DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

INEQUALITIES FOR POSITIVE SOLUTIONS OF
THE EQUATION ẏ(t) =− ∑ n
i=1 (a i + b i /t)y(t − τ i )
JOSEF DIBLÍK AND MÁRIA KÚDELČÍKOVÁ
The equation ẏ(t) =− ∑ n
i=1 (a i + b i /t)y(t − τ i ), where a i ,τ i ∈ (0,∞), i = 1,2,...,n, and
b i ∈ R are constants, is considered when t →∞under supposition that the transcendental
equation λ = ∑ n
i=1 a i e λτi has two real and different roots. The existence of a positive
solution is proved as well as its asymptotic behaviour.
Copyright © 2006 J. Diblík and M. Kúdelčíková. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
We consider equation
n∑ (
ẏ(t) =−
i=1
a i + b i
t
)
y ( t − τ i
) , (1.1)
where a i ,τ i ∈ R + := (0,∞), i = 1,2,...,n, andb i ∈ R are constants. The case when there
exist positive solutions is studied. In the supposition of existence of two real (positive)
different roots λ j , j = 1,2, λ 1