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

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

Peer-reviewed article not indexed in WoS or Scopus

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.

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.

Linear Volterra difference equation, asymptotic formula, asymptotic equivalence

Linear Volterra difference equation, asymptotic formula, asymptotic equivalence

DIBLÍK, J.; SCHMEIDEL, E.

2013

17.04.2012

Elsevier Science Publishing Co

USA

0096-3003

APPLIED MATHEMATICS AND COMPUTATION

2012

18

United States of America

9310

9320

11