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

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1867
Author(s):  
Ya-Ru Zhu ◽  
Zhong-Xuan Mao ◽  
Shi-Pu Liu ◽  
Jing-Feng Tian
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Scaling Technique ◽  
Oscillation Criteria ◽  
Nonlinear Dynamic Equation ◽  
Generalized Riccati Transformation

In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria.

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.

scholarly journals Kamenev-Type Oscillation Criteria for the Second-Order Nonlinear Dynamic Equations with Damping on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
M. Tamer Şenel
Keyword(s):  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Sufficient Conditions ◽  
Second Order ◽  
Oscillatory Solutions ◽  
Oscillation Criteria ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Dynamic Equations ◽  
Generalized Riccati Transformation ◽  
Type Oscillation

The oscillation of solutions of the second-order nonlinear dynamic equation(r(t)(xΔ(t))γ)Δ+p(t)(xΔ(t))γ+f(t,x(g(t)))=0, with damping on an arbitrary time scaleT, is investigated. The generalized Riccati transformation is applied for the study of the Kamenev-type oscillation criteria for this nonlinear dynamic equation. Several new sufficient conditions for oscillatory solutions of this equation are obtained.

scholarly journals Oscillation Criteria for Functional Dynamic Equations with Nonlinearities Given by Riemann-Stieltjes Integral

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yuangong Sun ◽  
Taher S. Hassan
Keyword(s):  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Second Order ◽  
Dynamic Equations ◽  
Stieltjes Integral ◽  
Oscillation Criteria ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Dynamic Equations

We present new oscillation criteria for the second order nonlinear dynamic equation[r(t)ϕγ(xΔ(t))]Δ+q0(t)ϕγ(x(g0(t)))+∫ab‍q(t,s)ϕα(s)(x(g(t,s)))Δζ(s)=0under mild assumptions. Our results generalize and improve some known results for oscillation of second order nonlinear dynamic equations. Several examples are worked out to illustrate the main results.

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

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.

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

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
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.

scholarly journals Oscillation of Nonlinear Fractional Dynamic Equations with Forcing Term

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
S. Manikandan ◽  
V. Muthulakshmi ◽  
S. Harikrishnan ◽  
Porpattama Hammachukiattikul
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Fractional Derivatives ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Interval Oscillation

In this paper, interval oscillation criteria for the nonlinear damped dynamic equations with forcing terms on time scales within conformable fractional derivatives are established. Our approach is determined from the implementation of generalized Riccati transformation, some properties of conformable time-scale fractional calculus, and certain mathematical inequalities. Also, we extend the study of oscillation to conformable fractional Euler-type dynamic equation. Examples are presented to emphasize the validity of the main theorems\enleadertwodots.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

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 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.

Sign in / Sign up

Export Citation Format

Share Document