Nonlinear Neutral Delay Dynamic Equations Research Articles

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.

Some oscillation results for second order nonlinear dynamic equations of neutral type

In this paper, some oscillation results for the second order nonlinear neutral delay dynamic equation ( r ( t ) ( ( y ( t ) + p ( t ) y ( t − τ ) ) Δ ) γ ) Δ + q ( t ) | y ( t − δ ) | γ sgn y ( t − δ ) = 0 are obtained on a time scale T , under the assumption ∫ t 0 ∞ ( 1 r ( t ) ) 1 γ Δ t = ∞ , where γ > 0 is a quotient of odd positive integers.

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.

On the Oscillation of Second Order Nonlinear Neutral Delay Dynamic Equations

Abstract In this paper we obtain some new oscillation criteria for the second order nonlinear neutral delay dynamic equation (𝑥(𝑡) – 𝑝(𝑡)𝑥(𝑡 – τ 1))ΔΔ + 𝑞(𝑡)𝑓(𝑥(𝑡 – τ 2)) = 0, on a time scale 𝕋. Moreover, a new sufficient condition for the oscillation sublinear equation (𝑥(𝑡) – 𝑝(𝑡)𝑥(𝑡 – τ 1))″ + 𝑞(𝑡)𝑓(𝑥(𝑡 – τ 2)) = 0, is presented, which improves other conditions and an example is given to illustrate our result.

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

In this article, we consider the second-order nonlinear neutral delay dynamic equation on a time scale and establish some new oscillation and nonoscillation criteria. Also from these we deduce the Leighton–Wintner, Hille–Kneser, Kamenev, and Philos types oscillation criteria. Our results are different and complement the existence oscillation results for neutral delay dynamic equations on time scales in (Agarwal et al. 2004, Oscillation criteria for second-order nonlinear neutral dynamic equations. Journal of Mathematical Analysis and Applications, 300, 203–217) and (S.H. Saker, 2006, Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. Journal of Computational and Applied Mathematics, 187, 123–141). Our results can be applied on the time scales , , , for h > 0, , , , and when where {t n } is the set of harmonic numbers, etc. Some examples are considered to illustrate the main results.

Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations

This paper is concerned with oscillation of second-order nonlinear neutral dynamic equations on time scales with a variable delay. By using the generalized Riccati technique and integral averaging technique, new oscillation criteria are obtained for all solutions of the equation. Some results extend known results for difference equations when the time scale is the set Z + of positive integers and for differential equations when the time scale is R . Several examples are given to illustrate the results of the paper.

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

In this paper, some sufficient conditions for oscillation of the second-order nonlinear neutral delay dynamic equation ( r ( t ) ( [ y ( t ) + p ( t ) y ( t - τ ) ] Δ ) γ ) Δ + f ( t , y ( t - δ ) ) = 0 , on a time scale T are established; here γ ⩾ 1 is an odd positive integer with r ( t ) and p ( t ) are rd -continuous functions defined on T . Our results as a special case when T = R and T = N , involve and improve some well-known oscillation results for second-order neutral delay differential and difference equations. When T = h N and T = q N = { t : t = q k , k ∈ N , q > 1 } , i.e., for generalized neutral delay difference and q -neutral delay difference equations our results are essentially new and also can be applied on different types of time scales, e.g., T = N 2 = { t 2 : t ∈ N } and T = T n = { t n : n ∈ N 0 } where { t n } is the set of harmonic numbers. Some examples illustrating our main results are given.

Oscillation criteria for second-order nonlinear neutral delay dynamic equations

In this paper we will establish some oscillation criteria for the second-order nonlinear neutral delay dynamic equation ( r ( t ) ( ( y ( t ) + p ( t ) y ( t − τ ) ) Δ ) γ ) Δ + f ( t , y ( t − δ ) ) = 0 on a time scale T ; here γ > 0 is a quotient of odd positive integers with r ( t ) and p ( t ) real-valued positive functions defined on T . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales, so this paper initiates the study.