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.

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

Abstract and Applied Analysis ◽

10.1155/2014/914264 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11

Author(s):

Li Gao ◽

Quanxin Zhang ◽

Shouhua Liu

Keyword(s):

Time Scales ◽

Sufficient Conditions ◽

Oscillatory Behavior ◽

Dynamic Equations ◽

Riccati Transformation ◽

Third Order ◽

Generalized Riccati Transformation ◽

Delay Dynamic Equations ◽

Inequality Technique ◽

Nonlinear Delay

A class of third-order nonlinear delay dynamic equations on time scales is studied. By using the generalized Riccati transformation and the inequality technique, four new sufficient conditions which ensure that every solution is oscillatory or converges to zero are established. The results obtained essentially improve earlier ones. Some examples are considered to illustrate the main results.

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Oscillation Theorems for Second-Order Half-Linear Neutral Delay Dynamic Equations with Damping on Time Scales

Abstract and Applied Analysis ◽

10.1155/2014/615374 ◽

2014 ◽

Vol 2014 ◽

pp. 1-12

Author(s):

Quanxin Zhang ◽

Shouhua Liu

Keyword(s):

Time Scales ◽

Second Order ◽

Dynamic Equations ◽

Riccati Transformation ◽

Neutral Delay ◽

Oscillation Criteria ◽

Generalized Riccati Transformation ◽

Delay Dynamic Equations ◽

Inequality Technique ◽

Oscillation Theorems

We establish the oscillation criteria of Philos type for second-order half-linear neutral delay dynamic equations with damping on time scales by the generalized Riccati transformation and inequality technique. Our results are new even in the continuous and the discrete cases.

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On the Oscillation for Second-Order Half-Linear Neutral Delay Dynamic Equations on Time Scales

Abstract and Applied Analysis ◽

10.1155/2014/321764 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Quanxin Zhang ◽

Xia Song

Keyword(s):

Time Scales ◽

Second Order ◽

Dynamic Equations ◽

Riccati Transformation ◽

Neutral Delay ◽

Oscillation Criteria ◽

Generalized Riccati Transformation ◽

Delay Dynamic Equations ◽

Inequality Technique

We discuss oscillation criteria for second-order half-linear neutral delay dynamic equations 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 of𝕋=ℝand𝕋=ℤ.

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Oscillation behavior of third-order quasilinear neutral delay dynamic equations on time scales

Filomat ◽

10.2298/fil1407425u ◽

2014 ◽

Vol 28 (7) ◽

pp. 1425-1436 ◽

Cited By ~ 3

Author(s):

Nadide Utku ◽

Mehmet Şenel

Keyword(s):

Sufficient Conditions ◽

Dynamic Equations ◽

Neutral Delay ◽

Third Order ◽

The Third ◽

Delay Dynamic Equation ◽

Integral Averaging Technique ◽

Neutral Delay Dynamic Equation ◽

Generalized Riccati Transformation ◽

Delay Dynamic Equations

The aim of this paper is to give oscillation criteria for the third-order quasilinear neutral delay dynamic equation [r(t)([x(t)+ p(t)x(?0(t))]??)?]? + q1(t)x?(?1(t)) + q2(t)x?(?2(t)) = 0; on a time scale T, where 0 < ? < ? < ?. By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.

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