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

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Hassan A. Agwa ◽  
Ahmed M. M. Khodier ◽  
Heba A. Hassan
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Oscillatory Potential ◽  
Delay Difference Equation ◽  
Riccati Technique ◽  
Oscillation Criteria ◽  
Nonlinear Dynamic Equation ◽  
Interval Oscillation ◽  
Delay Differential ◽  
Nonlinear Delay

We are concerned with the interval oscillation of general type of forced second-order nonlinear dynamic equation with oscillatory potential of the formrtg1xt,xΔtΔ+p(t)g2(x(t),xΔ(t))xΔ(t)+q(t)f(x(τ(t)))=e(t), on a time scaleT. We will use a unified approach on time scales and employ the Riccati technique to establish some oscillation criteria for this type of equations. Our results are more general and extend the oscillation criteria of Erbe et al. (2010). Also our results unify the oscillation of the forced second-order nonlinear delay differential equation and the forced second-order nonlinear delay difference equation. Finally, we give some examples to illustrate our 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 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 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 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 of Second-Order Nonlinear Delay Dynamic Equations with Damping on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
H. A. Agwa ◽  
Ahmed M. M. Khodier ◽  
Heba A. Hassan
Keyword(s):  
Continuous Functions ◽  
Second Order ◽  
Delay Difference Equation ◽  
Generalized Riccati Transformation ◽  
Delay Dynamic Equations ◽  
Inequality Technique ◽  
Delay Differential ◽  
Delay Difference ◽  
Nonlinear Delay Differential Equation ◽  
Nonlinear Delay

We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scaleT(r(t)g(x(t),xΔ(t)))Δ+p(t)g(x(t),xΔ(t)) + q(t)f(x(τ(t)))=0,wherer(t),p(t), andq(t)are positive right dense continuous (rd-continuous) functions onT. Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results.

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