Second order linear q -difference equations: nonoscillation and asymptotics

Pavel Řehák

Czechoslovak Mathematical Journal (2011)

  • Volume: 61, Issue: 4, page 1107-1134
  • ISSN: 0011-4642

Abstract

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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 ℕ 0 : = { q k : k ∈ 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 → ∞ of solutions to the q -difference equation D q 2 y ( t ) + p ( t ) y ( q t ) = 0 , where p : q ℕ 0 → 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.

How to cite

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Řehák, Pavel. "Second order linear $q$-difference equations: nonoscillation and asymptotics." Czechoslovak Mathematical Journal 61.4 (2011): 1107-1134. <http://eudml.org/doc/197009>.

@article{Řehák2011,
abstract = {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\}:=\lbrace q^k\colon k\in \mathbb \{N\}_0\rbrace $ 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\rightarrow \infty $ of solutions to the $q$-difference equation $D_q^2y(t)+p(t)y(qt)=0$, where $p\colon \smash\{q^\{\mathbb \{N\}_0\}\}\rightarrow \mathbb \{R\}$. 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.},
author = {Řehák, Pavel},
journal = {Czechoslovak Mathematical Journal},
keywords = {regularly varying functions; $q$-difference equations; asymptotic behavior; oscillation; regularly varying function; -difference equation; asymptotic behavior; oscillation},
language = {eng},
number = {4},
pages = {1107-1134},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Second order linear $q$-difference equations: nonoscillation and asymptotics},
url = {http://eudml.org/doc/197009},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Řehák, Pavel
TI - Second order linear $q$-difference equations: nonoscillation and asymptotics
JO - Czechoslovak Mathematical Journal
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 4
SP - 1107
EP - 1134
AB - 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}:=\lbrace q^k\colon k\in \mathbb {N}_0\rbrace $ 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\rightarrow \infty $ of solutions to the $q$-difference equation $D_q^2y(t)+p(t)y(qt)=0$, where $p\colon \smash{q^{\mathbb {N}_0}}\rightarrow \mathbb {R}$. 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.
LA - eng
KW - regularly varying functions; $q$-difference equations; asymptotic behavior; oscillation; regularly varying function; -difference equation; asymptotic behavior; oscillation
UR - http://eudml.org/doc/197009
ER -

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