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

English Title

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

Type

journal article - other

Language

en

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.

English 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

RIV year

2012

Released

17.04.2012

Publisher

Elsevier Science Publishing Co

Location

USA

ISBN

0096-3003

Periodical

APPLIED MATHEMATICS AND COMPUTATION

Year of study

2012

Number

18

State

US

Pages from

9310

Pages to

9320

Pages count

11

Documents

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",
annote="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.",
}