral differential difference equations.

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).

Oscillations of Second-Order Neutral Impulsive Differential Equations

Necessary and sufficient conditions are established for oscillation of second-order neutral impulsive differential equation , , where the coefficients ; , and .

Existence of Periodic Solutions for a Class of Second-Order Neutral Differential Equations with Multiple Deviating Arguments

Usando el teorema del punto fijo de Kranoselskii y el teorema de continuacion de Mawhin establecemos la existencia de soluciones periodicas de una ecuacion diferencial neutral de segundo orden con argumento de desviacion multiple.

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.

Asymptotic properties of solutions of second-order neutral difference equations

In this paper, we study asymptotic behavior of solutions of second-order neutral difference equation Δ 2 ( x n + px n - k ) + f ( n , x n ) = 0 . We present conditions under which all nonoscillatory solutions of the above equation have the property x n = cn + o ( n ) for some c ∈ R as well as sufficient conditions under which all nonoscillatory solutions are asymptotically linear.

Oscillation and Nonoscillation of Second Order Neutral Delay Difference Equations

Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation $$\Delta \left( {c_n \Delta \left( {y_n {\text{ + }}p_n y_n - k} \right)} \right) + q_n y_{n + 1 - m}^\beta = 0,n \geqslant n_0 $$

Oscillatory of unstable type second-order neutral difference equations

We consider the problem of oscillation and nonoscillation solutions for unstable type second-order neutral difference equation : Δ2(x(n) - p(n)x(n - τ)) = q(n)x(g(n)).In this paper, we obtain some conditions for the bounded solutions of Eq(1) to be oscillatory and for the existence of the nonoscillatory solutions.

Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima

In this paper, the asymptotic behavior of the non-oscillatory solutions of neutral difference equations with maxima is considered. Sufficient conditions for oscillation of all solutions are also obtained.

Oscillatory behavior of the second-order nonlinear neutral difference equations

In this paper, we consider the oscillation of the second-order neutral difference equation $$\Delta ^2 \left( {x_n - px_{n - \tau } } \right) + q_n f\left( {x_{n - \sigma _n } } \right) = 0$$ as well as the oscillatory behavior of the corresponding ordinary difference equation $$\Delta ^2 z_n + q_n f\left( {R\left( {n,\lambda } \right)z_n } \right) = 0$$ .

Existence and asymptotic behaviour of nonoscillatory solutions of second-order neutral differential equations with “maxima”

Bainov, D.D., V. Petrov and V. Proytcheva, Existence and asymptotic behaviour of nonoscillatory solutions of second order neutral differential equations with “maxima”, Journal of Computational and Applied Mathematics. In the paper, sufficient conditions for existence of positive and negative solutions of neutral equations with “maxima” are obtained and their asymptotic behaviour is investigated.

Asymptotic Properties of the Nonoscillatory Solutions of Second-Order Neutral Equations with a Deviating Argument

In the paper second-order linear neutral differential equations with a distributed delay are considered. The asymptotic properties of their nonoscillatory solutions are investigated.