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 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 and nonoscillation theorems of neutral dynamic equations on time scales

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yong Zhou ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Dynamic Equations ◽  
Nonoscillatory Solutions ◽  
Oscillation Criteria ◽  
Neutral Dynamic Equation ◽  
The Given ◽  
Neutral Dynamic Equations ◽  
Oscillation And Nonoscillation

Abstract We present the oscillation criteria for the following neutral dynamic equation on time scales: $$ \bigl(y(t)-C(t)y(t-\zeta )\bigr)^{\Delta }+P(t)y(t-\eta )-Q(t)y(t-\delta )=0, \quad t\in {\mathbb{T}}, $$ ( y ( t ) − C ( t ) y ( t − ζ ) ) Δ + P ( t ) y ( t − η ) − Q ( t ) y ( t − δ ) = 0 , t ∈ T , where $C, P, Q\in C_{\mathit{rd}}([t_{0},\infty ),{\mathbb{R}}^{+})$ C , P , Q ∈ C rd ( [ t 0 , ∞ ) , R + ) , ${\mathbb{R}} ^{+}=[0,\infty )$ R + = [ 0 , ∞ ) , $\gamma , \eta , \delta \in {\mathbb{T}}$ γ , η , δ ∈ T and $\gamma >0$ γ > 0 , $\eta >\delta \geq 0$ η > δ ≥ 0 . New conditions for the existence of nonoscillatory solutions of the given equation are also obtained.

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 Some Oscillation Criteria for Higher-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales

2018 ◽  
Vol 228 ◽  
pp. 01003
Author(s):  
Ying Sui ◽  
Yulong Shi ◽  
Yibin Sun ◽  
Shurong Sun
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Higher Order ◽  
Delay Equation ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Delay Dynamic Equations

New oscillation criteria are established for higher-order Emdn-Fowler dynamic equation $ q(v)x^{\beta } (\delta (v)) + (r(v)(z^{{\Delta ^{{n - 1}} }} (v))^{\alpha } )^{\Delta } = 0 $ on time scales, $ z(v): = p(v)x(\tau (v)) + x(v) $ Our results extend and supplement those reported in literatures in the sense that we study a more generalized neutral delay equation and do not require $ r^{\Delta } (v) \ge 0 $ and the commutativity of the jump and delay operators.

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 Solvability and Stability of Impulsive Set Dynamic Equations on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-19 ◽  
Author(s):  
Shihuang Hong ◽  
Jing Gao ◽  
Yingzi Peng
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Difference Equations ◽  
Time Scale ◽  
Dynamic Equations ◽  
Stability Of Solutions ◽  
Real Numbers ◽  
Generalized Derivative ◽  
Special Cases ◽  
Existence And Stability

A class of new nonlinear impulsive set dynamic equations is considered based on a new generalized derivative of set-valued functions developed on time scales in this paper. Some novel criteria are established for the existence and stability of solutions of such model. The approaches generalize and incorporate as special cases many known results for set (or fuzzy) differential equations and difference equations when the time scale is the set of the real numbers or the integers, respectively. Finally, some examples show the applicability of our results.

scholarly journals Periodic Solutions in Shifts for a Nonlinear Dynamic Equation on Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Erbil Çetin ◽  
F. Serap Topal
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Time Scales ◽  
Difference Equations ◽  
Dynamic Equation ◽  
Time Scale ◽  
Dynamic Equations ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Delay

Let be a periodic time scale in shifts . We use a fixed point theorem due to Krasnosel'skiĭ to show that nonlinear delay in dynamic equations of the form , has a periodic solution in shifts . We extend and unify periodic differential, difference, -difference, and -difference equations and more by a new periodicity concept on time scales.

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