INTEGRAL COMPARISON THEOREMS FOR SECOND ORDER LINEAR DYNAMIC EQUATIONS

2007 ◽  
Author(s):  
LYNN ERBE ◽  
ALLAN PETERSON ◽  
PAVEL ŘEHÁK
Keyword(s):  
Second Order ◽  
Comparison Theorems ◽  
Dynamic Equations ◽  
Order Linear ◽  
Linear Dynamic

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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