Second order linear $q$-difference equations: nonoscillation and asymptotics

Title:

Second order linear $q$-difference equations: nonoscillation and asymptotics

Creator:

Řehák, Pavel

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:e2a44e4e-2681-4cb7-81b2-bb8335354b8c
uuid:e2a44e4e-2681-4cb7-81b2-bb8335354b8c

Subject:

regularly varying functions, $q$-difference equations, asymptotic behavior, and oscillation

Type:

model:article and TEXT

Description:

The paper can be understood as a completion of the $q$-Karamata theory along with a related discussion on the asymptotic behavior of solutions to the linear $q$-difference equations. The $q$-Karamata theory was recently introduced as the theory of regularly varying like functions on the lattice $q^{\mathbb {N}_0}:=\{q^k\colon k\in \mathbb {N}_0\}$ with $q>1$. In addition to recalling the existing concepts of $q$-regular variation and $q$-rapid variation we introduce $q$-regularly bounded functions and prove many related properties. The $q$-Karamata theory is then applied to describe (in an exhaustive way) the asymptotic behavior as $t\to \infty $ of solutions to the $q$-difference equation $D_q^2y(t)+p(t)y(qt)=0$, where $p\colon \smash {q^{\mathbb {N}_0}}\to \mathbb {R}$. We also present the existing and some new criteria of Kneser type which are related to our subject. A comparison of our results with their continuous counterparts is made. It reveals interesting differences between the continuous case and the $q$-case and validates the fact that $q$-calculus is a natural setting for the Karamata like theory and provides a powerful tool in qualitative theory of dynamic equations.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

1107-1134

Source:

Czechoslovak Mathematical Journal | 2011 Volume:61 | Number:4

Harvested from:

CDK

Metadata only:

false