Asymptotic behavior of positive solutions of a discrete delayed equation

Asymptotic behavior of positive solutions of a discrete delayed equation

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Denote ${\Z}_s^q:=\{s,s+1,\dots,q\}$ where $s$ and $q$ are integers such that $s\leq q$. Similarly a set ${\Z}_s^{\infty}$ is defined. In the paper the scalar discrete equation with delay \begin{equation} \Delta x(n)=-\left(\frac{k}{k+1}\right)^k \frac{1}{k+1} \left[1+\omega(n)\right] x(n-k) \end{equation} is considered where function $\omega \colon {\Z}_a^{\infty}\to\R $ has a special form, $k\ge1$ is fixed integer, $n\in {\Z}_a^{\infty}$, and $a$ is a whole number. We prove that there exists a positive solution $x=x(n)$ of the equation and give its upper estimation.

Denote ${\Z}_s^q:=\{s,s+1,\dots,q\}$ where $s$ and $q$ are integers such that $s\leq q$. Similarly a set ${\Z}_s^{\infty}$ is defined. In the paper the scalar discrete equation with delay \begin{equation} \Delta x(n)=-\left(\frac{k}{k+1}\right)^k \frac{1}{k+1} \left[1+\omega(n)\right] x(n-k) \end{equation} is considered where function $\omega \colon {\Z}_a^{\infty}\to\R $ has a special form, $k\ge1$ is fixed integer, $n\in {\Z}_a^{\infty}$, and $a$ is a whole number. We prove that there exists a positive solution $x=x(n)$ of the equation and give its upper estimation.

discrete equation with delay, positive solution,upper estimation

discrete equation with delay, positive solution,upper estimation

BAŠTINEC, J.; DIBLÍK, J.

2018

31.01.2017

STU Bratislava

Bratislava

978-80-227-4650-2

Aplimat 2017, proceedings

63

68

6