scholarly journals Oscillation for Higher Order Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Taixiang Sun ◽  
Qiuli He ◽  
Hongjian Xi ◽  
Weiyong Yu
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Higher Order ◽  
Dynamic Equations ◽  
Positive Integers

We investigate the oscillation of the following higher order dynamic equation:{an(t)[(an-1(t)(⋯(a1(t)xΔ(t))Δ⋯)Δ)Δ]α}Δ+p(t)xβ(t)=0, on some time scaleT, wheren≥2,ak(t)  (1≤k≤n)andp(t)are positive rd-continuous functions onTandα,βare the quotient of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

Oscillation criteria for quasi-linear functional dynamic equations on time scales

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Samir Saker ◽  
Said Grace
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Linear Functional ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Dynamic Equations ◽  
Gamma Delta ◽  
T Tau ◽  
Positive Integers

AbstractThis paper is concerned with oscillation of the second-order quasilinear functional dynamic equation $(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$ on a time scale $\mathbb{T}$ where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on $\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}$ and $\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty $. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.

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 Advanced Dynamic Equations on Time Scales

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Shuhong Tang ◽  
Tongxing Li ◽  
Ethiraju Thandapani
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Second Order ◽  
Dynamic Equations ◽  
Oscillation Theorems

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation on an arbitrary time scale with sup , where and . Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify the oscillation of the second-order half-linear advanced differential equation and the second-order half-linear advanced difference equation. Three examples are included to illustrate the main results.

scholarly journals Nonoscillatory Solutions for Higher-Order Neutral Dynamic Equations on Time Scales

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi ◽  
Xiaofeng Peng ◽  
Weiyong Yu
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Dynamic Equations ◽  
Bounded Solutions ◽  
Nonoscillatory Solutions ◽  
Neutral Dynamic Equation ◽  
Necessary And Sufficient ◽  
Neutral Dynamic Equations

We study the higher-order neutral dynamic equation{a(t)[(x(t)−p(t)x(τ(t)))Δm]α}Δ+f(t,x(δ(t)))=0fort∈[t0,∞)Tand obtain some necessary and sufficient conditions for the existence of nonoscillatory bounded solutions for this equation.

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 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.

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.

scholarly journals Oscillatory behaviour of higher-order nonlinear neutral delay dynamic equations on time scales

Filomat ◽  
2018 ◽  
Vol 32 (7) ◽  
pp. 2635-2649
Author(s):  
M.M.A. El-Sheikh ◽  
M.H. Abdalla ◽  
A.M. Hassan
Keyword(s):  
Sufficient Conditions ◽  
Higher Order ◽  
Comparison Theorems ◽  
Dynamic Equations ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
Riccati Technique ◽  
Delay Dynamic Equations ◽  
Positive Integers ◽  
Oscillation Of Solutions

In this paper, new sufficient conditions are established for the oscillation of solutions of the higher order dynamic equations [r(t)(z?n-1(t))?]? + q(t) f(x(?(t)))=0, for t ?[t0,?)T, where z(t):= x(t)+ p(t)x(?(t)), n ? 2 is an even integer and ? ? 1 is a quotient of odd positive integers. Under less restrictive assumptions for the neutral coefficient, we employ new comparison theorems and Generalized Riccati technique.

Local-periodic solutions for functional dynamic equations with infinite delay on changing-periodic time scales

2018 ◽  
Vol 68 (6) ◽  
pp. 1397-1420 ◽  
Author(s):  
Chao Wang ◽  
Ravi P. Agarwal ◽  
Donal O’Regan
Keyword(s):  
Periodic Solutions ◽  
Time Scales ◽  
Time Scale ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Arbitrary Time ◽  
First Time ◽  
Composition Theorem

Abstract In this paper, by using the concept of changing-periodic time scales and composition theorem of time scales introduced in 2015, we establish a local phase space for functional dynamic equations with infinite delay (FDEID) on an arbitrary time scale with a bounded graininess function μ. Through Krasnoseľskiĭ’s fixed point theorem, some sufficient conditions for the existence of local-periodic solutions for FDEID are established for the first time. This research indicates that one can extract a local-periodic solution for dynamic equations on an arbitrary time scale with a bounded graininess function μ through some index function.

Oscillation criteria for nonlinear neutral functional dynamic equations on time scales

2013 ◽  
Vol 63 (2) ◽  
Author(s):  
I. Kubiaczyk ◽  
S. Saker ◽  
A. Sikorska-Nowak
Keyword(s):  
Difference Equations ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Dynamic Inequality ◽  
Delay Dynamic Equations ◽  
Delay Differential ◽  
Positive Functions

AbstractIn this paper, we establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation $$\left[ {r\left( t \right)\left[ {m\left( t \right)y\left( t \right) + p\left( t \right)y\left( {\tau \left( t \right)} \right)} \right]^\Delta } \right]^\Delta + q\left( t \right)f\left( {y\left( {\delta \left( t \right)} \right)} \right) = 0$$ on a time scale $$\mathbb{T}$$ which is unbounded above, where m, p, q, r, T and δ are real valued rd-continuous positive functions defined on $$\mathbb{T}$$. The main investigation of the results depends on the Riccati substitutions and the analysis of the associated Riccati dynamic inequality. The results complement the oscillation results for neutral delay dynamic equations and improve some oscillation results for neutral delay differential and difference equations. Some examples illustrating our main results are given.

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