Publication detail

# Large-time behavior of a class of positive solutions of discrete equation \Delta u(n + k) = -p(n)u(n) in the critical case.

BAŠTINEC, J. DIBLÍK, J. KLIMEŠOVÁ, M.

Original Title

Large-time behavior of a class of positive solutions of discrete equation \Delta u(n + k) = -p(n)u(n) in the critical case.

English Title

Large-time behavior of a class of positive solutions of discrete equation \Delta u(n + k) = -p(n)u(n) in the critical case.

Type

conference paper

Language

en

Original Abstract

It is well-known that the discrete delayed equation \Delta u(n+k)=-p_c u(n), where k is a positive integerand and p_c=\frac{k^k}{(k+1)^{k+1}} has a positive solution u=u(n), n=0,1,2,\dots. This is no longer true for the equation \Delta u(n+k)=-pu(n) where the constant p>p_c. In the paper, the delayed discrete equation \Delta (n+k)=-p^*(n)u(n) with a function p^*(n) positive for all sufficiently large n is studied. This function has a special form and satisfies the inequality p^*(n)>p_c. It is proved that, even in this case, there exists a class of positive solutions for n\to\infty and e two-sided estimates characterizing their behavior are derived.

English abstract

It is well-known that the discrete delayed equation \Delta u(n+k)=-p_c u(n), where k is a positive integerand and p_c=\frac{k^k}{(k+1)^{k+1}} has a positive solution u=u(n), n=0,1,2,\dots. This is no longer true for the equation \Delta u(n+k)=-pu(n) where the constant p>p_c. In the paper, the delayed discrete equation \Delta (n+k)=-p^*(n)u(n) with a function p^*(n) positive for all sufficiently large n is studied. This function has a special form and satisfies the inequality p^*(n)>p_c. It is proved that, even in this case, there exists a class of positive solutions for n\to\infty and e two-sided estimates characterizing their behavior are derived.

Keywords

discrete equation; large-time behaviour; critical case

Released

21.07.2017

Publisher

American Institute of Physics

Location

Rhodos

ISBN

978-0-7354-1538-6

Book

International Conference on Numerical Analysis and Applied Mathematics 2016 (ICNAAM-2016)

Pages from

480005-1

Pages to

480005-4

Pages count

4

URL

Documents

BibTex


@inproceedings{BUT138082,
author="Jaromír {Baštinec} and Josef {Diblík} and Marie {Klimešová}",
title="Large-time behavior of a class of positive solutions of discrete equation \Delta u(n + k) = -p(n)u(n) in the critical case.",
annote="It is well-known that the discrete delayed equation \Delta u(n+k)=-p_c u(n), where k is a positive integerand and p_c=\frac{k^k}{(k+1)^{k+1}}  has a positive solution u=u(n), n=0,1,2,\dots. This is no longer true for the equation \Delta u(n+k)=-pu(n) where the constant p>p_c. In the paper, the delayed discrete equation \Delta (n+k)=-p^*(n)u(n) with a function p^*(n) positive for all sufficiently large n is studied. This  function has a special form and satisfies the inequality p^*(n)>p_c. It is proved that, even in this case, there exists a class of positive solutions for  n\to\infty and e two-sided estimates characterizing their behavior are derived.",
}