scholarly journals On certain comparison theorems for half-linear dynamic equations on time scales

2004 ◽  
Vol 2004 (7) ◽  
pp. 551-565 ◽  
Author(s):  
Pavel Řehák
Keyword(s):  
Time Scales ◽  
Comparison Theorem ◽  
Dynamic Equation ◽  
Second Order ◽  
Comparison Theorems ◽  
Dynamic Equations ◽  
Discrete Case ◽  
Linear Dynamic ◽  
Suitable Function

We obtain comparison theorems for the second-order half-linear dynamic equation[r(t)Φ(yΔ)]Δ+p(t)Φ(yσ)=0, whereΦ(x)=|x|α−1sgn xwithα>1. In particular, it is shown that the nonoscillation of the previous dynamic equation is preserved if we multiply the coefficientp(t)by a suitable functionq(t)and lower the exponentαin the nonlinearityΦ, under certain assumptions. Moreover, we give a generalization of Hille-Wintner comparison theorem. In addition to the aspect of unification and extension, our theorems provide some new results even in the continuous and the discrete case.

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 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 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 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 Existence of Ψ-bounded Solutions for System of Linear Dynamic equations on Time Scales

2018 ◽  
Vol 7 (4.10) ◽  
pp. 698
Author(s):  
B. V Appa Rao ◽  
K. A S N V Prasad
Keyword(s):  
Dynamical System ◽  
Time Scales ◽  
Dynamic Equation ◽  
Bounded Solution ◽  
Dynamic Equations ◽  
Lebesgue Function ◽  
Continuous Systems ◽  
Bounded Solutions ◽  
Linear Dynamic ◽  
Asymptotic Nature

In this work, we develop the criteria for existence of Ψ- bounded solutions of system of linear dynamic equations on time scales. The advantage of results in this dynamical system is it unifies discrete as well as continuous systems. Initially, we develop if and only if conditions for the existence of at least one Ψ-bounded solution for linear dynamic equation y∆(τ ) = P (τ )y +g(τ ), for each Ψ- delta integrable  Lebesgue function g, on time scale T +. Later, we obtain asymptotic nature of Ψ-bounded solutions of dynamical system. Also we provided the examples for supporting the results.AMS Subject Classification: 74H20, 34N05, 34C11  

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