Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales

2011 ◽  
Vol 54 (4) ◽  
pp. 580-592 ◽  
Author(s):  
Jia Baoguo ◽  
Lynn Erbe ◽  
Allan Peterson
Keyword(s):  
Difference Equation ◽  
Dynamic Equation ◽  
Time Scale ◽  
Nonlinear Function ◽  
Second Order ◽  
Dynamic Equations ◽  
Coefficient Function ◽  
The Difference ◽  
Oscillation Theorems ◽  
Type Oscillation

AbstractConsider the second order superlinear dynamic equationwhere p ∈ C(, ℝ), is a time scale, ƒ : ℝ → ℝ is continuously differentiable and satisfies ƒ ′(x) > 0, and x ƒ (x) > 0 for x ≠ 0. Furthermore, f (x) also satisfies a superlinear condition, which includes the nonlinear function ƒ (x) = xα with α > 1, commonly known as the Emden–Fowler case. Here the coefficient function p(t) is allowed to be negative for arbitrarily large values of t. In addition to extending the result of Kiguradze for (∗) in the real case = ℝ, we obtain analogues in the difference equation and q-difference equation cases.

scholarly journals Oscillation Theorems for Second-Order Half-Linear Advanced Dynamic Equations on Time Scales

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Shuhong Tang ◽  
Tongxing Li ◽  
Ethiraju Thandapani
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Dynamic Equations ◽  
Oscillation Theorems

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation on an arbitrary time scale with sup , where and . Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify the oscillation of the second-order half-linear advanced differential equation and the second-order half-linear advanced difference equation. Three examples are included to illustrate the main results.

scholarly journals Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1897
Author(s):  
Taher S. Hassan ◽  
Yuangong Sun ◽  
Amir Abdel Menaem
Keyword(s):  
Dynamic Equation ◽  
Time Scale ◽  
Nonlinear Dynamic ◽  
Recent Literature ◽  
Second Order ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Arbitrary Time ◽  
Nonlinear Dynamic Equations ◽  
Type Oscillation

In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results confirm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the finding.

scholarly journals New Oscillation Results of Second-Order Damped Dynamic Equations withp-Laplacian on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Yang-Cong Qiu ◽  
Qi-Ru Wang
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Second Order ◽  
Dynamic Equations ◽  
Coefficient Function ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Function Classes ◽  
Generalized Riccati Technique

By employing a generalized Riccati technique and functions in some function classes for integral averaging, we derive new oscillation criteria of second-order damped dynamic equation withp-Laplacian on time scales of the form(rtφγ(xΔ(t)))Δ+ptφγ(xΔ(t))+f(t,x(g(t)))=0, where the coefficient functionp(t)may change sign. Two examples are given to demonstrate the obtained results.

scholarly journals On the Oscillation of Second Order Nonlinear Neutral Delay Dynamic Equations

2007 ◽  
Vol 14 (4) ◽  
pp. 597-606
Author(s):  
Hassan A. Agwo
Keyword(s):  
Dynamic Equation ◽  
Time Scale ◽  
Second Order ◽  
Dynamic Equations ◽  
Sufficient Condition ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Delay Dynamic Equation ◽  
Neutral Delay Dynamic Equation ◽  
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.

scholarly journals Oscillation Criteria of Second-Order Mixed Neutral Delay Dynamic Equations on Time Scales

2018 ◽  
Vol 228 ◽  
pp. 01006
Author(s):  
L M Feng ◽  
Y G Zhao ◽  
Y L Shi ◽  
Z L Han
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Second Order ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Dynamic Equation ◽  
Delay Dynamic Equations ◽  
Oscillation Theorems

In this artical, we consider a second-order neutral dynamic equation on a time scales. A number of oscillation theorems are shown that supplement and extend some known results in the eassay.

scholarly journals Amended Criteria of Oscillation for Nonlinear Functional Dynamic Equations of Second-Order

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1191
Author(s):  
Taher S. Hassan ◽  
Rami Ahmad El-Nabulsi ◽  
Amir Abdel Menaem
Keyword(s):  
Time Scale ◽  
Averaging Method ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Nonlinear Functional ◽  
Oscillation Criteria ◽  
Arbitrary Time ◽  
Type Oscillation

In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second-order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The obtained results demonstrate an improvement in Hille-type compared with the results reported in the literature. Some examples are provided to illustrate the significance of the obtained results.

scholarly journals Oscillation Behavior of a Class of Second-Order Dynamic Equations with Damping on Time Scales

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Weisong Chen ◽  
Zhenlai Han ◽  
Shurong Sun ◽  
Tongxing Li
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Nonlinear Dynamic Equation ◽  
Oscillation Theorems

By using a Riccati transformation and inequality, we present some new oscillation theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.

Oscillation criteria for quasi-linear neutral delay dynamic equations on time scale

2016 ◽  
Vol 66 (1) ◽  
Author(s):  
Qiaoshun Yang ◽  
Zhiting Xu ◽  
Ping Long
Keyword(s):  
Dynamic Equation ◽  
Time Scale ◽  
Second Order ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Dynamic Equation ◽  
Delay Dynamic Equations

AbstractIn this paper, we consider the oscillation for the second-order quasi-linear neutral dynamic equationon time scale 𝕋, where

scholarly journals A telescoping principle for oscillation of the second order half-linear dynamic equations on time scales

2009 ◽  
Vol 43 (1) ◽  
pp. 243-255
Author(s):  
Jiří Vítovec
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Second Order ◽  
Dynamic Equations ◽  
Linear Dynamic

Abstract . We establish the so-called “telescoping principle” for oscillation of the second order half-linear dynamic equation [r(t)Φ(x<sup>Δ</sup>)]<sup>Δ</sup> + c(t)Φ(x<sup>σ</sup>) = 0 on a time scale. This principle provides a method enabling us to construct many new oscillatory equations. Unlike previous works concerning the telescoping principle, we formulate some oscillation results under the weaker assumption r(t) ≠ 0 (instead r(t) > 0).

scholarly journals Oscillation Criteria for Second Order Impulsive Delay Dynamic Equations on Time Scale

2021 ◽  
Vol 45 (4) ◽  
pp. 531-542
Author(s):  
GOKULA NANDA CHHATRIA ◽  
Keyword(s):  
Time Scale ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Oscillation Criteria ◽  
Delay Dynamic Equations ◽  
Impulsive Delay

In this work, we study the oscillation of a kind of second order impulsive delay dynamic equations on time scale by using impulsive inequality and Riccati transformation technique. Some examples are given to illustrate our main results.

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