scholarly journals Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
J. Diblík ◽  
M. Růžičková ◽  
Z. Šmarda ◽  
Z. Šutá
Keyword(s):  
Time Scales ◽  
Discrete Time ◽  
Dynamic Equation ◽  
Time Scale ◽  
Asymptotic Convergence ◽  
Main Attention ◽  
All Solutions ◽  
Convergent Solution

The paper investigates a dynamic equationΔy(tn)=β(tn)[y(tn−j)−y(tn−k)]forn→∞, wherekandjare integers such thatk>j≥0, on an arbitrary discrete time scaleT:={tn}withtn∈ℝ,n∈ℤn0−k∞={n0−k,n0−k+1,…},n0∈ℕ,tn<tn+1,Δy(tn)=y(tn+1)−y(tn), andlimn→∞tn=∞. We assumeβ:T→(0,∞). It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent forn→∞. The results are presented as inequalities for the functionβ. Examples demonstrate that the criteria obtained are sharp in a sense.

scholarly journals A new conformable nabla derivative and its application on arbitrary time scales

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohamad Rafi Segi Rahmat ◽  
M. Salmi M. Noorani
Keyword(s):  
Power Function ◽  
Time Scales ◽  
Exponential Function ◽  
Discrete Time ◽  
Dynamic Equation ◽  
Time Scale ◽  
Gronwall’S Inequality ◽  
Factorial Function ◽  
Uniqueness Of The Solution ◽  
Nabla Derivative

AbstractIn this article, we introduce a new type of conformable derivative and integral which involve the time scale power function $\widehat{\mathcal{G}}_{\eta }(t, a)$ G ˆ η ( t , a ) for $t,a\in \mathbb{T}$ t , a ∈ T . The time scale power function takes the form $(t-a)^{\eta }$ ( t − a ) η for $\mathbb{T}=\mathbb{R}$ T = R which reduces to the definition of conformable fractional derivative defined by Khalil et al. (2014). For the discrete time scales, it is completely novel, where the power function takes the form $(t-a)^{(\eta )}$ ( t − a ) ( η ) which is an increasing factorial function suitable for discrete time scales analysis. We introduce a new conformable exponential function and study its properties. Finally, we consider the conformable dynamic equation of the form $\bigtriangledown _{a}^{\gamma }y(t)=y(t, f(t))$ ▽ a γ y ( t ) = y ( t , f ( t ) ) , and study the existence and uniqueness of the solution. As an application, we show that the conformable exponential function is the unique solution to the given dynamic equation. We also examine the analogue of Gronwall’s inequality and its application on time scales.

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 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 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.

scholarly journals Asymptotic Convergence of the Solutions of a Discrete Equation with Two Delays in the Critical Case

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
L. Berezansky ◽  
J. Diblík ◽  
M. Růžičková ◽  
Z. Šutá
Keyword(s):  
Critical Case ◽  
Critical Value ◽  
Discrete Equation ◽  
Asymptotic Convergence ◽  
All Solutions ◽  
Two Delays ◽  
Convergent Solution

A discrete equationΔy(n)=β(n)[y(n−j)−y(n−k)]with two integer delayskandj, k>j≥0is considered forn→∞. We assumeβ:ℤn0−k∞→(0,∞), whereℤn0∞={n0,n0+1,…},  n0∈ℕandn∈ℤn0∞. Criteria for the existence of strictly monotone and asymptotically convergent solutions forn→∞are presented in terms of inequalities for the functionβ. Results are sharp in the sense that the criteria are valid even for some functionsβwith a behavior near the so-called critical value, defined by the constant(k−j)−1. Among others, it is proved that, for the asymptotic convergence of all solutions, the existence of a strictly monotone and asymptotically convergent solution is sufficient.

scholarly journals A telescoping principle for oscillation of the second order half-linear dynamic equations on time scales

2009 ◽  
Vol 43 (1) ◽  
pp. 243-255
Author(s):  
Jiří Vítovec
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Second Order ◽  
Dynamic Equations ◽  
Linear Dynamic

Abstract . We establish the so-called “telescoping principle” for oscillation of the second order half-linear dynamic equation [r(t)Φ(x<sup>Δ</sup>)]<sup>Δ</sup> + c(t)Φ(x<sup>σ</sup>) = 0 on a time scale. This principle provides a method enabling us to construct many new oscillatory equations. Unlike previous works concerning the telescoping principle, we formulate some oscillation results under the weaker assumption r(t) ≠ 0 (instead r(t) > 0).

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.

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.

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