Publication detail

On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence

DIBLÍK, J. SCHMEIDEL, E.

Original Title

On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence

Type

journal article - other

Language

English

Original Abstract

Schauder's fixed point technique is applied to asymptotical analysis of solutions of a linear Volterra difference equation $$ x(n+1)=a(n)+b(n)x(n)+\sum\limits^{n}_{i=0}K(n,i)x(i) $$ where $n\in \bN_0$, $x\colon\bN_0\to\bR$, $a\colon \bN_0\to\bR$, $K\colon\bN_0\times\bN_0\to \bR$, and $b\colon\bN_0 \to \bR\setminus\{0\}$ is $\omega$-periodic. In the paper, sufficient conditions are derived for the validity of a property of solutions that, for every admissible constant $c\in \bR$, there exists a solution $x=x(n)$ such that $$ {x(n){\sim}}\left(c+\sum\limits_{i=0}^{n-1}\frac{a(i)}{\beta(i+1)}\right)\beta(n),$$ where $\beta(n)=\prod\limits_{j=0}^{n-1}b(j)$, for $n\to\infty$ and inequalities for solutions are derived. Relevant comparisons and illustrative examples are given as well.

Keywords

Linear Volterra difference equation, asymptotic formula, asymptotic equivalence

Authors

DIBLÍK, J.; SCHMEIDEL, E.

RIV year

2012

Released

17. 4. 2012

Publisher

Elsevier Science Publishing Co

Location

USA

ISBN

0096-3003

Periodical

APPLIED MATHEMATICS AND COMPUTATION

Year of study

2012

Number

18

State

United States of America

Pages from

9310

Pages to

9320

Pages count

11

BibTex

@article{BUT90950,
  author="Josef {Diblík} and Ewa {Schmeidel}",
  title="On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence",
  journal="APPLIED MATHEMATICS AND COMPUTATION",
  year="2012",
  volume="2012",
  number="18",
  pages="9310--9320",
  issn="0096-3003"
}