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.",
address="Elsevier Science Publishing Co",
chapter="90950",
institution="Elsevier Science Publishing Co",
number="18",
volume="2012",
year="2012",
month="april",
pages="9310--9320",
publisher="Elsevier Science Publishing Co",
type="journal article - other"
}