Second-order Nonlinear Delay Differential Equation Research Articles

Oscillation of second-order nonlinear delay dynamic equations on time scales

By using the generalized Riccati transformation and the inequality technique, we establish one new oscillation criterion for the second-order nonlinear delay dynamic equations on time scales. Our results not only extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales.

Oscillation of second-order nonlinear delay dynamic equations with damping on time scales

By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations with damping on a time scale. Our results extend and improve some known theorems, but also unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping.

Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

Some oscillation criteria for the second order nonlinear neutral delay differential equations[a(t)x(t)+∑li=1ci(t)x(t-τi(t))]″+∑mi=1pi(t)fi(x(t-δi(t))) -∑ni=1qi(t)gi(x(t-σi(t)))=0,t0.(E)are established in this paper.we give sufficient conditions for the oscillation of all soiutions.And the results of some references have been extended.

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.

On the existence of positive solutions on the half-line to nonlinear two-dimensional delay differential systems

AbstractThis paper is concerned with a boundary-value problem on the half-line for nonlinear two-dimensional delay differential systems with positive delays. A theorem is established, which provides sufficient conditions for the existence of positive solutions. The application of this theorem to the special case of second-order nonlinear delay differential equations is given. Also, the application of the theorem to two-dimensional Emden–Fowler-type delay differential systems with constant delays is presented. Moreover, some general examples demonstrating the applicability of the theorem are included.

Asymptotic behavior of solutions of second-order nonlinear delay differential equations with impulses

This paper is concerned with second-order nonlinear delay differential equations with impulses of the form ( r ( t ) x ′ ( t ) ) ′ + p ( t ) x ′ ( t ) + f ( t , x ( t − δ ) ) = 0 , t ≠ t k , x ′ ( t k + ) = I k ( x ′ ( t k ) ) , x ( t k + ) = J k ( x ( t k ) ) , t = t k , k = 1 , 2 , … , t ≥ t 0 . By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear delay differential equations with impulses are obtained. Some examples are also inserted to illustrate the impulsive effect.

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.

Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations on a time scale ; here is a quotient of odd positive integers with and real-valued positive rd-continuous functions defined on . Our results not only extend some results established by Hassan in 2008 but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.

Bounded oscillation criteria for second-order nonlinear delay difference equations of unstable type

This paper deals with some sufficient conditions for the bounded oscillation criteria for the second-order nonlinear delay difference equation Δ2xn=∑i=1mpixn−ki+f(n,xn−l1,…,xn−ls),n=0,1,2,…. In addition, we generalize and improve the existing results, and then give some examples of applications.

Existence and Uniqueness of Solutions for a Second-Order Delay Differential Equation Boundary Value Problem on the Half-Line

This paper is concerned with the existence and uniqueness of solutions for the second-order nonlinear delay differential equations. By the use of the Schauder fixed point theorem, the existence of the solutions on the half-line is derived. Via the Banach contraction principle, another result concerning the existence and uniqueness of solutions on the half-line is established. The main results in this paper extend some of the existing literatures.

Oscillation by impulses for a second-order delay differential equation

We consider a certain second-order nonlinear delay differential equation and prove that the all solutions oscillate when proper impulse controls are imposed. An example is given.

Bounded oscillation for second-order nonlinear delay differential equation in a critical state

In this paper we establish sufficient conditions for bounded oscillation of the second-order nonlinear delay differential equation x ″ ( t ) = ∑ i = 1 m p i x ( t - τ i ) + f ( t , x ( t - σ 1 ( t ) ) , … , x ( t - σ n ( t ) ) ) under a critical state by the second-order ordinary differential equation y ″ ( t ) + ( 1 - ε ) e - λ 0 t f ( t , e λ 0 ( t - σ 1 ( t ) ) y ( t ) , … , e λ 0 ( t - σ n ( t ) ) y ( t ) ) = 0 . At the same time, necessary conditions for the above bounded oscillation equation are obtained.

The stability problem in the Kaldor–Kalecki business cycle model

Abstract We consider the Kaldor–Kalecki model of the business cycle which is the modified Kaldor model with the Kalecki time delay in investment. The model is formulated in terms of a second-order nonlinear delay differential equation with a negative feedback. We investigate the problem of stability of cycles caused by retarded action. The method of a centre manifold is used to find the conditions for the Hopf bifurcation. The conditions for stability of limit cycles on the centre manifold is given.

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.

Oscillation criteria for second-order impulsive delay difference equations

We investigate the oscillation of a kind of very extensively studied second-order nonlinear delay difference equations under impulsive perturbations, some interesting results are obtained, and some examples which illustrate that impulses play a very important role in giving rise to the oscillations of equations are also included.

Oscillation theorems for second-order nonlinear delay difference equations

By means of Riccati transformation technique, we establish some new oscillation criteria for second-order nonlinear delay difference equation $$\Delta (p_n (\Delta x_n )^\gamma ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;\;n = 0,1,2,...,$$ when \(\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }\). When \(\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} < \infty }\) we present some sufficient conditions which guarantee that, every solution oscillates or converges to zero. When \(\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }\) holds, our results do not require the nonlinearity to be nondecreasing and are thus applicable to new classes of equations to which most previously known results are not.

Oscillation criteria for second-order nonlinear differential equations with impulses

The present paper is devoted to the investigation of the oscillatory behavior of a kind of extensively studied second-order nonlinear delay differential equations with impulses. Some interesting results are obtained, which illustrate that impulses play a very important role in giving rise to the oscillations of equations.

Oscillation Criteria for Certain Nonlinear Differential Equations

We present new oscillation criteria for certain classes of second-order nonlinear differential equations and delay differential equations obtained by using an integral averaging technique. Our theorems complement a number of existing results and handle the cases which are not covered by known criteria.