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 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 Criteria for Some New Generalized Emden-Fowler Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Haidong Liu ◽  
Puchen Liu
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Analytical Techniques ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Special Cases

By means of novel analytical techniques, we have established several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale𝕋, that is,(r(t)|ZΔ(t)|α-1ZΔ(t))Δ+f(t,x(δ(t)))=0, with respect to the case∫t0∞r-1/α(s)Δs=∞and the case∫t0∞r-1/α(s)Δs<∞, whereZ(t)=x(t)+p(t)x(τ(t)),  αis a constant,|f(t,u)|⩾q(t)|uβ|,βis a constant satisfyingα⩾β>0, andr,p, andqare real valued right-dense continuous nonnegative functions defined on𝕋. Noting the parameter valueαprobably unequal toβ, our equation factually includes the existing models as special cases; our results are more general and have wider adaptive range than others' work in the literature.

Oscillation criteria for higher order nonlinear delay dynamic equations on time scales

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Xin Wu ◽  
Taixiang Sun
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Higher Order ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
Arbitrary Time ◽  
Delay Dynamic Equation ◽  
Delay Dynamic Equations ◽  
Nonlinear Delay

AbstractIn this paper, we study the oscillation criteria of the following higher order nonlinear delay dynamic equationon an arbitrary time scalewith

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 Some Remark on Oscillation of Second Order Impulsive Delay Dynamic Equations on Time Scales

2020 ◽  
Vol 76 (1) ◽  
pp. 115-126
Author(s):  
Gokula Nanda Chhatria
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Second Order ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Oscillation Criteria ◽  
Delay Dynamic Equations ◽  
Impulsive Delay

AbstractThis article deals with the oscillation criteria for a very extensively studied second order impulsive delay dynamic equations on time scale by using the Riccati transformation technique. Some examples are given to show the effect of impulse and to illustrate our main 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 Oscillation Criteria of Third-Order Nonlinear Damped Dynamic Equations on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Yang-Cong Qiu
Keyword(s):  
Time Scales ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Third Order ◽  
Oscillation Criteria ◽  
Function Classes ◽  
Generalized Riccati Transformation

We establish oscillation criteria of third-order nonlinear damped dynamic equations on time scales of the formr1tr2txΔtγΔΔ+ft, xt,xσt, xgt, xΔt=0by employing functions in some function classes and the generalized Riccati transformation. Two examples are given to show the significance of the conclusions.

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 criteria of certain fractional partial differential equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Di Xu ◽  
Fanwei Meng
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivatives ◽  
Sufficient Conditions ◽  
Riccati Transformation ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation

Abstract In this article, we regard the generalized Riccati transformation and Riemann–Liouville fractional derivatives as the principal instrument. In the proof, we take advantage of the fractional derivatives technique with the addition of interval segmentation techniques, which enlarge the manners to demonstrate the sufficient conditions for oscillation criteria of certain fractional partial differential equations.

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

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.

Sign in / Sign up

Export Citation Format

Share Document