scholarly journals Trench's Perturbation Theorem for Dynamic Equations

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Martin Bohner ◽  
Stevo Stevic
Keyword(s):  
Difference Equations ◽  
Dynamic Equation ◽  
Time Scale ◽  
Dynamic Equations ◽  
Linear Perturbation ◽  
Perturbation Theorem ◽  
Linear Dynamic ◽  
Special Cases ◽  
Perturbed Equation ◽  
Unperturbed Equation

We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equation. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench.

Abel dynamic equations of first and second kind

2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Martin Bohner ◽  
Sabrina H. Streipert
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Dynamic Equation ◽  
Time Scale ◽  
Dynamic Equations ◽  
Specific Class ◽  
Special Cases ◽  
Abel Differential Equations ◽  
General Time

AbstractThis paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations. By using appropriate transformations, expressions of Abel dynamic equations of second kind are derived on the general time scale. This also leads to a specific class of Abel dynamic equations of first kind. Finally, the canonical Abel dynamic equation is defined and examined.

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 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.

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.

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).

scholarly journals Oscillation criteria for first order forced dynamic equations

2009 ◽  
Vol 3 (2) ◽  
pp. 253-263 ◽  
Author(s):  
Lynn Erbe ◽  
Taher Hassan ◽  
Allan Peterson
Keyword(s):  
Difference Equations ◽  
Time Scale ◽  
Dynamic Equations ◽  
Oscillation Criteria ◽  
First Order ◽  
Forcing Term ◽  
Oscillatory Function ◽  
Special Cases

We obtain some new oscillation criteria for solutions to certain first order forced dynamic equations on a time scale T of the form x?(t) + r(t)??(x? (t)) + p(t)?? (x? (t)) + q(t)??(x?(t)) = f(t); with ??(u) :=?u?n-1, ?>0. > 0. Here r(t); p (t) ; q(t) and f (t) are rdcontinuous functions on T and the forcing term f(t) is not required to be the derivative of an oscillatory function. Our results in the special cases when T = R and T = N involve and improve some previous oscillation results for first-order differential and difference equations. An example illustrating the importance of our results is also included.

scholarly journals New Inequalities of Opial's Type on Time Scales and Some of Their Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Samir H. Saker
Keyword(s):  
Time Scales ◽  
Second Order ◽  
Dynamic Equations ◽  
Linear Dynamic ◽  
Special Cases ◽  
Discrete Inequalities ◽  
Zeros Of Solutions ◽  
Yield Conditions ◽  
New Inequalities

We will prove some new dynamic inequalities of Opial's type on time scales. The results not only extend some results in the literature but also improve some of them. Some continuous and discrete inequalities are derived from the main results as special cases. The results will be applied on second-order half-linear dynamic equations on time scales to prove several results related to the spacing between consecutive zeros of solutions and the spacing between zeros of a solution and/or its derivative. The results also yield conditions for disfocality of these equations.

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 Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales

2018 ◽  
Vol 51 (1) ◽  
pp. 198-210 ◽  
Author(s):  
Douglas R. Anderson ◽  
Masakazu Onitsuka
Keyword(s):  
Time Scales ◽  
Dynamic Equations ◽  
Ulam Stability ◽  
Step Size ◽  
First Order ◽  
Linear Dynamic ◽  
Discrete Step ◽  
Special Cases ◽  
Parameter Values ◽  
Homogeneous Linear

Abstract We establish theHyers-Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, which include the continuous (step size zero) and the discrete (step size constant and nonzero) dynamic equations as important special cases. In particular, for certain parameter values in relation to the graininess of the time scale, we find the minimum HUS constants. A few nontrivial examples are provided. Moreover, an application to a perturbed linear dynamic equation is also included.

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