scholarly journals Interval Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Yang-Cong Qiu ◽  
Qi-Ru Wang
Keyword(s):  
Time Scales ◽  
Nonlinear Dynamic ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Function Classes ◽  
Nonlinear Dynamic Equations ◽  
Generalized Riccati Technique ◽  
Interval Oscillation

Using functions in some function classes and a generalized Riccati technique, we establish interval oscillation criteria for second-order nonlinear dynamic equations on time scales of the form(p(t)ψ(x(t))xΔ(t))Δ+f(t,x(σ(t)))=0. The obtained interval oscillation criteria can be applied to equations with a forcing term. An example is included to show the significance of the results.

scholarly journals Kamenev-Type Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Yang-Cong Qiu ◽  
Qi-Ru Wang
Keyword(s):  
Time Scales ◽  
Nonlinear Dynamic ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Function Classes ◽  
Nonlinear Dynamic Equations ◽  
Generalized Riccati Technique ◽  
Type Oscillation

Using functions from some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the form(p(t)ψ(x(t))k∘xΔ(t))Δ+f(t,x(σ(t)))=0. Two examples are included to show the significance of the results.

scholarly journals Oscillation Criteria for Second-Order Nonlinear Dynamic Equations on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Shao-Yan Zhang ◽  
Qi-Ru Wang
Keyword(s):  
Time Scales ◽  
Nonlinear Dynamic ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Nonlinear Dynamic Equations ◽  
Generalized Riccati Technique ◽  
Averaging Techniques

This paper is concerned with oscillation of second-order nonlinear dynamic equations of the form on time scales. By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation criteria which handle some cases not covered by known criteria.

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 Interval Oscillation Criteria for Second-Order Forced Functional Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shao-Yan Zhang ◽  
Qi-Ru Wang
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Generalized Riccati Technique ◽  
Interval Oscillation ◽  
Averaging Techniques

This paper is concerned with oscillation of second-order forced functional dynamic equations of the form(r(t)(xΔ(t))γ)Δ+∑i=0n‍qi(t)|x(δi(t))|αisgn  x(δi(t))=e(t)on time scales. By using a generalized Riccati technique and integral averaging techniques, we establish new oscillation criteria which handle some cases not covered by known criteria.

scholarly journals Oscillation Criteria of Second-Order Dynamic Equations with Damping on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yang-Cong Qiu ◽  
Qi-Ru Wang
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Function Classes ◽  
Generalized Riccati Technique ◽  
Type Oscillation

Using functions in some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order dynamic equations with damping on time scales of the form(r(t)(xΔ(t))γ)Δ+p(t)(xΔ(t)γ)+f(t,x(g(t)))=0. Two examples are included to show the significance of the results.

scholarly journals Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yibing Sun ◽  
Zhenlai Han ◽  
Shurong Sun ◽  
Chao Zhang
Keyword(s):  
Positive Integer ◽  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Oscillation Criteria ◽  
Nonlinear Dynamic Equations ◽  
Interval Oscillation ◽  
Forced Term

By using the Riccati transformation technique and constructing a class of Philos-type functions on time scales, we establish some new interval oscillation criteria for the second-order damped nonlinear dynamic equations with forced term of the form(r(t)xΔ(t))Δ+p(t)xΔσ(t)+q(t)(xσ(t))α=F(t,xσ(t))on a time scale𝕋which is unbounded, whereαis a quotient of odd positive integer. Our results in this paper extend and improve some known results. Some examples are given here to illustrate our main results.

scholarly journals New Oscillatory Behavior of Second-Order Nonlinear Dynamic Equations with Damping on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Shouhua Liu ◽  
Quanxin Zhang
Keyword(s):  
Time Scales ◽  
Nonlinear Dynamic ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Nonlinear Dynamic Equations

We establish four new oscillation criteria of Grace-type for the second-order nonlinear dynamic equations with damping. These criteria extend known criteria for corresponding dynamic equations. Our results are new even in the continuous and the discrete cases.

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