IJPAM: Volume 41, No. 4 (2007)


Jaromír Baštinec$^1$, Josef Diblík$^2$, Zdenek Šmarda$^3$
$^{1,3}$Department of Mathematics
Faculty of Electrical Engineering and Communication
Brno University of Technology
8, Technická, Brno, 616 00, CZECH REPUBLIC
$^1$e-mail: bastinec@feec.vutbr.cz
$^3$e-mail: smarda@feec.vutbr.cz
$^2$Institute of Mathematics and Descriptive Geometry
Faculty of Civil Engineering
Brno University of Technology
17, Zezkova, Brno, 616 00, CZECH REPUBLIC
e-mails: diblik@feec.vutbr.cz, diblik.j@fce.vutbr.cz

Abstract.The main goal of this paper is to give a new criterion for the existence of positive solutions for delayed discrete equations

\begin{displaymath}\Delta u(n+k)=f(n,u(n),u(n+1),\dots,u(n+k)).\end{displaymath}

Sufficient conditions with respect to $f$ are formulated in order to guarantee the existence of a positive solution for $n\to \infty$. The upper estimate for it is given as well. We show that the result presented generalizes the previous results in this direction. As an example, the result obtained is applied to a linear difference equation with delay.

Received: October 10, 2007

AMS Subject Classification: 39A10, 39A11

Key Words and Phrases: discrete equation, bounded solution, retract, retraction

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 41
Issue: 4