Second-order Nonlinear Delay Dynamic Equations Research Articles

Oscillation of second-order nonlinear delay dynamic equations on time scales with a nonpositive neutral term

Oscillation of second-order nonlinear delay dynamic equations on time scales with a nonpositive neutral term

Oscillation of second-order nonlinear delay dynamic equations on time scales with a nonpositive neutral term

In this paper, we establish some new oscillation criteria for second- order nonlinear neutral delay dynamic equation of the form (r(t)((m(t)y(t) ? p(t)y(τ(t)))∆)γ )∆ + ƒ (t, y(?(t))) = 0 on a time scale

New oscillation criteria for second-order neutral delay dynamic equations with a non-positive neutral term on time scales

In this paper, we investigate the oscillatory behavior of solutions of a class of secondorder nonlinear neutral delay dynamic equations with non-positive coefficient where α ≥ β are ratios of odd positive integers and z(t) := x(t) - p(t)x(τ (t)). The obtained results guarantee that all solutions are oscillatory. Illustrative examples are provided.

Oscillation criteria for second-order nonlinear delay dynamic equations of neutral type

We investigate oscillatory behavior of solutions to a class of second-order nonlinear neutral delay dynamic equations with nonpositive neutral coefficients. In particular, we study the corresponding noncanonical neutral differential equations. New oscillation criteria are established that complement and improve related contributions to the subject. An example is given to illustrate the main results.

Oscillation Criteria for Second-Order Nonlinear Neutral Delay Dynamic Equations with Damping on Time Scales

In this paper, we discuss oscillation criteria for second-order nonlinear neutral delay dynamic equations with damping on time scales by using the generalized Riccati transformation and the inequality technique. Under certain conditions, we establish four new oscillation criteria. Our results in this paper are new even for the cases of T = R and T = Z.

English

English

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.

Kamenev-type criteria for nonlinear damped dynamic equations

We establish a new Kamenev-type theorem for a class of second-order nonlinear damped delay dynamic equations on a time scale by using the generalized Riccati transformation technique. The criterion obtained improves related contributions to the subject. An example is provided to illustrate assumptions in our theorem are less restrictive.

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.

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.

Some oscillation results for second-order nonlinear delay dynamic equations

This paper is concerned with oscillatory behavior of a class of second-order delay dynamic equations on a time scale. Two new oscillation criteria are presented that improve some known results in the literature. The results obtained are sharp even for the second-order ordinary differential equations.

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.

On The Bounded Oscillation of Certain Second-order Nonlinear Neutral Delay Dynamic Equations with Oscillating Coefficients

We investigate the bounded oscillation of the second-order nonlinear neutral delay dynamic equation with oscillating coefficients \[\Big(r(t)\Big|\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big|^{\alpha-1}\big[x(t)+p(t)x(\tau(t))\big]^\Delta\Big)^\Delta+q(t)|x(t)|^{\beta-1}x(t)=0 \] on an arbitrary time scale $\mathbb{T}$, where $p$ is an oscillating function defined on $\mathbb{T}$ and $\alpha,\beta>0$ are constants, and obtain several sufficient conditions for the oscillation of all bounded solutions of the equation when $\beta>\alpha, \beta=\alpha$ and $\beta<\alpha$, respectively. Our results extend and complement some known results where $p(t)\equiv 0$ and $\alpha,\beta$ are quotients of odd positive integers.

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.

Abstract Differential and Difference Equations

This special issue of Advances in Difference Equations is devoted to highlight some recent developments in abstract differential equations, fractional differential equations, and difference equations and their applications to mathematical physics, engineering, and biology. It consists of 20 papers carefully selected through a rigorous peer review. The first category of papers deals with the asymptotic and oscillatory behavior of solutions to various abstract differential equations and fractional differential equations. Periodic problems involving the scalar p-Laplacian equation on time scales, or n-species nonautonomous food chains with harvesting terms are studied using the Mawhin’s continuation theorem. Some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations and nonlinear delay dynamic equations on a time scale T, are established, improving some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. The study of almost automorphic functions in Bochner’s sense have attracted several mathematicians since the publication of N’Guerekata’s book in 2001. Recently, their applications to fractional differential equations have become an emerging field. A new and general existence and uniqueness theorem of almost automorphic solutions for the semilinear fractional differential equation

Oscillation Criteria for Second-Order Quasilinear Neutral Delay Dynamic Equations on Time Scales

We establish some new oscillation criteria for the second-order quasilinear neutral delay dynamic equations on a time scale , where , . Our results generalize and improve some known results for oscillation of second-order nonlinear delay dynamic equations on time scales. Some examples are considered to illustrate our main results.

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.

Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation $$ [r(t)[y(t) + p(t)y(\tau (t))]^\Delta ]^\Delta + q(t)f(y(\delta (t))) = 0 $$ , on a time scale \( \mathbb{T} \). The results improve some oscillation results for neutral delay dynamic equations and in the special case when \( \mathbb{T} \) = ℝ our results cover and improve the oscillation results for second-order neutral delay differential equations established by Li and Liu [Canad. J. Math., 48 (1996), 871–886]. When \( \mathbb{T} \) = ℕ, our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh [Comp. Math. Appl., 36 (1998), 123–132]. When \( \mathbb{T} \) =hℕ, \( \mathbb{T} \) = {t: t = qk, k ∈ ℕ, q > 1}, \( \mathbb{T} \) = ℕ2 = {t2: t ∈ ℕ}, \( \mathbb{T} \) = \( \mathbb{T}_n \) = {tn = Σk=1n\( \tfrac{1} {k} \), n ∈ ℕ0}, \( \mathbb{T} \) ={t2: t ∈ ℕ}, \( \mathbb{T} \) = {√n: n ∈ ℕ0} and \( \mathbb{T} \) ={\( \sqrt[3]{n} \): n ∈ ℕ0} our results are essentially new. Some examples illustrating our main results are given.