Publication detail

An asymptotic analysis of nonoscillatory solutions of q-difference equations via q-regular variation

ŘEHÁK, P.

Original Title

An asymptotic analysis of nonoscillatory solutions of q-difference equations via q-regular variation

English Title

An asymptotic analysis of nonoscillatory solutions of q-difference equations via q-regular variation

Type

journal article in Web of Science

Language

en

Original Abstract

We do a thorough asymptotic analysis of nonoscillatory solutions of the $q$-difference equation $D_q(r(t)D_q y(t))+p(t)y(qt)=0$ considered on the lattice $\{q^k:k\in\mathbb{N}_0\}$, $q>1$. We classify the solutions according to various aspects that take into account their asymptotic behavior. We show relations among the asymptotic classes. For every positive solution we establish asymptotic formulae. Several discrepancies are revealed, when comparing the results with their existing differential equations or difference equations counterparts; however, it should be noted that many of our observations in the $q$-case have not their continuous or discrete analogies yet. Important roles in our considerations are played by the theory of $q$-regular variation and various transformations. The results are illustrated by examples.

English abstract

We do a thorough asymptotic analysis of nonoscillatory solutions of the $q$-difference equation $D_q(r(t)D_q y(t))+p(t)y(qt)=0$ considered on the lattice $\{q^k:k\in\mathbb{N}_0\}$, $q>1$. We classify the solutions according to various aspects that take into account their asymptotic behavior. We show relations among the asymptotic classes. For every positive solution we establish asymptotic formulae. Several discrepancies are revealed, when comparing the results with their existing differential equations or difference equations counterparts; however, it should be noted that many of our observations in the $q$-case have not their continuous or discrete analogies yet. Important roles in our considerations are played by the theory of $q$-regular variation and various transformations. The results are illustrated by examples.

Keywords

q-difference equation; nonoscillatory solution; monotone solution; asymptotic formula; regular variation

Released

24.05.2017

Pages from

829

Pages to

882

Pages count

54

URL

Documents

BibTex


@article{BUT136766,
  author="Pavel {Řehák}",
  title="An asymptotic analysis of nonoscillatory solutions of q-difference equations via q-regular variation",
  annote="We do a thorough asymptotic analysis of nonoscillatory solutions of the $q$-difference 
equation $D_q(r(t)D_q y(t))+p(t)y(qt)=0$ considered on the lattice $\{q^k:k\in\mathbb{N}_0\}$, $q>1$. We classify the solutions according to various aspects that take into account their asymptotic behavior. We show relations among the asymptotic classes. For every positive solution we establish asymptotic formulae. Several discrepancies are revealed, when comparing the results with their existing differential equations or difference equations counterparts; however, it should be noted that many of our observations in the $q$-case have not their continuous or discrete analogies yet. Important roles in our considerations are played by the theory of $q$-regular variation and various transformations. The results are illustrated by examples.",
  chapter="136766",
  doi="10.1016/j.jmaa.2017.05.034",
  howpublished="online",
  number="2",
  volume="454",
  year="2017",
  month="may",
  pages="829--882",
  type="journal article in Web of Science"
}