Second-order Nonlinear Delay Differential Equation Research Articles

Oscillation criteria for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term

We obtain oscillation conditions for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term. To cope with non-canonical types of equations, we propose new oscillation criteria for the main equation when the neutral coefficient does not satisfy any of the conditions that call it to either converge to \(0\) or \(\infty\). Our approach differs from others in that we first turn into the non-canonical equation to a canonical form and as a result, we only require one condition to weed out non-oscillatory solutions in order to induce oscillation. The conclusions made here are new and have been condensed significantly from those found in the literature. For the sake of confirmation, we provide examplesthat cannot be included in earlier works.

Nonoscillation and Oscillation Criteria for a Class of Second-Order Nonlinear Neutral Delay Differential Equations with Positive and Negative Coefficients

In this paper, we investigate some nonoscillatory and oscillatory solutions for a class of second-order nonlinear neutral delay differential equations with positive and negative coefficients. By means of the method of contraction mapping principle and some integral inequality techniques, we extend the recent results provided in the literature.

Asymptotic Stability of Solutions to a Class of Second-Order Delay Differential Equations

We consider a class of second-order nonlinear delay differential equations with periodic coefficients in linear terms. We obtain conditions under which the zero solution is asymptotically stable. Estimates for attraction sets and decay rates of solutions at infinity are established. This class of equations includes the equation of vibrations of the inverted pendulum, the suspension point of which performs arbitrary periodic oscillations along the vertical line.

Oscillation of second-order mixed nonlinear neutral delay differential equation

In this paper, we study oscillatory behavior of a class of second-order mixed nonlinear neutral delay differential equation. By using of extended Riccati transformation, integral averaging and inequality technique, a new oscillation criteria of the equation were established, which had extended the results in the recent literatures.

On oscillatory second-order nonlinear delay differential equations of neutral type

On oscillatory second-order nonlinear delay differential equations of neutral type

Interval oscillation criteria for second-order nonlinear delay differential equations with damping

Interval oscillation criteria for second-order nonlinear delay differential equations with damping

Oscillation theorems for second-order nonlinear delay differential equations of neutral type

In this paper, new sufficient conditions are obtained for oscillation of second-order neutral delay differential equations of the form \[\frac{d}{dt}\bigg[r(t)\frac{d}{dt}[x(t)+p(t)x(\tau(t))]\bigg]+q(t)G\bigl(x(\sigma(t))\bigr)=0\: for\: t\geq t_{0},\] under the assumptions $\int^{\infty}\frac{1}{r(\eta)}d\eta=\infty$ and $\int^{\infty}\frac{1}{r(\eta)}d\eta<\infty$ for various ranges of the bounded neutral coefficient $p$. Unlike most of the previous results, $\tau^{\prime}$ is allowed to be oscillatory. Further, some illustrative examples showing applicability of the new results are included.

Stability and boundedness of solutions of certain nonlinear delay differential equations of second order

In this paper, we give some sufficient conditions to guarantee the asymptotic stability and uniform boundedness of certain vector second-order nonlinear delay differential equations with a continuous deviating argument by using a Lyapunov function as basic tool. In doing so we extends some of the existing results in the literature.

Interval Oscillation Criteria for Forced Second-Order Nonlinear Delay Dynamic Equations with Damping and Oscillatory Potential on Time Scales

We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the formrtg1xt,xΔtΔ+p(t)g2(x(t),xΔ(t))xΔ(t)+q(t)f(x(τ(t)))=e(t), on a time scaleT. We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our results.

On the Oscillation of Second-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Based on Riccati transformation and the inequality technique, we establish some new sufficient conditions for oscillation of the second-order neutral 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. At the end of this paper, we give an example to illustrate the main results.

Asymptotic integration of second-order nonlinear delay differential equations

Abstract We study the asymptotic integration problem for second-order nonlinear delay differential equations of the form ( p ( t ) x ′ ( t ) ) ′ + q ( t ) x ( t ) = f ( t , x ( g ( t ) ) ) . It is shown that if u and v are principal and nonprincipal solutions of equation ( p ( t ) x ′ ) ′ + q ( t ) x = 0 , then there are solutions x 1 ( t ) and x 2 ( t ) of the above nonlinear equation such that x 1 ( t ) = a u ( t ) + o ( u ( t ) ) , t → ∞ and x 2 ( t ) = b v ( t ) + o ( v ( t ) ) , t → ∞ .

Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients

We present several oscillation criteria for a second-order nonlinear delay differential equation with a nonpositive neutral coefficient. Two examples are given to illustrate the main results.

BOUNDED OSCILLATION FOR SECOND-ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

Two necessary and sufficient conditions for the oscillation of the bounded solutions of the second-order nonlinear delay differential equation <TEX>$$(a(t)x^{\prime}(t))^{\prime}+q(t)f(x[{\tau}(t)</TEX><TEX>]</TEX><TEX>)=0$$</TEX> are obtained by constructing the sequence of functions and using inequality technique.

Non-oscillatory solutions for a nonlinear neutral delay differential equation

We study the existence of non-oscillatory bounded solutions for a second-order nonlinear neutral delay differential equation. Some sufficient conditions for the existence of non-oscillatory bounded solutions with positive lower bound or negative upper bound are established, and the non-oscillatory solutions with decay as t→∞ are discussed.

Oscillation Theorems for Second-Order Nonlinear Neutral Delay Differential Equations

We analyze the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations. Our theorems improve a number of related results reported in the literature.

New Oscillation Criteria of Second-Order Nonlinear Delay Dynamic Equations on Time Scales

By using the generalized Riccati transformation and the integral averaging technique, the paper establishes some new oscillation criteria for the second-order nonlinear delay dynamic equations on time scales. The results in this paper unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation on time scales. The Theorems in this paper are new even in the continuous and the discrete cases.

Oscillation of Second-Order Nonlinear Delay Dynamic Equations with Damping on Time Scales

We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scaleT(r(t)g(x(t),xΔ(t)))Δ+p(t)g(x(t),xΔ(t)) + q(t)f(x(τ(t)))=0,wherer(t),p(t), andq(t)are positive right dense continuous (rd-continuous) functions onT. Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results.

Stability and bifurcation analysis in the delay-coupled nonlinear oscillators

This paper investigates the dynamical behavior of two oscillators with nonlinearity terms, which are coupled with finite delay parameters. Each oscillator is a general class of second-order nonlinear delay-differential equations. The system of delay differential equations is analyzed by reducing the delay equations to a system of ordinary differential equations on a finite-dimensional center manifold, the corresponding to an infinite-dimensional phase space. In addition, the characteristic equation for the linear stability of the trivial equilibrium is completely analyzed and the stability region is illustrated in the parameters space. Our analysis reveals necessary coefficients of the reduced vector field on the center manifold for studying the bifurcations of the trivial equilibrium such as transcritical, pitchfork, and Hopf bifurcation. Finally, we consider the delay-coupled van der Pol equations.

Asymptotic behavior of solutions of nonlinear delay differential equations with impulse

This paper studies the asymptotic behavior of solutions of the second-order nonlinear delay differential equations with impulses: and some sufficient conditions are obtained. Key words: Asymptotic behavior, second-order nonlinear delay differential equation, impulses.

ON THE OSCILLATION OF SECOND-ORDER NONLINEAR DELAY DYNAMIC EQUATIONS ON TIME SCALES

By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criterion for the second-order nonlinear delay dynamic equations on a time scale , here is the ratio of two positive odd integers with and real-valued positive right-dense continuous functions defined on . Our results not only extend and improve some known results, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.