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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
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" }