Detail publikace

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

Originální název

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

Anglický název

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

Jazyk

en

Originální abstrakt

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.

Anglický abstrakt

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.

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.",
}