Two Weighted Norm Dynamic Inequalities with Applications on Second Order Half-Linear Dynamic Equations

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Samir H. Saker ◽  
Mahmoud M. Osman ◽  
Douglas R. Anderson
Keyword(s):  
Second Order ◽  
Dynamic Equations ◽  
Weighted Norm ◽  
Linear Dynamic

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 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.

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