Nonlinear Neutral Delay Dynamic Equations Research Articles

Oscillatory Behavior of Solutions of Third-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

Oscillatory Behavior of Solutions of Third-Order Nonlinear Neutral Delay Dynamic Equations on Time Scales

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 of second-order nonlinear neutral dynamic equations with “Maxima” on time scales

This paper is concerned with the oscillation of solutions of the second order nonlinear neutral delay dynamic equation with “Maxima” of the formon a time scale 𝕋. The obtained results improve and generalize some recent published results. Examples are given to illustrate the main results.

Oscillatory behavior of higher order nonlinear homogeneous neutral delay dynamic equations with positive and negative coefficients

Abstract In this paper, we derive some sufficient conditions for the oscillatory and asymptotic behavior of solutions of the higher order nonlinear neutral delay dynamic equation with positive and negative coefficients. The results of this paper extend and generalize the results of [S. Panigrahi and P. Rami Reddy, Oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20 2013, 143–163] and [S. Panigrahi, J. R. Graef and P. Rami Reddy, Oscillation results for fourth order nonlinear neutral dynamic equations, Commun. Math. Anal. 15 2013, 11–28]. Examples are included to illustrate the validation of the results.

Oscillation of Emden–Fowler type nonlinear neutral delay dynamic equation on time scales

In this paper, we consider the following Emden–Fowler type nonlinear neutral delay dynamic equation of the form $$\begin{aligned} \left( r(t)\left( (y(t)+p(t)y(\tau (t)))^\Delta \right) ^\alpha \right) ^\Delta +q(t)y^\beta (m(t))=0 \end{aligned}$$ on time scales, where $$\alpha $$ and $$\beta $$ are ratios of two odd positive integers. Specially, $$\alpha $$ , $$\beta $$ are arbitrary and without any conditional restrictions, which are completely new compared with previous references. Some new oscillatory and asymptotic properties are obtained by means of inequality technique and Riccati transformation. The results obtained generalize and improve some of the results of Agarwal and Džurina et al.

On the oscillation for n th-order nonlinear neutral delay dynamic equations on time scales

On the oscillation for n th-order nonlinear neutral delay dynamic equations on time scales

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.

Oscillatory behaviour of higher-order nonlinear neutral delay dynamic equations on time scales

In this paper, new sufficient conditions are established for the oscillation of solutions of the higher order dynamic equations [r(t)(z?n-1(t))?]? + q(t) f(x(?(t)))=0, for t ?[t0,?)T, where z(t):= x(t)+ p(t)x(?(t)), n ? 2 is an even integer and ? ? 1 is a quotient of odd positive integers. Under less restrictive assumptions for the neutral coefficient, we employ new comparison theorems and Generalized Riccati technique.

English

English

On the oscillation for third-order nonlinear neutral delay dynamic equations on time scales

In this paper, we consider the third-order nonlinear neutral delay dynamic equations $$\begin{aligned} \left( b(t)\left( \left( ((x(t)-p(t)x(\tau (t)))^\Delta )^{\alpha _1}\right) ^\Delta \right) ^{\alpha _2}\right) ^\Delta +f(t,x(\delta (t)))=0 \end{aligned}$$ on a time scale \(\mathbb {T}\), where \(\alpha _i\) are quotients of positive odd integers, \(i=1\), 2, \(|f(t,u)|\ge q(t)|u|\), \(b,\ p\) and q are real-valued positive rd-continuous functions defined on \(\mathbb {T}\). By using the Riccati transformation technique and integral averaging technique, some new sufficient conditions which ensure that every solution oscillates or tends to zero are established. Our results are new for third-order nonlinear neutral delay dynamic equations and extend many known results for oscillation of third order dynamic equations. Some examples are given here to illustrate our main results.

A Philos-type theorem for second-order neutral delay dynamic equations with damping

We establish a Philos-type oscillation theorem for a class of nonlinear second-order neutral delay dynamic equations with damping on a time scale by using the Riccati transformation and integral averaging technique. An illustrative example is provided to show that our theorem has practicability and maneuverability.

Generalized Philos-type oscillation criteria for second order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation w(s) and showing that [H(σ(t),s)w(s)]Δs≤0, we establish new Philos-type oscillation criteria for second order nonlinear dynamic equations on time scales. The obtained results essentially generalize and improve the well-known oscillation results for half-linear dynamic equations such as Philos-type and Kamenev-type oscillation criteria. We illustrate the versatility of our results by means of examples.

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.

Oscillation of fourth-order nonlinear neutral delay dynamic equations

In this work we establish some new sucient conditions for oscillation of fourth-order nonlinear neutral delay dynamic equations of the form (a(t)([x(t) p(t)x(h(t))] ) ) +q(t)x (g(t)) = 0; t2 [t0;1)T; where and are quotients of positive odd integers with .

On oscillatory and asymptotic behavior of fourth order nonlinear neutral delay dynamic equations with positive and negative coefficients

Abstract In this paper, oscillatory and asymptotic properties of solutions of nonlinear fourth order neutral dynamic equations of the form $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = 0(H)$ and $(r(t)(y(t) + p(t)y(\alpha _1 (t)))^{\Delta ^2 } )^{\Delta ^2 } + q(t)G(y(\alpha _2 (t))) - h(t)H(y(\alpha _3 (t))) = f(t),(NH)$ are studied on a time scale $\mathbb{T}$ under the assumption that $\int\limits_{t_0 }^\infty {\tfrac{t} {{r(t)}}\Delta t = \infty } $ and for various ranges of p(t). In addition, sufficient conditions are obtained for the existence of bounded positive solutions of the equation (NH) by using Krasnosel’skii’s fixed point theorem.

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

The oscillation for certain third-order nonlinear neutral delay dynamic equations on time scales is discussed in this article. By using the generalized Riccati transformation and the inequality technique, three new different sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially generalize and improve earlier ones.

Necessary and sufficient conditions on the asymptotic behavior of second‐order neutral delay dynamic equations with positive and negative coefficients

In this paper, we establish necessary and sufficient conditions for the solutions of a second‐order nonlinear neutral delay dynamic equation with positive and negative coefficients to be oscillatory or tend to zero asymptotically. We consider three different ranges of the coefficient associated with the neutral part in one of which it is allowed to be oscillatory. Thus, our results improve and generalize the existing results in the literature to arbitrary time scales. Some examples on nontrivial time scales are also given. Copyright © 2013 John Wiley & Sons, Ltd.

On oscillatory and asymptotic behavior of fourth order non-linear neutral delay dynamic equations

In this paper, oscillatory and asymptotic property of solutions of a class of nonlinear fourth order neutral dynamic equations of the form ( H ) ( r ( t ) ( y ( t ) + p ( t ) y ( α ( t ) ) ) Δ 2 ) Δ 2 + q ( t ) G ( y ( β ( t ) ) ) = 0 and ( N H ) ( r ( t ) ( y ( t ) + p ( t ) y ( α ( t ) ) ) Δ 2 ) Δ 2 + q ( t ) G ( y ( β ( t ) ) ) = f ( t ) for t ∈ [ t 0 , ∞ ] T , t 0 ⩾ 0 , where T is a time scale such that sup T = ∞ , t 0 ∈ T have been studied under the assumption ∫ t 0 ∞ t r ( t ) Δ t = ∞ for various ranges of p ( t ) . Sufficient conditions are obtained for the existence of bounded positive solutions of ( N H ) by using Schauder’s fixed point theorem.