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

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

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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.

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.

discrete equation; large-time behaviour; critical case

discrete equation; large-time behaviour; critical case

BAŠTINEC, J.; DIBLÍK, J.; KLIMEŠOVÁ, M.

2018

21.07.2017

American Institute of Physics

Rhodos

978-0-7354-1538-6

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

0094-243X

AIP conference proceedings

Spojené státy americké

480005-1

480005-4

4

http://dx.doi.org/10.1063/1.4992641